zotero-db/storage/2VVCFGN5/.zotero-ft-cache

THE GEOMETRIES OF VISUAL SPACE
MARK WAGNER

The Geometries of Visual Space

The Geometries of Visual Space
Mark Wagner Wagner College

2006

LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS

Mahwah, New Jersey

London

Camera ready copy for this book was provided by the author.

Copyright © 2006 by Lawrence Erlbaum Associates, Inc.
All rights reserved. No part of this book may be reproduced in any form, by photostat, microform, retrieval system, or any other means, without prior written permission of the publisher.
Lawrence Erlbaum Associates, Inc., Publishers 10 Industrial Avenue Mahwah, New Jersey 07430 www.erlbaum.com

Cover design by Kathryn Houghtaling Lacey

Library of Congress Cataloging-in-Publication Data

Wagner, Mark. The geometries of visual space / Mark Wagner. p. cm.

Includes bibliographical references and index.

ISBN 0-8058-5252-2(cloth : alk. paper) ISBN 0-8058-5253-0 (pbk. : alk. paper) 1. Space perception. 2. Visual perception. BF469.W35 2005 152.14’2—dc22

I. Title.
2005051008 CIP

Books published by Lawrence Erlbaum Associates are printed on acid-free paper, and their bindings are chosen for strength and durability.

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

v
This book is dedicated to John C. Baird —mentor, collaborator, friend, and psychophysical wise man. It is also dedicated to Susan Bernardo and Katie MacDonald,
for their love and support over the years.

vii
Contents

Preface

ix

1 Introduction: Contrasting Visual, Experiential,

1

and Physical Space

2 Traditional Views of Geometry and Vision

12

3 Synthetic Approaches to Visual Space Perception

30

4 An Analytic Approach to Space and Vision

50

5 Effects of Context on Judgments of Distance,

74

Area, Volume, and Angle

6 Factors Affecting Size Constancy

103

7 The Metrics of Visual Space: Multidimensional

143

Approaches to Space Perception

8 Cognitive Maps, Memory, and Space Perception

189

9 The Geometries of Visual Space: Conclusion

223

References

231

Author Index

265

Subject Index

271

Preface
When most people think of space, they think of physical space. Physical space, which is primarily the concern of physicists and geometers, is defined in reference to objective, physical measures such as rulers and protractors. As a perceptual psychologist, I am interested in another sort of space: Visual space. Visual space concerns space as we consciously experience it, and it is studied through subjective measures, such as asking people to use numbers to estimate perceived distances, areas, angles, or volumes.
Space perception is an important area to consider for a number of reasons. First, the study of space perception has a long pedigree. Many of the greatest philosophers and scientists in history including Descartes, Reid, Berkeley, Hume, and Kant have examined how well our perceptions of space match physical reality. The space perception problem has concerned some of the greatest minds in the history of psychology as well, including Helmholtz, Luneburg, Titchener, Wundt, James, and Gibson. Space, together with time, is the fundamental basis of all sensible experience. Understanding the nature of our spatial experience, then, addresses one of the most basic intellectual problems. Second, psychology began as the study of conscious experience. Behaviorism arose in the 1920s by asserting the proposition that it is impossible to say anything significant about conscious experience. Behaviorism is just part of a larger materialist philosophy that pervades modern science and medicine. I believe this materialist philosophy is over emphasized, and that consciousness is at least as fundamental and important as the physical world. This work on space perception is an attempt to show that one can develop a sophisticated and coherent understanding of conscious experience. Finally, there are potential practical applications of work on this topic. In the real world, predictable errors in spatial perception can have very real consequences, from landing planes badly to driving mistakes that can cost lives.
Numerous studies have found that physical space and visual space can be very different from each other. This past work has demonstrated that mismatches between physical and visual space are not isolated occurrences, but that large, systematic mismatches regularly occur under ordinary circumstances. This book reviews work that explores this mismatch between perception and physical reality. In addition, this book describes the many factors that influence our perception of space including the meaning we assign to geometric concepts like distance, the judgment method we use to report our experience, the presence or absence of cues to depth, the orientation of a stimulus with respect to our point of view, and many other factors.
Previous theorists have often tried to test whether visual space is best described by a small set of traditional geometries, such as the Euclidean geometry most of us studied in High School or the hyperbolic and spherical geometries introduced by 19th-century mathematicians. This “synthetic” approach to defin-
ix

x

PREFACE

ing visual space relies on laying out a set of axioms characteristic of a geometry and testing the applicability of the axioms. This book describes this sort of research and demonstrates that the synthetic approach has largely failed because the empirical research commonly does not support the postulates or axioms these geometries assume.
I take a different approach based on what mathematicians call metric functions; that is, I attempt to specify the measurable properties of visual space, such as distances, angles, and areas, using functions that take into account the location of a stimulus in physical space and other psychological factors. The main theme of this book is that no single geometry describes visual space, but that the geometry of visual space depends on stimulus conditions and mental shifts in the subjective meaning of size and distance. Yet, despite this variation, our perceptions are predictable based on a set of relatively simple mathematical models.
Although this work is primarily intended for scholars in perception, mathematical psychology, and psychophysics, I have done my best to make this discussion accessible to a wider audience. For example, chapter 2 reviews the mathematical, philosophical, and psychophysical tools on which this book relies at what I believe is a very readable level. Because of this, I believe this book would also make for a good graduate-level textbook on space perception.

Plan of the book. The first two chapters contain philosophical, mathematical, and psychophysical background material. Visuals space is defined, and I explain why the problem is important to study. These chapters trace the history of philosophical work on space perception, which antedates psychology. They also explain how mathematicians approach geometry, describe some of the most important and widely known geometries, and discuss the psychophysical techniques used to explore visual space.
Chapter 3 looks at synthetic approaches to space perception including work on hyperbolic, spherical, and Euclidean geometries. I lay out the axioms for geometries of constant curvature and consider the extent to which these axioms are supported by empirical work. Chapter 4 proposes an alternative way to investigate the geometry of visual space, the analytic approach. Here, geometries are defined by using coordinate equations to express the metric properties of the space, such as distance, angle, area, and volume. I describe ways of assigning coordinates to visual space, talk about the origin of visual space —the egocenter, and talk about the general form of equations to describe metrics. Finally, I demonstrate that visual space violates the assumptions of one of the most general types of geometries, metric spaces.
The next three chapters review the three other major domains of psychophysical research on space perception. Chapter 5 presents a meta-analysis of studies that ask observers to directly estimate size, distance, area, angle, and volume. This meta-analysis examines how judgments of the measurable properties of visual space depend on contextual factors such as instructions, cue conditions, memory vs. direct judgment, the range of stimuli, judgment method, and so on. Chapter 6 looks at the size constancy literature in which observers are asked to adjust a comparison stimulus to match a variety of standards at different distances away. This chapter discusses the history of this literature and con-

PREFACE

xi

siders the effects of many variables on size constancy judgments such as instructions, cue conditions, age, and stimulus orientation. Chapter 7 discusses research that takes a multi-dimensional approach toward studying visual space. These studies look at how size and angle judgments change when stimuli are oriented horizontally, vertically, or in-depth. In all three chapters, mathematical models are presented that integrate data presented in the literature reviews.
Chapter 8 talks about how spatial experience is influenced by memory. In particular, I review factors that affect the development and structure of cognitive maps, including individual difference variables such as age, navigational experience, gender, and personality. In addition, it describes the types judgment errors that are unique to cognitive maps. Chapter 9 summarizes and synthesizes the data and theories discussed in the earlier chapters of the book. In addition, this chapter discusses spatial experience arising from modalities other than vision.

Acknowledgments. This book would not have been possible without the assistance of many people. First of all, I would like to thank my editors at Lawrence Erlbaum Associates, Bill Webber and Lori Stone, who helped guide me through the publication process in a gentle and professional way. In addition, I wish to thank Nadine Simms for her help with the production process. I’d also like to thank Jim Brace-Thompson of Sage Publications for the encouragement and help he provided shortly after I began writing this book.
I am also grateful to Elaina Shapiro, my undergraduate assistant, who helped me track down many hundreds of the articles that have gone into this work. She always approached this monumental task with good humor and enthusiasm. In addition, I would like to express my appreciation to Evan Feldman and Heather Kartzinel, whose research has contributed to this book.
I’d like to thank Stephen W. Link and several anonymous reviewers who gave me sage advice that helped improve the quality of the final product. Steve, in particular, gave me incredibly detailed feedback and much needed encouragement. Thanks, Steve!
I’d also like to acknowledge Susan Bernardo and Katie MacDonald for their love, tolerance, and support. Finally, I’d like to thank John C. Baird, my mentor, collaborator, and friend. This book would not have been possible without the vision and professionalism he displayed throughout our many collaborations.

—Mark Wagner

1
Introduction Contrasting Visual, Experiential,
and Physical Space
Time and space are the two pure forms of all sensible perception, and as such they make a priori synthetic propositions possible. Immanuel Kant, Critique of Pure Reason
Contrasting Conceptions of Space
This book will investigate the properties of our visual perceptions of space. The concept of space has been an object of speculation and dispute throughout the history of philosophy and science. Great philosophers and scientists—Immanuel Kant, Thomas Reid, Henri Poincaré, Issac Newton, and Albert Einstein (to name a few)—have considered space (together with time) to be one of the cornerstones on which existence is based and from which philosophy and science arise.
At the outset, two terms need to be defined and distinguished: physical space and visual space. While it is tempting to distinguish between the two by saying that the latter reflects conscious experience and the former does not, I believe one must resist this temptation. Both concepts reflect aspects of our experience of the world. But the attitude we take toward that experience differs between physical and visual space. Perhaps, I risk offending some readers by reviving the shadows of Wundt and Titchener, but in perception at least, Titchener was accurate in observing:
All human knowledge is derived from human experience; there is no other source. But human experience, as we have seen may be considered from different points of view.... First, we will regard experience as altogether independent of any particular person; we will assume that it goes
1

2

CHAPTER 1

on whether or not anyone is there to have it. Secondly, we will regard experience as altogether dependent on the particular person; we will assume that it goes on only when someone is there to share it. (Titchener, 1900/1909, p. 6)

However, conscious experience can be a rather slippery conceptual fish to grapple with. A scientist can only theorize based on solid data, and consciousness does not itself make marks on paper nor does it directly leave other physical traces that can be studied at leisure.
Harvey Carr once noted, “Consciousness is an abstraction that has no more independent existence than the grin of a Cheshire cat” (Carr, 1925). To open our experience to scientific investigation, we must rely on objectively observable behaviors and verbal reports that attempt to capture some aspect of experience, and we must resort to operational definitions of our concepts in order to render them concrete enough to use.
With this in mind, let me define my terms. By physical space I mean the space revealed to us by measuring devices such as rulers and protractors. Physical space is objectively defined; that is, the properties of physical space are largely observer independent. By visual space, I mean the space revealed by the psychophysical judgments of an observer. Visual space is not objectively defined; that is, the properties of visual space may depend critically on certain aspects of the observer, such as location in physical space, experimental conditions, and the mindset of the observer.
Defining visual space this way sidesteps the central issue: do the judgments people give accurately reflect their subjective experience of the world? Are the introspective reports that people generate a fair reflection of what is really witnessed internally? No doubt I would be wisest to simply drop the issue; however, I am too much of a philosopher to pass on without venturing an opinion.
Let me boldly state my own equivocal belief. While I believe that observers do attempt to base their judgments on their subjective experience of the world and I believe they really do try to be accurate, it is impossible to say how well they accomplish their goal. It is impossible to independently verify what is really in the subjective experience of an observer. The closest proxies we have are the judgments themselves.
Of course, if we did not believe that the numbers generated in psychophysical experiments reflected something of a person’s internal experience, we would quickly lose interest in the subject. Why would one really care about mere number generating responses? A true behaviorist should find perception boring.

Geometry and space. A variety of geometries have been employed to describe physical space at different levels of scale. When the distances under consideration are large, Einstein (1922) pointed out that a hyperbolic geometry might best describe physical space. When the slightly less grandiose distances of

INTRODUCTION

3

the earth’s surface are considered, a spherical (or elliptical) geometry makes sailing or flying around the world quicker and more efficient. Yet, if we confine ourselves to that range of distances which humans commonly experience; that is, if we confine ourselves to the ecological level of analysis mentioned by Gibson (1979); then any curvature in the earth’s surface or in the fabric of space itself is small enough to be ignored. The world is Euclidean. When distances are measured by a ruler, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the two legs to a high degree of approximation—just as Euclidean geometry would predict. When angles are measured by a protractor, the sum of the angles of any triangle is always very, very, very close to 180˚—just as Euclidean geometry would predict.
The same definite conclusions cannot be made regarding visual space. People are capable of thinking about geometric concepts in different ways. By a simple mental shift, we can think of the distance from home to work as the crow flies, as the length of the path to get there, as the time it takes to drive, or as a segment of the “great circle” that intersects the two points. We can think of distance as the physicist sees it or take the artist’s perspective and see distance as the amount of canvas lying between two objects in a painting. One time we can use category estimation to judge distance and try to keep differences between categories subjectively identical while another time we use magnitude estimation and try to reflect the ratio of the subjective sizes of targets; and emphasizing these different mathematical aspects of the situation leads to very different psychophysical functions. Which of the geometries of visual space that result from these different perspectives is correct? I believe it is best to simply admit that no single view is correct, but that they all are. All may be valid descriptions of our varying subjective experience.
In addition, our experience is influenced by the situation we find ourselves in. Trying to judge the distance to an on-coming car is more difficult at night than it is during the day. Things that are far away can seem different than when they are brought close to us, and the angle from which we regard an object can make a difference to our perceptions of it. The world can seem large in the mind of a child, but the adult who returns to the old neighborhood is struck by how small and underwhelming things seem.
As we will see later, many have attempted to specify the geometry of visual space, but in my view that enterprise is hopeless from the outset. There is no single geometry that describes visual space, but there are many. The geometries of visual space vary with experience, with mental set, with conditions, and with time.
The purpose of this book is to determine how the geometry of visual space changes along with conditions. In addition, as part of that, this book will look at the changing relationship that exists between physical space and our visual perceptions of it.

4

CHAPTER 1

Memory and space. The foregoing hints that physical and visual spaces are not the only ones of interest to the psychologist. What of memorial space, space as we remember it based on a past viewing of an object or setting? Even if one believed that space as it is directly perceived is both accurate and Euclidean as a Gibsonian would suggest, a psychologist would have good reason to suppose that the process of memory would distort our judgments into a very nonEuclidean form. Memories are incomplete and reconstructed.
Cognitive maps are another step away from direct perception. Cognitive maps refer to our mental representations of the layout of our surrounding environment. Cognitive maps generally concern large-scale environments that are too big to ever be seen at one time (except perhaps from an airplane or a space ship); so, cognitive maps are constructed across time based on our unfolding experience. As we will see later, cognitive maps are riddled with holes (that represent unexperienced territories), distortions, discontinuities, and non-spatial associations. A complete characterization of cognitive space is not only nonEuclidean; it is probably non-Riemannian. In fact, there may be no simple mathematical system that could ever fully characterize the richly chaotic nature of our cognitive maps. Cognitive maps may consist of a patchwork of loosely connected parts.
From a psychological standpoint, memorial space and cognitive maps certainly deserve our attention, and this book will describe something of their nature. Once more, the family of geometries that describe human experiences expands. Who could think there might be only one?

Experiential space. Of course, one need not stop here. A more general conception than visual space is that of experiential space. By experiential space I refer to our experience of space of any kind. By its very nature, the term visual space excludes spatial perceptions based on the other senses. Yet, clearly we do perceive space in extra-visual ways. Not only does it make sense to speak of visual space, but one may also meaningfully speak of auditory space, haptic space, gustatory space, kinesthetic space, proprioceptive space, and olfactory space. This book will largely confine itself to vision because the vast majority of research studies on spatial perception concern visual stimuli, but I will have a few words to say about these other spaces in various places in this book.

Why Is This Problem Important?

A noble intellectual pedigree. The problem of space perception is one with a long and prestigious pedigree. According to Wade (1996), ancient Greek philosophers including Aristotle and Euclid recognized that spatial perception did not always correspond to physical measures and that variables such as binocularity, aerial perspective, and distance to the stimulus can alter size estimates. Roman era thinkers including Galen, Lucretius, and Ptolemy noted that variables

INTRODUCTION

5

like linear perspective and the orientation of a stimulus can lead to breakdowns in size constancy. The great 11th Century Islamic philosopher, Ibn al-Haytham, spoke of the effects of stereopsis and familiar size on spatial perception. Leonardo da Vinci reiterated the importance of binocularity and aerial perspective on size perception.
Philosophers throughout the modern era often wrote about spatial experience as part of their systems of philosophy. Wade (1996) mentions Francis Bacon’s and René DesCartes’s interest in the problems of space perception. As will be discussed at length in Chapter 2, Berkeley, Hume, Kant, Reid, Poincaré, and Husserl all held well-developed views on the geometric character of our spatial experience.
Interest in the problem of space perception also played an integral role in the development of psychology as a discipline. Helmholtz (1868/1921) extensively wrote about space perception and empirically investigated the problem as part of his assault on Kantian philosophy. Weber’s studies of two-point limen in touch were largely motivated by his wish to understand how humans develop our sense of space. Other early founders of psychology, including Titchener and James, wrote extensive chapters (or even multiple chapters) on space perception in their foundational works on psychology. In fact, the longest single chapter in James’s two-volume The Principles of Psychology is dedicated to the subject. Wundt, whom some consider the founder of psychology, was so dedicated to studying the nature of space perception that James said of him: “Wundt has all his life devoted himself to the elaboration of space theory” (James, 1890, p. 276). (By the way, I tend to agree with Link (1994, 2002) that Fechner is a better candidate for the role of psychology’s founder than Wundt. While Wundt may have been better at self-promotion, psychology was alive and well before he ever came on the scene.)
Harvey Carr (1935), the great American Functionalist, wrote an entire book on space perception. In addition, when G. Stanley Hall was granted the first Ph.D. ever awarded in psychology in America, his dissertation was on (you guessed it) space perception (Boring, 1950).
In short, some of the greatest philosophers and psychologists in history focused considerable attention on the problems of space perception. The present book follows this rich tradition and reconceptualizes our spatial experience in the light of the massive body of empirical research performed in more recent years.

Space is foundational. These great minds devoted so much of their attention of spatial experience for a very good reason. Space is foundational. The universe itself may represent little more than the interplay of space, time, and energy. Modern physics seeks to explain gravity, black holes, and the expansion of the universe in terms of alterations in the fabric of space.

6

CHAPTER 1

Psychologically, space is one of the fundamental building blocks of human experience. Without a conception of space, object perception and meaningful interaction with the world would be impossible. One literally could not live without some ability to sense the layout of the world. At times, one literally cannot live when this perception is in error at a critical time.
Kant (1781/1929) firmly believed that spatial experience served as the base out of which our phenomenal experience grows. In his words:

Space is a necessary a priori representation, which underlies all outer intuitions. We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent on them. (Kant, 1781/1929, p. 24)

Like Kant, I feel that spatial experience represents something particularly fundamental that deserves detailed study. Unlike Kant, I believe that explicating the nature of visual space is an empirical, rather than a logical, a priori issue. This book describes the nature of visual space as revealed by the research literature.

A paradigm for measuring mind. Fechner (1860) and Wundt (1874/1904) attempted to apply mathematical tools and the scientific method to the study of consciousness, and for a while all of psychology focused on the study of conscious mind. But as time passed, psychology became ever less interested in consciousness and ever more interested in behavior. Why did this happen? Some believe early Structural Psychology died due to its methodological defects. Carr (1925), who did not wholly reject the introspective method of the Structuralists, pointed out the defects of introspection. He felt that introspection was too difficult to do to give much detailed information about consciousness, that introspective reports were not subject to independent verification, and that Structuralists tended to rely on trained observers whose observations were too easily influenced by their knowledge of the research hypotheses—what James (1890) referred to as the Psychologist’s Fallacy.
A more fatal line of attack on introspection came from Watson (1914, 1919, 1924, 1925). Watson felt that it was impossible to make any real progress with a science based on introspection and that the whole enterprise could be dismissed as irrelevant. “The psychology begun by Wundt has thus failed to become a science and has still more deplorably failed in contributing anything of a scientifically usable kind to human nature” (Watson, 1919, p. 3).
While I realize that modern psychology has lost much of its behaviorist character, Watson’s challenge is still one I take very seriously. Is it possible to take introspective reports and develop them into an organized, sophisticated, devel-

INTRODUCTION

7

oping body of knowledge? If Watson is right, then it is not only difficult to study the mind, but mind becomes a mere wisp or vapor of no importance.
But, one can develop a sophisticated science based on introspective reports. And I believe no area of psychology is fitter to demonstrate this point than the spatial perception literature. Space perception can be seen as a paradigm of success in the study of mind. This book is an attempt to answer Watson’s charge.
More recently, a second serious charge was leveled against the whole enterprise of psychophysics. Lockhead (1992) accused psychophysicists of generating a sterile discipline that consists of a series of unidimensional investigations that fail to adequately grapple with the effects of context on judgments. I see the present book as a lengthy refutation of Lockhead’s charge. When taken together the spatial perception literature paints a rich, multidimensional picture that dynamically changes as a function of contextual variables.

Practical applications of visual space perception. James (1907/1964) felt that scientists could be divided into two groups based on their temperaments. The forgoing justifications might appeal to those with what James referred to as a “tender-minded make-up,” but might not convince those with a more “toughminded make-up.” A final justification for the study of space perception might even satisfy readers of the hard-nosed persuasion. Space perception research can have many practical applications.
For example, Kong, Zhang, Ding, and Huikun (1995) found that accidentprone railroad drivers had poorer spatial perception skills, particularly those related to depth perception, than safe railroad drivers. Another group of Chinese researchers divided drivers into excellent, regular, relatively poor, and accident prone groups based on driving test scores and accident records and found that the worst drivers had significantly poorer visual depth perception (Zhang, Huang, Liu, & Hou, 1995). Hiro (1997) noted that the faster people drive, the more they underestimate the distance to the car ahead of them. Given that it takes more time to stop at faster speeds, this underestimation of distance could prove to be fatal.
Another skill that drivers need is the ability to read maps accurately. Gillan, Schmidt, and Hanowski (1999) found that contextual variables such as MüllerLyer Illusion elements in the map can lead to map reading errors.
Pilots need to perceive accurately spatial layout in order to land their planes safely. Lapa and Lemeshchenko (1982) found that pilots who use an egocentric coordinate system have slower reaction times and make more errors in judging layout than those using a geocentric coordinate system. Of course, these pilot errors can cost lives.
Other pilot tasks involving spatial perception include searching for places or landing fields, flying in formation, aerial refueling, collision avoidance, weapons targeting, and low-level flight (Harker & Jones, 1980). Westra, Simon, Collyer, and Chambers (1982) found that landing on aircraft carriers depended

8

CHAPTER 1

more on a pilot’s spatial abilities and training than on equipment factors. Unfortunately, distance judgments made from up in the air often lack many of the cues to depth usually found for terrestrial observers. Roscoe (1979) found that spatial perception was particularly difficult at dusk or in the dark, when flying over water, and when coming out of a bank of clouds. Roscoe (1982, 1985) also found that inaccuracies in spatial perception occur when pilots accommodate to their dark focus depth or on the cockpit window rather than on objects external to the cockpit.
If pilots have difficulty judging spatial layout because cues to depth are often absent, astronauts are likely to experience even more difficulty in determining the location of objects external to their capsule since many cues to depth are totally absent in space. Understanding spatial perception in outer space can be an important area of future research to assist the development of projects such as the International Space Station.
Space perception can also be critical for sports performance. For example, Issacs (1981) found that poor depth perception was an important variable in free-throw shooting in basketball. Similarly, McBeath, Shaffer, and Kaiser (1995) and Shaffer, Karauchunas, Eddy, and McBeath (2004) have shown how complicated the simple process of catching a baseball can be. In fact, Oudejans, Michaels, Bakker, and Dolne (1996) indicate that stationary observers are very poor at judging the catchableness of a baseball compared to moving observers. Obviously, this suggests that a running start may be an essential trick to being a good outfielder.
Another place where the ability to accurately perceive spatial layout is important is in surgery. Reinhardt and Anthony (1996) found the ability to engage in remote operation procedures involving internal cameras depended on the adequacy of depth and distance information. Conflicts between monocular and stereoscopic cues proved particularly problematic.
In another recent medical study, Turano and Schuchard (1991) found spatial perception deficits often result from macular and extramacular-peripheral visual field loss. (Although some subjects with quite extensive loss showed normal space perception.) What is more, these perceptual deficits occurred even outside of the damaged areas and when visual acuity is good.
Inaccuracy in distance estimates can also be an issue in some court cases. At least as far back as Moore (1907) legal scholars have been aware of a multitude of variables that negatively affect the reliability of witness testimony regarding spatial layout and the speed of movement. These variables include the amount of time witnesses observe a layout, the passage of time since the incident, the emotional state of the witness, motion in the object observed, darkness, and whether the incident is seen through water or air. At times witness estimates of layout can be critical information in courtroom testimony.
Because our ability to correctly perceive spatial layout is necessary for proper performance in so many areas, it is important to know which factors lead

INTRODUCTION

9

to spatial estimation errors so that we may engage in actions that may eliminate those errors. This book will examine many of these factors.

The Plan of the Book

All of the chapters of this book are directed at two central purposes: to describe our perceptions of visual space and to compare these perceptions to physical layout. The domain delimited by these objectives still covers a vast amount of material, because these two problems have many facets and can be approached from many different directions. The remainder of this chapter describes the various approaches this book takes toward addressing these central objectives. It lays out the basic plan of the book, briefly describing the contents of each of the chapters that follow.
Chapter 2. Like all issues in psychology, the questions discussed in this book arise within a larger historical context. As someone who has a deep interest in the history of psychology—I even co-edited a book on American Functionalism (Owens & Wagner, 1992)—I believe it is important to set up the discussion that follows by providing a bit of this historical background. Chapter 2 also describes some of the mathematical and psychophysical tools that may be used to characterize the geometries of visual space.
In particular, this chapter first discusses the ways mathematicians addressed geometry across history. Secondly, like the rest of psychology, the study of visual space grew out an attempt to apply scientific methods to a long-standing philosophical problem. This chapter speaks about early philosophical approaches to space perception. Finally, Chapter 2 discusses how the study of space perception fits into the wider domain of psychophysics, which provides the basic techniques necessary to paint a picture of visual space.

Chapter 3. Mathematicians define a space in two ways: synthetically and analytically. In the synthetic approach to geometry, the mathematician lays out a set of postulates that define a geometry and deduces theoretical statements that are the consequence of these statements. Chapter 3 describes the work of psychologists who applied this synthetic approach to visual space, particularly emphasizing Luneburg’s hyperbolic model. Theoretical works proposing Euclidean and other more exotic geometries are also mentioned. Chapter 3 discusses the assumptions made by these theorists, the predictions made by each theory, and the degree to which empirical research supports these synthetic models.
Chapter 4. My approach to describing visual space is analytic. In an analytic geometry, a set of coordinates is assigned to a space and equations are used to describe the measurable properties of the space like distance, angles, and area. This chapter discusses the advantages of the analytic approach. It describes

10

CHAPTER 1

methods for assigning coordinates to visual space, the location of the origin of visual space, and general formulas for distance, area, and volume judgments. The chapter also talks about how visual space sometimes fails to satisfy the axioms of a metric space, one of the most general forms of an analytic geometry, and describes some dramatic consequences of this failure.

Chapter 5. An expansive literature shows that judgments of the measurable properties of visual space depend on contextual factors such as instructions, cue conditions, memory vs. direct judgment, the range of stimuli, judgment method, etc. Chapter 5 performs a meta-analysis of the effects of these factors on the parameters of psychophysical functions and on the goodness of fit of these psychophysical functions. This meta-analysis is based on over seven times as many studies and experimental conditions than any previously published meta-analysis on space perception. Multiple regression analyses of this data produce a set of general psychophysical equations for distance, area, and volume judgments as a function of contextual conditions. Angle judgments are also briefly examined.

Chapter 6. A second spatial perception literature concerns the perception of size constancy. In this literature, a near comparison is adjusted to match the size of standard stimuli at varying distances from the observer, at varying orientations, and under varying cue conditions. This chapter discusses the history of this literature and considers the effects of many variables on size constancy judgments. It also develops a theory to explain the results that is a generalization of the classic Size-Distance-Invariance Hypothesis. The virtue of the present theory is that it allows one to unify the size constancy literature and the psychophysical literature addressed in the previous chapter. Finally, the chapter briefly talks about the link between size-constancy and the moon illusion.

Chapter 7. The vast majority of the psychophysical literature is based on unidimensional judgments, where depth and egocentric distance are looked at independently from frontal size perception. Chapter 7 talks about a few exceptions to this unidimensional rule that look at spatial judgments as a function of two or even three dimensions simultaneously. I describe two of my own studies of this type and present several models to describe this data. Here, at last, we create models that fully specify the geometry of visual space under a given set of conditions. The rest of the chapter discusses other work of this type. These studies look at changes in visual space as a function of distance from the observer, elevation of gaze, and the presence of context-defining objects. The chapter also mentions evidence for the presence of multiple visual systems, one that guides motion and the other that produces visual experience.

Chapter 8. Memory adds yet another layer of complexity to the analysis of spatial experience. This chapter contrasts the data and theoretical approaches

INTRODUCTION

11

produced by the direct perception and memory literatures, particularly focusing on the cognitive mapping literature. It describes the structural elements of cognitive maps and how cognitive maps are acquired across time. I look at the affect of individual difference factors on cognitive maps such as age, navigational experience, gender, and personality. I also look at the nature of the errors that cognitive maps contain. After this, Chapter 8 compares the psychophysical judgments of size and distance that observers give under direct perception, memory, and cognitive mapping conditions. The chapter also develops a theoretical framework for understanding these data. Finally, I list a few objections to the cognitive-science paradigm that pervades much of this literature and mention an alternative way to think about memory.
Chapter 9. The final chapter summarizes and synthesizes the data and theories discussed in the earlier chapters of the book. In addition, Chapter 9 will touch on spatial experience arising from modalities other than vision. Finally, the chapter discusses the ecological, philosophical, and practical implications of the spatial perception literature.

The End of the Beginning
In summary, a wealth of data that are discussed in the following chapters indicates that visual space is different from physical space. In fact, the geometry that best describes visual space changes as a function of experimental conditions, stimulus layout, observer attitude, and the passage of time.
In addition, the problem of human spatial perception is one of great antiquity, long-standing philosophical import, and considerable practical significance. The spatial perception literature is well enough developed to convincingly show that a sophisticated science can be based on the introspective reports of observers.

2
Traditional Views of Geometry and Vision
Like most psychological problems, the problem of space perception exists within a context larger than itself. This chapter provides a bit of this context. In particular, this chapter examines the historical background of the problem and looks at the empirical and analytical tools available to describe visual space.
The first leg of this contextual tour looks at the approaches mathematicians use to define the geometry of a space. Following this, we discuss the works of early philosophers whose views about visual space naturally led to more recent psychological developments. Finally, this chapter briefly discusses the psychophysical methods that are employed to empirically measure visual space perception.
Geometry as the Mathematician Sees It
Because the problem of visual space perception is explicitly geometric in nature, a logical place to begin searching for tools to work on the problem is with geometry. How do mathematicians define a space? According to Kline (1972), there are two general approaches to geometry. One is synthetic, and the other is analytic.
Synthetic approaches. Ancient Babylonia and Egypt possessed forms of geometry; however, these geometries were concrete, primitive, and lacked unifying principles. This early work focused on solving practical problems associated with flood control, building, and trade. They relied on approximation rather than exact numbers. For example, was thought to be three. While these ancient mathematicians anticipated many important elements of geometry (such as the Pythagorean Theorem), their works were empirically derived. They lacked the modern concept of proof, and the various mathematical findings were not integrated into a coherent structure.
The first real sophistication in mathematics began with the classical Greeks, who created many geometry theorems. The earliest proof is generally attributed 12

TRADITIONAL VIEWS OF GEOMETRY AND VISION

13

to Thales about 600 BC. Over the next few centuries, the Pythagoreans and others added many new geometrical proofs. Euclid theorems organized these theorems into a coherent structure in his book Elements.
In this book, Euclid laid out proofs for 465 geometrical propositions. Euclid’s method, however, was of far greater importance than this impressive number of proofs. Euclid began his development by making a list of definitions and postulates. His ten postulates consisted of global, rather than algebraic, assumptions. The most famous example is the Parallel Postulate that states “Through a point P not on a line L, one and only one parallel to L can be drawn.” Euclid (and others subsequently) then deduced his many theorems based on these definitions and postulates. Such a geometry, consisting of global definitions, postulates, and theorems, is called a synthetic geometry.
For over a millennium, mathematicians believed that Euclid’s geometry was the only one possible. Asking what geometry best describes visual space would have made no sense to them. Visual space could only be Euclidean.
The self-evident certainty of Euclidean geometry crumbled in the early 19th century as a result of mathematical investigations of the Parallel Postulate. Euclid’s Parallel Postulate had always been unsatisfactory to mathematicians. In 1733, the Jesuit mathematician Saccheri vainly attempted to prove the Parallel Postulate based on the other nine postulates. While other mathematicians largely rejected the “proof” he generated, his work induced others to take an interest in the problem.
Finally, in 1829, Lobatchevsky demonstrated not only that the Parallel Postulate could not be proved but that a perfectly consistent geometry could be constructed from the assumption that more than one parallel exists to a line through a point not on the line. A few years later Bolyai (1833) published his work demonstrating the same point. (Gauss’s notes indicated that he had developed similar proofs earlier than Lobatchevsky and Bolyai, but he never published the work).
The geometry defined by this new form of parallel postulate is called a hyperbolic geometry. A hyperbolic geometry has many properties that are different from those in Euclidean geometry. In a hyperbolic geometry, the sum of the angles of a triangle is less than 180˚. The “straight” lines of the space are shaped like hyperbolas. No infinity exists in the hyperbolic space; that is, the space is bounded.
In 1854, Riemann invented a third type of synthetic geometry that arises from another variant of the Parallel Postulate. In this case, Riemann assumed that no parallels to a line could be drawn through a point not on the line. Such a geometry is called a spherical geometry. A simple example of a spherical geometry is the surface of the earth. Here, “straight lines” are circles whose centers are coincident with the center of the earth. All lines (known as Great Circles) defined this way must intersect at two points.
A spherical geometry has a number of other interesting characteristics. First of all, all lines have a finite length. Because pairs of lines intersect at two points, spherical geometry violates Euclid’s postulate that two straight lines cannot en-

14

CHAPTER 2

close a space. The sum of the angles of a triangle is always greater than 180˚ (but less than 540˚). The perimeter and area of all figures cannot exceed a maximum size.

Analytic approaches. Geometry was almost exclusively synthetic in nature until the 17th century. In 1637, René Descartes introduced analytic geometry. He established what we now call the Cartesian coordinate system (although he only defined the first or positive quadrant). He demonstrated that many hitherto unsolved geometric problems were solvable by means of analytic geometry. Three key ideas separate analytic from synthetic geometry. First, numbers are associated with the locations or coordinates of points. Second, equations are associated with curves. Third, coordinate equations are used to define distance and other metric properties.
Descartes’s ideas proved to be extremely important. Algebra and geometry merged into one discipline. As geometry could now be quantitative, mathematicians put more effort into the study of algebra. The calculus became possible. In short, mathematics became far more flexible and powerful. This analytic approach to mathematics and geometry made many of the profound discoveries of Newtonian physics possible.
The analytic approach can be used to describe all of the synthetic geometries we just mentioned. For a time after Lobatchevsky, synthetic and analytic geometry contested for supremacy. In the end, analytic geometry won the battle. In 1854, Riemann introduced an extremely general form of geometry. The synthetic Euclidean, hyperbolic, and spherical geometries were simply special cases of this more general analytic geometry. Where synthetic geometry had only introduced a handful of possible geometries, Riemannian geometry allowed for a potentially infinite variety. In addition, analytic geometries can make use of powerful tools such as algebra and calculus. Since Riemann’s time, synthetic geometry gradually faded from the mathematical world. In fact, one of the few places where it still finds adherents is in psychology (as we will see later).
In Riemann’s terms, Euclidean, hyperbolic, and spherical geometries are referred to as geometries of constant curvature. A Euclidean geometry is considered flat and has a curvature of zero. A spherical geometry has a constant positive curvature, and a hyperbolic geometry has a constant negative curvature.

Metric spaces. A Riemannian geometry has two essential characteristics (Eisenhart, 1925): First, the space must be a manifold. That is, there must be some way to assign coordinates to the points in the space, and functions assigning these coordinates must be smooth. (There should be no discontinuities between the coordinates of points lying close to each other.) Second, the nature of the space is critically related to the distance function that is defined on the space. Different distance functions are indicative of different spaces. In fact, in 1871

TRADITIONAL VIEWS OF GEOMETRY AND VISION

15

Klein showed that Euclidean, hyperbolic, and spherical geometries essentially only differ in their respective distance functions.
Riemann’s ideas are stated in their most general form in modern topology and real analysis. One of the most general types of distance-defined spaces is called a metric space. A metric space consists of two parts. First, there must be a non-empty set of points (X). Second, there is a function (d) defined on the set which assigns a distance to any pair of points (x,y). Such a distance function, called a metric, must satisfy four conditions:

Let x, y, and z be elements of set X, then d(x,y) is a metric on X if

(1) Distance is always non-negative. That is,

d(x,y) 0.

(2.1)

(2) Non-identical points have a positive distance. That is,

d(x,y) = 0 if and only if x = y.

(2.2)

(3) Distance is symmetric. That is,

d(x,y) = d(y,x).

(2.3)

(4) Distance is the shortest path between points. In other words, a path between two points which is traced through a third point can never be shorter than the distance between the two points. (There are no short cuts.) This property is often called the triangle inequality. That is,

d(x,y) d(x,z) + d(z,y).

(2.4)

These metric axioms express much of what is essential to our every day concept of distance.
The metric axioms are also quite general. An infinite variety of possible metric spaces exist. Three of the most well known metrics are the Euclidean, city block, and Minkowski metrics. Let’s look at these three metrics as examples of metric functions.
In a two dimensional Cartesian coordinate system, the Euclidean distance between two points P1 and P2 located at the coordinates (x1,y1) and (x2,y2) respectively is

d(P1,P2) = (x1-x2)2+(y1-y2)2 .

(2.5)

This is the typical “straight line” distance between two points. The metric space defined in this way has all the properties of a Euclidean space.

16

CHAPTER 2

Distance need not be defined in a Euclidean manner. Other metrics may be more natural under various circumstances. For instance, imagine that you are in New York City at 96th street and 1st Avenue, and you want to walk to a diner at 90th street and 3rd Avenue. Unless you can walk through buildings, the distance you would need to walk would not be the Euclidean, as-the-crow-flies distance. In this case a more realistic conception of distance is that your destination is six blocks downtown and two blocks cross-town. In other words, the diner is eight blocks away. The metric we have just described is appropriately called the city block metric. It is expressed mathematically as

d(P1,P2) = x1-x2 + y1-y2 .

(2.6)

A final example shows how general and powerful metrics can be. A third type of metric is the Minkowski metric, defined as

d(P1,P2) = [x1-x2 R + y1-y2 R]1/R.

(2.7)

Here, R may take on any positive value greater than or equal to one. If R=1, the city block metric results and we have Equation 2.6. If R=2, the Euclidean metric results and we have Equation 2.5. Clearly, R can take on an infinite number of values resulting in a infinite number of potential metric spaces. Metrics also exist for hyperbolic and spherical geometries.
People are capable of looking at distance, of creating metrics, in more than one way, and stimulus conditions can also influence the metric used. As a simple example, most city dwellers tend to think of the distance between places in terms of a driving time metric. Because different roads travel at different speeds at different times of the day, this driving time metric would be quite complicated and very non-Euclidean.
A second more whimsical metric should be familiar to anyone who lives in a cold climate. I might call it the pain metric. People are often willing to walk a bit out of their way as long as they can stay out of the cold. The pain metric, then, would be the path that produces the minimum amount of pain from the cold.
An interesting account of metrics and their applications can be found in Shreider (1974). In terms of the psychology literature reported later in this book, we will see that the metric of visual space under laboratory conditions varies depending on which instructions are given to the subject and stimulus conditions. One of the primary themes of this book is that there is no single metric that describes visual space.
Metrics are sometimes stated in a differential form. Here we assume that if the metric function is true on a large scale it is also true on an infinitesimal scale. In terms of differentials, we would express the Euclidean metric (Equation 2.5) as

TRADITIONAL VIEWS OF GEOMETRY AND VISION

17

ds2 = dx2 + dy2 .

(2.8)

Here, dx indicates a very small change in the x-dimension, dy indicates a very small change in the y-dimension, and ds indicates the change in distance. This form of the distance equation makes it possible to use differential equations and tensors when working with geometrical problems. You will notice throughout this book that I will tend to prefer algebraic expressions most of the time, because I believe that clarity demands using the simplest form of an equation possible.

Banach spaces and other more esoteric variations. Another way to employ spatial coordinates is to think of them as representing vectors. The length of vectors is expressed in terms of norms. In a normed-linear space, the norm is a function that assigns a number ||x|| to each vector such that:

(1) The norm is always non-negative and only zero if the vector has no length. That is,

||x|| 0, and ||x|| = 0 if and only if x is the zero vector.

(2.9)

(2) The triangle inequality holds

||x+y|| ||x|| + ||y||.

(2.10)

(3) Multiplying the coordinates by a scalar increases the norm by that scalar

||ax|| = |a| ||x||.

(2.11)

A normed-linear space that is also a complete metric space is called a Banach space. Thus, a Banach space is a metric space that assumes that distances have a ratio property in addition to the other assumptions. The distance between two points in a Banach space is simply ||x-y||.
Even more general spaces exist such as Hausdroff spaces, Normal spaces, T1 spaces, and topological spaces. For the most part, these spaces are of lesser importance to the present enterprise because the concepts of size and distance are not as central in these spaces. The most general form of geometric space is a topological space. Topology looks at the properties of a space that remain invariant under continuous transformations. A continuous transformation is one in which two points that are close to each other to begin with will still be close to each other after the transformation, i.e., a bend or a stretch, but not a break or a tear. We will examine some of the topological properties of visual space later.

18

CHAPTER 2

Quasimetrics. Angles and areas are also types of measures on a space. Many of their properties may be described by metric-like axioms. Angles may be defined on three points x, y, and z (where y is the vertex) by the function A(x,y,z) where 0˚ A 180˚. The area of a triangle defined by three points x, y, and z may be expressed by the function T(x,y,z). Thinking in terms of a Euclidean space as an example, both of these functions should have the property of being non-negative. Angles are symmetric in that

A(x,y,z) = A(z,y,x).

(2.12)

Areas are entirely symmetric in that all six possible orders for x, y, and z produce the same area. A form of the triangle inequality also holds. That is, including a fourth point t,

A(x,y,z) A(x,y,t) + A(t,y,z).

(2.13)

And T(x,y,z) T(x,y,t) + T(x,t,z) + T(t,y,z).

(2.14)

I will call measures that satisfy metric-like axioms quasimetrics. The quasimetrics and metrics of a space taken together will be referred to as the metric properties of the space. Geometry is largely the study of distances, angles and areas and there interrelationships. By defining the metric properties of a space, we can set down a reasonably complete description of the geometry of that space.
I will also use the term quasimetric in another sense. Distance functions may satisfy some, but not all of axioms of a metric space. That is, we will find that peoples’ judgments of size and distance do not always satisfy all of the axioms of a metric space, but still express peoples’ perceptions of these quantities. I will also call distance functions that don’t quite satisfy metric assumptions, but still reflect our mental conceptions of distance, quasimetrics. Failures to meet the metric axioms will make us question how well visual space fits together into a coherent structure.

Geometry as transformation. Because the same objects exist in both physical and visual spaces, points that exist in one space should also exist in the other. Once approach to understanding visual space then is to ask how points in physical space are transformed or mapped onto visual space. In 1872, Felix Klein, a German mathematician, proposed that different geometries can be defined by the transformations they permit and the properties that remain invariant in an object’s structure after these transformations. (See Tittle, Todd, Perotti, and Norman (1995) for a excellent review of Klein’s concepts and an empirical investigation of how they apply to visual space.) For example, Euclidean geometry allows an infinite variety of translations and rotations of coordinates which still preserve the distance between any two points in a space.

TRADITIONAL VIEWS OF GEOMETRY AND VISION

19

One of the simplest possible transformations between visual space and physical space would be a similarity transformation. Here, visual space would simply be a larger or smaller version of physical space, while the basic structure of physical space would be preserved in all other ways. After a similarity transformation, observers would report the absolute sizes of objects in visual space to be larger or smaller than the corresponding size in physical space, but they would be able to accurately report on ratios of distances, angles should be accurately perceived, and factors such as orientation and position in space should have no affect on judgments.
Another transformation that could be involved in transforming physical space to visual space would be a conformal transformation. The geometries of constant curvature mentioned earlier are of this type. While these geometries allow a considerable degree of distortion to exist between the two spaces on the macro level, they assume that both spaces are locally Euclidean. That is distance and angle relations in sufficiently small, local regions of the space are essentially Euclidean. On the macro level, however, straight lines in physical space may appear curved in visual space.
Another sort of transformation that proves to be very relevant in describing visual space is an affine transformation (c.f., Tittle, Todd, Perotti, & Norman, 1995; Todd & Bressan, 1990; Todd & Norman, 1991; Wagner, 1985; Wagner & Feldman, 1989). An affine transformation would allow space to be stretched in one direction compared to physical space. After such a transformation, the curvature constants of the space would remain the same and parallel lines would still look parallel. The geodesic, or the shortest path between two points, would remain the same. Observers would be able to correctly judge distance ratios for stimuli with the same orientation; however, the judged size of an object would systematically change along with changes in stimulus orientation.
A final transformation allowed in a Klein geometry is a topological transformation. A topological transformation would allow very complicated distortions to exist in visual space relative to visual space subject to the constraint that points adjacent to each other in physical space must still be adjacent to each other in the transformed space. Visual space is almost certainly of this type; however, the same cannot be said for memorial space or cognitive maps. Cognitive maps have many gaps and discontinuities that make them difficult to describe in geometric terms.

Philosophical Approaches to Visual Space Perception

The concept of visual space has a long history that antedates psychology as a discipline. Physical space, experiential space, and mathematical space occupied the attention of many great philosophers. In a recent article reviewing the history of a broad range of perceptual phenomena, Wade (1996) repeatedly demonstrates that pre-20th century philosophers and scientists often anticipated the

20

CHAPTER 2

observations of their modern counterparts. Experiential geometry is no exception to this rule.
According to Kline (1972), the ancient Greeks recognized a difference between the mathematical space described by their geometry and experiential space. Many subsequent philosophers lost this subtle distinction.
Newton pointed out that all mathematicians up to his time were convinced that Euclidean geometry correctly represented the properties of physical (and experiential) space. Isaac Barrow listed eight reasons for the absolute truth of Euclidean geometry. Hobbes, Locke, and Leibniz believed that Euclidean geometry is inherent in the design of the universe.
Bishop Berkeley (1709/1910) rediscovered the distinction between physical space, mathematical space, and visual space. Berkeley believed that nothing exists outside of conscious experience. Berkeley went on, however, to distinguish between the space defined by touch (“tangible extension”) and the space defined by vision (“visible extension”). The tangible space was thought to be primary and is equivalent to the space revealed by measuring devices (what I call physical space). Visible space, on the other hand, was thought to be derived through associations between vision and touch. However, Berkeley clearly declared that tangible space, visible space, and the abstract space of mathematicians need not be the same.
Having made this bold statement, Berkeley then retreated into showing why the two types of space are equivalent. Berkeley believed that the eye itself is incapable of depth perception because objects at different distances can fall on the same point of the retina. Visual sensations of distance were said to arise because “when we look at a near object with both eyes, according as it approaches or recedes from us, we alter the disposition of the eyes, by lessening or widening the interval between the pupils. This disposition or turn of the eyes is attended with a sensation ... ” from which perceptions of distance arise. These perceptions of distance are not direct, however, but rather arise from the repeated association of tangible experience with visible experience. Thus, Bishop Berkeley concluded that visual space ends up being identical to physical or tangible space. That is, visual space is Euclidean.
On the other hand, Berkeley argued that visual space is not continuous. He believed there exists a minima visibilia or a minimum size visible that we can perceive. Visual experience is composed of these small, but finitely-sized components. This viewpoint would imply that visual space does not satisfy one of the axioms of a Riemannian geometry of which Euclidean space is a subset. As Berkeley lived before Riemann’s time, he did not see this implication. Once again, Berkeley believed visual space is Euclidean and is equivalent to tangible space.
Berkeley’s idea that touch is the basis of all spatial knowledge and that minima visibilia exist would later influence psychology through the work of Weber. Weber’s creation of the just noticeable difference (JND) concept came

TRADITIONAL VIEWS OF GEOMETRY AND VISION

21

from his studies of touch and how touch is the basis for our concept of space. Weber later influenced Fechner, and Fechner influenced Helmholtz.
The first major philosopher to deviate from Berkeley’s position was David Hume in his Treatise of Human Nature (1739/1896). Hume did not deny the existence of a world outside of human experience. He did feel, however, that no one could justify such a belief. Thus, all statements made by philosophers and scientists concerning space must be in regard to experiential space. Hume also felt that there were no self-evident truths or universal laws. As such, Euclidean geometry could not be thought of as a certain truth. Like any other question, the validity of Euclidean geometry is subject to empirical investigation. Even if scientific investigation proved to support the Euclidean position, there is no guarantee that the universe will always be Euclidean or is Euclidean everywhere.
Immanuel Kant reacted violently to Hume’s nihilism. In his Critique of Pure Reason (1781/1929) Kant asserted that the Euclidean nature of space is a synthetic a priori truth. In other words, it is a truth which goes beyond logic per se but which nevertheless is certain. This certainty does not arise from knowledge of a pre-experiential world. Instead, space is a fundamental element of human experience; it is a necessary precondition for a person to experience anything at all.
In Kant’s words:

Space is not an empirical concept which has been derived from outer experiences. For in order that certain sensations be referred to something outside me (that is, to something in another region of space from that in which I find myself), and similarly in order that I may be able to represent them as outside and alongside one another, and accordingly as not only different but as in different places, the representation of space must be presupposed. The representation of space cannot, therefore, be empirically obtained from the relations of outer appearance. On the contrary, this outer experience is itself possible only through that representation. (Kant, 1781/1929, p. 23)

Kant made no distinction between space as the individual experiences it and as the physical scientist experiences it. Both represent phenomenal experience as opposed to the “noumenal world” that lies beyond experience. The geometry of the noumenal world is unknown. The Euclidean geometry reflected in our experience and in our physical science is a function of the organizing power of the human mind. The visual world has three dimensions not because this is the true nature of the universe but because our mind gives it to us in this form.
Helmholtz (1869/1921) was drawn to the study of geometry and experience by his opposition to Kantian nativism. While he did agree with Kant that some sort of basic concept or intuition of space must exist before we can perceive the world spatially at all, Helmholtz believed that determining the specific geometries that best describe visual experience and the physical universe were em-

22

CHAPTER 2

pirical problems to be settled by observation rather than philosophical analysis. Because more than one consistent geometry can be imagined, Euclidean spatial axioms can not be considered necessary for us to conceive of space. A general spatial sense can be transcendentally given without the axioms that govern it being so. The axioms that are expressed in spatial experience can be empirically investigated.
Helmholtz (1868/1921) believed that he could establish three axioms which described visual and physical space. In particular, he believed the following three characteristics describe our spatial experience: 1) Space is a three dimensional manifold. That is space can be coordinatized, and motion produces a continuous change in the coordinates. 2) Rigid objects display the property of free mobility such that moving from one part of space to another does not change the structure of the object. 3) Space is monodromic. A complete rotation of an object around any axis will produce a figure that is congruent to the original figure. Helmholtz believed that these three axioms taken together imply that experiential space is a geometry of constant curvature. That is, experiential space must be either Euclidean, spherical, or hyperbolic.
Helmholtz (1868/1921) felt that one could show empirically that physical space is Euclidean, at least within the limits of measurement possible for terrestrial measures. Experiential space, on the other hand, could be consistent with any of the geometries of constant curvature. Helmholtz went on to describe how the world would be experienced if our perceptions were spherical or hyperbolic. Which of these geometries best describe visual space is an open empirical question.
Meanwhile, back on the British Isles, Thomas Reid, a Scottish philosopher, published a book entitled Inquiry into the Human Mind (1764/1813). With this book, the study of visual space per se began.
Thomas Reid was interested in “what the eye alone can see” without regard to deeper processing due to cognition, experience, or motion. Indeed, Reid rejected the representational view of mind altogether (Ben-Zeev, 1989,1990). With this view of perception in mind, Reid developed a geometry for this visible space which was synthetic in nature. As summarized by Daniels (1974), Reid’s geometry was presented in four steps: (1) He used standard notions of points, lines, angles, etc.; (2) he applied these notions to what an idealized eye would see; (3) he claimed that the eye itself is incapable of depth perception for much the same reason as Berkeley; as such, he stated that his “visible” space can be represented by a sphere of arbitrary radius encompassing the space; (4) he deduced some central theorems of the geometry. (For example, no parallel lines exist is the space. The sum of the angles of a triangle is greater than 180˚. A straight line in visual space will cut the space in two and come around to meet itself if followed through its whole circuit.) Mathematically, these assumptions are consistent with a spherical geometry (technically called a doubly elliptic Riemannian geometry).

TRADITIONAL VIEWS OF GEOMETRY AND VISION

23

Reid conducted a thought experiment with a hypothetical race called the Idomenians who could see, but not touch the world. He believed that such creatures would possess no conception of depth. Like Berkeley, Reid believed that depth perception arises from our sense of touch.
Thomas Reid’s theory is significant for several reasons. First, his statement of a non-Euclidean geometry antedates Lobatchevsky’s mathematically rigorous work. Second, Reid’s geometry is explicitly applied to a visual space as opposed to physical space (which Reid also believed existed) and as opposed to a purely mathematical space. Third, Reid was also interested in how observers were capable of depth perception. He may be the first to submit that frontal and in-depth perceptions of extent might come from different sources. If frontal and in-depth perceptions of extent might come from different sources, we might conjecture that it is possible that the two spatial dimensions could be perceived differently.
Henri Poincaré, one of the greatest mathematical philosophers of the 20th century, wrote extensively about the nature of space. Poincaré concluded that there is no “true” geometry. Many geometries may be applied to experience. Some geometries are simpler and more convenient than others, however.
Poincaré spoke explicitly about visual space in one section of his Science and Hypothesis (1905). In several respects, Poincaré’s views were similar to Reid’s. The eye itself only receives a two dimensional view of the world. This visual space is clearest at the center of the field of vision. More importantly, Poincaré reiterated that perception of depth arises from “sensations quite different from the visual sensations which have given us the concept of the first two dimensions. The third dimension will therefore not appear to us as playing the same role as the two others. What may be called complete visual space is not therefore an isotropic space” (p. 53). Once again, the untested possibility exists that frontal and in-depth perception of space may be quite different.
The viewpoint that visual space is represented by a spherical geometry is common among more modern philosophers who have been influenced by Husserl’s (1910) phenomenological approach. Husserl felt that during most of their lives people assume a natural attitude in which the focus of our awareness is turned outward toward the world and we experience it without examining our thoughts. In contrast, the phenomenological attitude of Husserl involves turning ones mind inward and attempting to “bracket out” all presuppositions arising from our knowledge of and theories about the outer world.
In terms of space perception, Husserl believed that our experience of space is layered (Scheerer, 1985). His first layer is the visual field. The visual field is similar to an artist’s canvas on which sensations are spread out left and right, and up and down, but the visual field lacks depth. Objects only present one face to the observer. In other words, we see what Idhe (1986) referred to as the manifest profile of an object. His second layer involves eye movement. In this occulomotor field, visual and kinesthetic experiences are integrated to produce a wider area of clear visual experience. The third layer involves head movement; this kephalomotor field exposes us to a hemisphere of visual experience. Up to

24

CHAPTER 2

this point, Husserl’s analysis produces an experience similar to being inside a sphere of arbitrary size. The third dimension of depth is only added to our experience through motion. In this locomotor field, the horizon of my experience shifts with the movements of my body and the properties of three-dimensional space begin to emerge. Locomotion within the field produces an experience of expansion of approaching objects and contraction of receding objects. This expansion and contraction of objects is correlated with our perception of distance and depth. In other words, we experience visual flow patterns similar to those Gibson (1950) describes.
This description of the world reminds me of Helmholtz (1868/1921) who described the experiences of a person living in a hyperbolic space (which he termed a “pseudospherical” space) in a similar fashion to this analysis. According to Helmholtz, if the experiences of a person were hyperbolic they would “give him the same impression as if he were at the center of Beltrami’s spherical image. He would believe he could see all round himself the most distant objects of this space at a finite distance, let us for example assume a distance of 100 feet. But if he approached these distant objects they would expand in front of him, and indeed more in depth than in area, while behind them they would contract” (p. 21).
More recently, Daniels (1974) and Angell (1974) have proposed that visual space is phenomenologically best represented by a spherical geometry, just as Reid (1764/1813) suggested. Along similar lines, French (1987) published an excellent philosophical treatise on space perception that also takes a phenomenological point of view. French contrasts his conception of “visual space” with a naive realist view of space. To French, visual or phenomenal space is defined by an attitude shift from the outward-oriented view of a physical scientist to an inward-oriented view of the world which examines the experience of space that is phenomenally immediately present.
French believes that phenomenological space is continuous and bounded within a region of about 170˚ horizontally and 120˚ vertically. Phenomenal space is two-dimensional because it does not possess thickness. In fact, to French, phenomenal space is like seeing a projection onto part of a sphere where the viewer is located at the center of the sphere. Depth perception is a distinctly different topic and may be thought of as a separate dimension, separately determined. On the other hand, depth does influence our perception. French believes that objects at different distances are projected onto spheres of varying curvature and that this curvature is a function of distance from the observer.
In my opinion, the various philosophers who argue that visual space is spherical are not wrong, but they are incomplete. Under monocular conditions where binocular depth cues are unavailable (and head and body movement are not allowed) and when the observer is asked to assume a phenomenal attitude in which cognitive cues to depth are ignored, the spherical model probably does describe our perceptions of visual space. Although these conditions seem rather

TRADITIONAL VIEWS OF GEOMETRY AND VISION

25

narrowly drawn, humans are capable of assuming such an attitude under such conditions and therefore the spherical models deserves a legitimate place among the geometries of visual space. However, the phenomenological attitude is not the only attitude that may be included sensibly within the definition of visual space. Our ordinary understanding of space as we manipulate it to achieve our ends is certainly a very different experience of space than the phenomenological view, but it is none-the-less an experience of space. In fact, it is the more common and useful experience of space. Neither of these sorts of spatial experience need correspond to space as physical measures reveal it to us. A complete understanding of visual space then, must take into account the varying attitudes that people are able to assume when thinking about space. We will speak more of this matter when we discuss the effects of instructions on spatial judgments in Chapters 5 and 6.

Opening the Psychophysical Toolbox to Build a Window into Visual Space

Physical space can be measured using a variety of physical instruments such as rulers and protractors. The results of these measures are objective in that more than one observer can simultaneously examine the results of the measurement and different observers should come up with precisely the same results through repeated measurement. Unfortunately, experiential space (and visual space in particular) is subjective by its very nature. It is impossible to open up a person’s conscious experience to allow multiple observers to make observations. The only route available to investigate visual space is to allow people to report on their own experiences using the tools of psychophysics.
The purpose of the last section of this chapter is to briefly introduce the tools available for examining visual space perception and to say a few words about their weaknesses and the meaning that can be assigned to the results they produce. Subsequent chapters will contain a more complete discussion of the issues and techniques raised here, but saying a few words now will make it easier to develop those other points later.

Psychophysical tools and their limitations. Psychophysical methods are of three basic types: numeric estimation, magnitude production, and sensitivity measures. Numeric estimation techniques require an observer to assign a number to a perceived intensity or extent. Two commonly employed numeric estimation methods are magnitude estimation and category estimation. In magnitude estimation observers attempt to preserve ratios, that is, if one stimulus seems twice as large another, the observer is asked to assign a number that is twice as big. In category estimation, observers are asked to assign each stimulus to a limited number of perceptually equal categories. In space perception, numeric estimation techniques have been used to estimate perceived areas, volumes, an-

26

CHAPTER 2

gles, and other metric characteristics of a space, but they have been employed primarily for estimating size and distance.
Magnitude production techniques require an observer to match perceptions. In a perceptual matching task, the observer may be asked to adjust the intensity or extent of a stimulus under standard conditions until it is perceived to be equivalent to a fixed stimulus under varying conditions or they may be asked to pick which of a series of standard stimuli best matches their perception of a comparison stimulus. Alternatively, the observer might be asked to draw or physically produce their impressions of a stimulus arrangement. For example, the observer might be asked to draw a map of the perceived layout of a set of stimuli. This mapping technique is most commonly employed in cognitive mapping studies that involve large-scale environment whose spatial layout is learned across time. It is seldom used for the perception of smaller-scale stimuli that are physically present and simultaneously observable at the time of the testing (although there are exceptions).
Sensitivity measures ascertain the smallest physical difference between stimuli that observers can discriminate. The most common sensitivity measurement techniques are the method of constant stimuli, the method of limits, and the method of adjustment. These techniques are typically directed at small-scale stimuli such as the perception of line length and seldom employed to measure perceptions of large-scale environments. In general, sensitivity measures give two sorts of information (Ono, Wagner, & Ono, 1995; Wagner, Ono, & Ono, 1995). First, one can determine the accuracy of a perception; that is, how much a perception deviates from some objective value. This is particularly useful in quantifying spatial illusions. Second, one can determine the precision of perceptual processing; that is, how little must a stimulus change before the observer notices a difference. These just noticeable differences may be thought of as measures of perceptual sensitivity where sensitivity is highest when the just noticeable difference is small.
All of these psychophysical methods have been applied extensively to the spatial domain. From a geometric point of view, this extensive body of literature suffers from several weaknesses. First of all, psychophysical work has been concerned almost exclusively with unidimensional stimuli (Lockhead, 1992), although there are exceptions to this rule (Wagner, 1992). Distance perception, for instance, has been extensively investigated. Many studies explore perception of distance in-depth (stretching away from the observer). Other studies explore distance perception in the frontal plane. Yet, few studies systematically explore depth and frontal perception simultaneously. The infinite number of possible orientations for distances in between frontal and in-depth orientations is virtually an untapped void. We will mention a few studies that apply psychophysical methods to multidimensional stimuli in Chapter 7, but these studies are rare. Second, the psychophysical literature on space perception is largely disorganized. Providing a multidimensional geometric model for visual space may allow

TRADITIONAL VIEWS OF GEOMETRY AND VISION

27

us to fit unrelated studies into a more coherent framework. This is the goal of Chapters 5, 6, and 7. Third, it is rare for any researcher to report results from more than one method in the same paper. This makes it impossible to know whether the psychophysical functions found in any given study are due to perception itself or due to response processes inherent in the method. A few studies will be mentioned later that do apply several alternative techniques at once. Commonalties resulting from these converging measures are more likely to represent aspects of visual space that are independent of the methods employed. Yet, few studies bother to take this step.
Fechner, Stevens, and the power function. When people are asked to report on their spatial perceptions using numeric estimation methods, the relationship between judged (J) and actual (D) distance is most often described by a power function (Baird, Wagner, & Noma, 1982; Cadwallader, 1979):

J= D

(2.15)

where is an exponent and is usually thought of as a scaling constant (although it has other possible meanings such as indicating the existence of an illusion as we will see later). In psychophysics, this equation is sometimes known as Steven’s Law and it is commonly thought to describe the relationship between magnitude estimates and almost any unidimensional stimulus continua (Stevens, 1957, 1975). Theoretically, category estimates are said to follow a logarithmic function (Galanter, 1962), but a many researchers (Krueger, 1989; Wagner, 1982, 1989) feel that Equation 2.15 also describes these judgments and is the more useful and meaningful formulation.
If our perceptions perfectly matched reality, we would expect that the exponent for the power function should be exactly 1.0. Yet, past research has shown that there is no single exponent that holds under all conditions (Wiest & Bell, 1985). In fact, there are times when the exponent is considerably higher than 1.0, and conditions under which it is considerably lower than 1.0. The exponent in perceptual studies depends on instructions, the richness of depth information, and stimulus orientation (Baird, 1970). Similarly, when people are asked to judge distance from memory, as in cognitive mapping studies, exponents are typically less than one and often range widely depending on factors such as the familiarity of the environment, the presence of barriers, and the informational density of the environment (Wagner, 1998; Wiest & Bell, 1985). In summary, the exponent is not a constant, nor is it generally equal to 1.0 in either the direct perception or recall of spatial information.
What does the exponent to the power function mean? Wagner (1998) purposed that the exponent for the power function is related to the concept of uncertainty reduction. Uncertainty about spatial layout in turn should influence

28

CHAPTER 2

judgment precision. Conditions that make our knowledge of stimuli less precise should in theory lead to a decline in the exponent.
Baird and Noma (1978) and Teghtsoonian (1971) believe the exponent reflects the relative sensitivity of the subject to the response and stimulus dimensions. Teghtsoonian (1971) also believed that the exponent is directly related to the stimulus range. When a broad range of stimuli are presented the exponent should decline compared to presenting a narrow range of stimuli.
Warren (1958, 1969, 1981) proposed a Psychophysical Correlate Theory. He believed that subjects do not always directly estimate the stimulus dimension that the experimenter intends them to, but rather base their judgments on some other aspect of the stimulus. For example, instead of actually judging area, estimates may actually reflect the size of a single dimension of the stimulus. Variations in exponents result when the experimenter attempts to compare the subject’s estimates to the wrong stimulus dimension.
Steven’s (1970, 1971) felt variations in the exponent result from the varying degrees of transformation that occur in the process of transducing stimulus energy into neural firing and in the process of carrying that information up to and through the brain. Similarly, Baird’s Complementrity Theory (1996) posits that the exponent arises from the competition of two processes, a sensory process that depends on the activation of neural populations by stimuli and a second cognitive process that reflects the subject’s uncertainty when giving a response on a given trial.
Whatever the true meaning of the exponent, we may think of the process of arriving at a psychophysical judgment as having two stages. First, the perceptual process produces a sensation that we experience, and then a judgment process operates on this sensation to produce a number that represents our judgment. So the exponent of the power function really has two parts and Equation 2.15 can be rewritten as in the following equation:

sr
J = (D )

(2.16)

Where s indicates that some portion of the exponent is determined by sensory factors, and r indicates that some portion of the exponent is determined by response factors. (For some even this may be too strong a formulation. Perhaps such a reader will accept that the exponent is some function of sensory and response factors.)
The Holy Grail for a psychophysicist interested in space perception would be to determine the equation that truly reflects a person’s experience of world as a function of physical layout. Unfortunately, if Equation 2.16 or something like it is correct, this Holy Grail will be forever out of reach. We will never know the degree to which the judgments observers produce reflect their conscious experience as opposed to judgment factors related to number usage or production errors.

TRADITIONAL VIEWS OF GEOMETRY AND VISION

29

However, because this issue cannot be resolved, the theorist is forced to take the data as it is and define visual space in terms of the data. The geometry of visual space is really the geometry of the judgments subjects produce.
There is one final issue I would like to discuss concerning psychophysical functions. Generally, the parameters of Equation 2.15 are thought of as being constants. Lockhead (1992) criticized psychophysical work for not sufficiently taking into account context when generating psychophysical equations, and I have agreed with him that it is important to take such things into account (Wagner, 1992). In truth the parameters of Equation 2.15 should be thought of as being functions of these contextual conditions, not constants. Perhaps Equation 2.14 is better rewritten as

( , , , . . .)
J = ( , , , . . .) D

(2.17)

where , ,and represent varying experimental conditions. Here, the “scaling constant” and the exponent are no longer thought of as constants, but as functions of judgment conditions such as instructions, cue conditions, etc. This final equation reflects the essential spirit of this book. There is no single geometry for visual space perception, but the geometry of the space is a function of conditions.

3
Synthetic Approaches to Visual Space Perception
As is often the case with problems in perception, psychology took over where philosophy left off. The Kantian (1781/1923) notion that only one geometry for phenomenal space was possible, because this way of seeing the world is one of the organizing structures of the mind without which perception is impossible, gave way to Helmholtz’s (1869/1921) view. According to Helmholtz, because more than one geometry may be consistently apprehended by the mind, the nature of space as it is experienced is an empirical question best left to scientific investigation.
Visual Space as a Hyperbolic Geometry
The empirical investigation of visual space also began with Helmholtz (1867, 1896/1925). (Although a good case could be made for Götz Martius’s (1889) work on size constancy which we will discuss in Chapter 6.) Helmholtz found that when an observer is asked to arrange three luminous points in the horizontal plane in a straight line, the resulting arrangement is not always physically straight. The configuration can be physically concave toward the observer for near points, physically straight at a small range of intermediate distances, or convex relative to the observer for physically distant triplets.
In another classic experiment, Hillebrand (1902) asked observers in a darkened room to arrange pairs of luminous points stationed at various distances from the observer to form an alley with walls equidistant from each other. The resulting arrangement was not physically straight, but both walls of the alley curved outward with increasing distance from the observer.
Blumenfeld (1913) replicated Hillebrand’s work and extended it in important ways. Like Hillebrand, Blumenfeld asked observers to arrange two rows of luminous points to form an alley whose walls were equidistant. The most distant pair of points was fixed. The resulting “distance alley” arrangement was neither physically straight nor were the walls parallel in a Euclidean sense. As before, the physical distance between pairs of points gradually increased with increasing distance of the pair from the observer. Blumenfeld also asked the observers to arrange the points to form straight lines parallel to each other. The resulting “parallel alley” arrangement is similar to that obtained for the “distance alleys.” The distance between pairs of points gradually increases as the pairs lie farther from the observer. The parallel alleys were consistently different from the dis30

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 31

tance alleys in one respect, however. The parallel alleys were always located inside the distance alleys; that is, the distance between pairs of points is always smaller for parallel alleys than it is for distance alleys. Blumenfeld’s experiments have been replicated many times by other observers (Hardy, Rand, & Ritter, 1951; Hardy, Rand, Rittler, Blank, & Boeder, 1953; Higashiyama, Ishikawa, & Tanaka, 1990; Indow, Inoue, & Matsushima, 1962b, 1963; Indow & Watanabe, 1984a). However, attempts to extend this work to the fronto-parallel plane have generally found distance and parallel alleys to coincide and not display significant curvature (Indow, 1988; Indow & Watanabe, 1984b, 1988).

Luneburg’s theory of binocular space perception. Luneburg (1947, 1948, 1950) used Helmholtz and Blumenfeld’s demonstrations as evidence that visual space is hyperbolic. Luneburg’s approach was synthetic in nature. He began by listing a series of axioms about visual space similar to those Helmholtz proposed (which we discussed in the previous chapter). Based on these axioms, Luneburg concluded that visual space is a hyperbolic geometry.
After Luneburg’s death in 1949, a number of talented mathematical psychologists have kept Luneburg’s work alive by proposing modified versions of the theory. Blank (1953, 1957, 1958, 1959) refined Luneburg’s theory somewhat along the same synthetic path. Blank more explicitly laid out the axioms and hypotheses on which the theory was based and accounted for the experimental evidence that existed at the time. Following Blank, Indow (1967, 1974, 1979, 1990, 1995) became the primary proponent of the theory, producing a series of mathematically sophisticated papers and empirical tests of the theory. More recently, Aczél, Boros, Heller, & Ng (1999) and Heller (1997a, 1997b) have written clear papers that present additional refinements of the theory.
Luneburg’s theory was based on a fairly sizeable set of axioms. First, Luneburg assumed that visual space is a metric space as defined in the last chapter. Second, the theory assumes that visual space is convex. This axiom says that for any two points (P1 and P3) on a line a third point (P2) exists on the line between them such that

D(P1,P2) + D(P2, P3) = D(P1,P3)

(3.1)

where D is the metric for the space. Third, the theory assumes that visual space is compact. This means that for
any point P1 and any number , there exists another point P2 such that

0 < D(P1,P2) <

(3.2)

This axiom allows one to assume that the metric is continuous and wellbehaved. It also allows one to express the metric in a differential form, because the distance function is defined at the smallest levels.
Fourth, the theory assumes that visual space is locally Euclidean. This means that, within a sufficiently small neighborhood, metric relationships between points are essentially Euclidean, even though the space as a whole may not be.

32

CHAPT ER 3

For example, the surface of the earth may best be described by a spherical geometry, but within any local region we get along rather well using Euclidean concepts of measurement to perform our daily tasks.
Fifthly, the theory assumes that visual space is Desarguesian. This says that for any two points on a visual plane, the geodesic (or shortest path between the points) does not depart from the plane.
Finally, Luneburg assumed visual space has the property of free mobility. That is a rigid structure moved through visual space should retain its distance and angular relationships. In other words, visual space is homogenous, and one can construct visually congruent configurations at any location and orientation.
Taken together, these axioms imply that visual space is a Riemannian geometry of constant curvature. Riemannian geometries of constant curvature are of three types: Euclidean, elliptical (spherical), and hyperbolic. Blumenfeld’s demonstration that more than one set of “parallels” could be produced by different instructions led Luneburg to conclude that visual space is non-Euclidean. The fact that the “parallel alleys” tend to be inside the “distance alleys” allowed Luneburg to conclude that visual space is hyperbolic.
Knowing the visual space is a Riemannian geometry of constant curvature also defines the metric function that allows us to specify the distance between points in space. Using differential form, the length of a line element ds is

ds2 = dx2 + dy2 + dz2 [1 + K (X2+Y2+Z2)] 4

(3.3)

where dx, dy, and dz represent small changes in the x, y, and z dimensions of a point located at coordinates X, Y, and Z. Here, K is the curvature of the space. If K is positive, the space is spherical. If K is zero, the space is Euclidean. In Luneburg’s theory, K takes on the negative value of a hyperbolic space.
In this hyperbolic space, points lying along the same Vieth-Müller circle are perceived as being equidistant from the observer. Here, Vieth-Müller circles are circles that contain the center of each eye as part of their circumference. (See Figure 3.1.) Under idealized assumptions about the structure of the eye, the Vieth-Müller circle consists of stimulus locations that are projected on corresponding places on the two retinas when the observer fixates on one point of the circle (Howard & Rogers, 1995). The lines in the hyperbolic space consist of Hillebrand hyperbolae. These lines should be perceived to have a constant visual direction. If the observer fixates on a point along an Hillebrand hyperbola, other points along the hyperbola are projected on retinal points that deviate the same extent from the fovea in both eyes but in opposite directions.
Luneberg’s theory makes several testable predictions outside of those already mentioned. First of all, the sum of the angles of any triangle should always be less than 180˚. Second, Luneburg established a correspondence between physical space and visual space. This correspondence yields a metric formula (in terms of physical coordinates) that relates physical space to distance in visual space. This formula could be tested empirically. Third, the mapping from physical space to visual space is conformal. That is, an angle in physical space is mapped onto an

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 33 equal angle in visual space. (The validity of each of these predictions will be discussed in detail in Chapter 7.) In addition, Luneburg’s formulation requires visual space to be bounded. The most distant objects in physical space should seem to lie at a finite distance from the observer.
Figure 3.1. Vieth-Müller circles and hyperbolae of Hillebrand. Points lying along Vieth-Müller circles are theoretically perceived as being equidistant from the observer, while points on Hillebrand hyperbolae are theoretically perceived as having the same visual direction.

34

CHAPT ER 3

A beautiful theory, but does it work? Anyone who has worked with Luneburg’s theory is impressed by its brilliance, precision, and scope. The theory stands as one of the most sophisticated and beautiful contributions ever made to psychology. (Interestingly, Luneburg was actually a physicist rather than a psychologist.) This very elegance may account for the fact that the theory has survived over 50 years. As Blank (1959) said “There is something more to be said for an apt mathematical model. So elegant was Luneburg’s theoretical development that it was not abandoned despite the initial failure of experimental results to conform to theory” (p. 398). Despite the brilliance of Luneburg’s theory, empirical results have not always supported it.
Of course, the weakest point of any well-developed axiomatic system is in the axioms themselves. Luneburg’s theory is no exception. Luneburg’s first axiom is that visual space is a metric space. A number of researchers have disputed this claim. For example, Cadwallader (1979), Codol (1985), and Burroughs and Sadalla (1979) report that distance judgments are not always symmetrical; that is, the distance from point A to point B is not always the same as the distance from point B to point A. Baird, Wagner, and Noma (1982) demonstrated that distance judgments often violate the triangle inequality. Baird et al. also show that another of Luneburg’s assumptions, that visual space is convex, routinely fails to describe distance judgments. We’ll speak more about the logic and evidence behind these statements and the consequences of their violations in the next chapter of this book.
Convexity and metricity are not the only axioms of Luneburg’s system that may be open to doubt. For example, is visual space really compact? This axiom requires that for any detectable distance percept, yet finer discriminations must be possible. In other words, there should be no limit to our ability to discriminate stimuli. However, all psychophysical dimensions have a threshold for detection and a finite just noticeable difference for distinguishing between stimuli. Similarly, there is a minimum size to allow detection of an object, and there is a minimum difference between sizes that is distinguishable. For example, the Weber fraction for line lengths is about .04 (Baird & Noma, 1978). While this is very good, it is not good enough to satisfy the compactness axiom. Bertrand Russell (1971) expressed this conclusion eloquently

We must do one of two things: either declare that the world of one man’s sense-data is not continuous, or else refuse to admit that there is any lower limit to the duration and extension of a single sense-datum. The later hypothesis seems untenable, so that we are apparently forced to conclude that the space of sense-data is not continuous... (p. 107)

The compactness axiom is necessary to allow the expression of distance metrics in differential form. Luce and Edwards (1958) argue that existence of finite sized just noticeable differences makes the use of differential form illegitimate. If taken seriously, this objection is general enough to lead us to reject more than Luneburg’s hyperbolic model; without the compactness axiom, we also need to conclude that visual space is not even Riemannian.

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 35
Similarly, the rejection of the compactness axiom casts into doubt Luneburg’s axiom that visual space is locally Euclidean. The local Euclidean property states that metric relations are Euclidean in a sufficiently small neighborhood. Yet, if one can’t speak of differential sized regions, one also can’t speak of the metric relations within such a region. Under such circumstances, it is hard to assign any meaning to the locally Euclidean axiom.
Another critical axiom is that visual space is Desarguesian. According to this axiom, the visual geodesic (the shortest distance) connecting any two points in a perceptual plane should not depart anywhere from that plane. Foley has tested this property. Foley’s early work (1964a, 1964b) supported the Desarguesian property, while his later work (1972) denies it.
Foley’s (1972) work also casts into doubt the free mobility axiom. Foley asked observers to construct two separate triangles that seemed identical to the observer in different positions in space. Having constructed the triangles, observers make judgments about the corresponding portions of the triangle. The judgments of corresponding sides were not congruent, showing that what seemed to be the same configuration as a whole was not the same in terms of its parts.
Similarly, according to this axiom, it should be possible to move figures through visual space without distortion. Moving an object through space should preserve distances and angles. Angles in particular should be conformal; an angle in physical space should correspond to the same angle in visual space. Wagner (1985) showed that distance and angle judgments for physically similar objects changed dramatically depending on the location and orientation of the object in space. In particular, objects oriented frontally with respect to the observer were seen as much larger than those oriented in depth, and angles facing toward or away from an observer where seen as much larger than those seen to the side. Thus, it would seem that the location and orientation of an object in space will influence our perceptions of it, and the free mobility axiom does not hold.
Based on Foley (1972) and Wagner (1985), Suppes (1995) concluded that describing visual space would require radically different assumptions than those of Luneburg. Suppes believed that visual space was unlikely to be any of the geometries of constant curvature. Instead, Suppes believed that visual space is not unitary, but that different geometries may be required to describe different experimental results.
Cuijpers, Kappers, and Koenderink (2001) also challenged the free-mobility axiom using a different method. The authors presented reference targets at various orientations and locations relative to the observer. Observers were asked to rotate a comparison target until it appeared parallel to the reference target. Contrary to the Luneberg’s predictions (or that of any geometry of constant curvature), the comparison’s orientation systematically deviated from the reference orientation and that the degree of deviation varied as a function of relative location of the stimuli in space. They also found that the degree of deviation was influenced by how the stimuli were oriented compared to the walls of the room in which the experiment was conducted. Thus, the angular judgments not only varied with position in space, but also were influenced by the presence of reference stimuli. Cuijpers et al. hold out the possibility that free mobility might hold in an environment free of reference stimuli, such as a completely darkened room (condi-

36

CHAPT ER 3

tions similar to the parallel alley experiments that Luneberg’s proponents have used to support the Hyperbolic model), but the authors feel that geometries of constant curvature can not account for their data when reference information is present. Because most ordinary perception takes place under information rich conditions with many reference stimuli present, Cuijpers et al’s data rules out the applicability of Luneberg’s theory under most ordinary conditions.
Taken together Luneburg’s axioms imply that visual space is a Riemannian geometry of constant curvature. What is more, Luneburg used the alley experiments to conclude that this curvature was negative. Is visual space a geometry of constant negative curvature? Some evidence indicates that this may not be the case. Based on alley experiments, Hardy, Rand, and Rittler (1951) and Ishii (1972) found that for about half of their observers, the curvature was negative and for the remainder it was positive. Higashiyama, Ishikawa, and Tanaka (1990) found that the parallel and distance alleys did not differ under a variety of conditions, indicating a curvature of zero. On the other hand, Indow, Inoue, and Matsuchima (1962a, 1962b, 1963), Hagino and Yoshika (1976), Higashiyama (1976), and Zajaczkowska (1956a, 1956b), have found negative curvature for the great majority of their observers. If one steps away from alley experiments, however, the picture changes. Higashiyama (1981, 1984) asked observers to move a light until it generated either a right triangle or an equilateral triangle relative to a second point and the observer’s location. Based on this experiment, he concluded that the curvature of visual space is variable, depending on how far points range from the median plane and how far they are away. Koenderink, van Doorn, and Lappin (2000, 2003) also used triangle adjustment to determine that the curvature of visual space was positive for near stimuli and negative for distant stimuli. The curvature ranged from very negative to very positive. Similarly, Ivry and Cohen (1987) determined that the curvature varied from positive to negative for different stimulus configurations. Therefore, it would seem that visual space does not consistently display negative curvature. In fact, the curvature may not be constant at all, just as French (1987) proposed.
On the other hand, Cuijpers, Kappers, and Koenderink (2003) derived metric functions for visual and haptic space using a parallelity task. Their model found that a model assuming zero curvature fit data related to fronto-parallel horopters, parallel alleys, and distance judgments (based on data from Gilinsky (1951) and Wagner (1985)). A zero curvature would signal a Euclidean space rather than a hyperbolic one.
Luneburg’s theory predicts that the sum of the angles of a triangle should be less than 180˚. Moar and Bower (1983) had subjects estimate the direction from a location to two other locations, and calculated the angle between these direction estimates. They then asked the subjects to make similar judgment from each of the other two locations. The sum of the derived angles tended to be consistently greater than 180˚, a result that is inconsistent with both the Euclidean and hyperbolic geometry formulations, and more consistent with a spherical geometry model. Of course, the direction estimates were based on memory; so, it is possible that they do not apply to direct perception. Lucas (1969) made a similar observation from a phenomenological standpoint. Based on careful introspection he

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 37
observed that the sum of the angles of a quadrilateral appear to sum to more than 360˚, once again consistent with a spherical geometry and not a hyperbolic one.
Luneburg’s theory asserts that visual space is bounded; that is, the most distant objects should seem to lie at a finite distance from the observer. While no definite conclusions can be reached about the boundedness of visual space, a few studies bear on this issue somewhat. Plug (1989) investigated the measurement systems of ancient astronomers. He concluded that the Babylonians, Arabs, Greek, and Chinese believed the stars to lay no more than 10 to 40 meters away from the observer. Similarly, Rock and Kaufmann (1962) attempted to explain the moon illusion through a similar conception that the night sky is perceived as being a finite length away. In the case, of Rock and Kaufmann, the sky was thought to have the shape of an upside-down soup bowl. Baird and Wagner (1982) refuted this claim. They had subjects use magnitude estimation to have observers judge the distance to the night sky at various elevations. They found that some observers saw the zenith sky as being closer than the horizon just as Rock and Kaufmann predicted. On the other hand, slightly more observers saw it the other way around, the zenith was seen as being farther away than the horizon. Other subjects were in the middle, with the horizon and zenith being perceived as equally far away. What is more, the judgments of distance to the horizon sky were highly correlated with the physical distances to objects located along the horizon. Baird and Wagner concluded that the night sky is not perceived as a surface located at a specific distance away from the observer. As the article summarized “To the query ‘How high is the sky?’ we suggest the retort, ‘What a meaningless question?’“ (p. 303).
If the distance to celestial objects is indeterminate except in terms of our perceptions of distance along the ground, the next question that needs to be asked is whether our perceptions of distance along the ground are bounded. The exponent for egocentric distance judgments is typically greater than one (Wiest & Bell, 1985). This would indicate that the distance to physically far away stimuli is actually overestimated, and this over-estimation increases with increasing distance from the observer. Thus, not only do distance judgments not approach a limit, but the precise opposite appears to obtain. In addition, while there are circumstances where judgments of egocentric distance from the observer do show a tendency for judgments to distant objects to be compressed relative to what we would expect based on the judgments made toward nearer objects (because the exponent for the power function relating perceived distance to physical distance is often less than one), even these judgments do not approach a limit.
I might offer one observation that I have made introspectively that shows there to be no obvious limit to our ability to perceive that one object along the ground is farther away than another object. Anyone who has ever climbed a mountain knows that as one ascends the distance one can see continues to expand. From the top of the mountain, one can see that one hill is farther away than another even though the hills are many miles away. If visual space is bounded, the boundary must lie further away than one can see along the surface of the earth.
Another aspect of the theory can also be questioned. According to the theory, points along the Vieth-Müller circles should be perceived as being an equal dis-

38

CHAPT ER 3

tance from the observer. Yet, Hardy et al. (1953), Foley (1966), and Higashiyama (1984) have shown that distance judgment deviate systematically from this prediction. In particular, perceived equidistance lies between the Vieth-Müller circle and physical equidistance, where this deviation from prediction is greatest for stimuli close at hand. A number of theorists have attempted to modify Luneburg’s theory (Aczél et al., 1999; Blank, 1978; Heller, 1997) or proposed their own theories (Foley, 1980) to account for these deviations. None of these attempts have been completely satisfactory (Higashiyama, 1984).
Battro, di Piero Netto, and Rozenstraten (1976) found highly variable frontoparallels under full-cue conditions that varied as a function of distance from the observer and varied between observers considerably. These results also challenge Luneburg’s conceptualization.
Koenderink, van Doorn, Kappers, and Lappin (2002) present an even greater challenge to the traditional view of the Vieth-Müller circles. They asked observers to adjust a central point until it appeared to form a straight line with two flanking points along the fronto-parallel plane. The adjustments observers m ade were slightly concave toward the observer, but the curvature was very small. (The radius of curvature was 21 m for stimuli 2 m from the observer and 178 m for stimuli 10 m from the observer.) A second pointing task actually found fronto-parallel planes that were concave toward the observer, a result directly the opposite of the traditional form of Vieth-Müller circles.
Finally, I’d like to add a theoretical objection on top of the empirical disconfirmation just mentioned. Our perceptions of visual space are the result of a long process that begins in the retina, follows along ganglion cells, and continues into the visual cortex. At no stage do we expect the end product of perception to be identical to the physical instantiation of the information. We do not perceive the world as being upside-down like the image of light on the retina. We do not perceive the world as being larger in the middle of the field of vision than in the periphery even though the visual cortex devotes more space to the middle of the visual field. Why would we expect equal perceived distance from the observer to fall along the Vieth-Müller circle or for parallel lines in the space to follow Hillebrand hyperbolea just because such circles and lines fall along corresponding points on the retina. Visual space is a unified whole, not the image on the retina. Similarly, French (1987) feels that visual space must be spherical because stimulation falls on a circular retina, and geometrically this can’t be projected onto a flat surface without adding distortion. Because no distortion is evident, French feels this is evidence for a spherical geometry for visual space. To both Luneburg and French I have the same basic objection: visual space is the end product of a process which includes a great deal of cognitive embellishment and which is designed to give the person access to the most veridical picture of the world possible. We should not expect it to coincide with the physical structures of the eyes or the brain. They are means to an end, but not an end in themselves.

Alternative accounts for the alley experiments. Although many objections can be raised to Luneburg’s theory, it does have the virtue of accounting for the data from the often-repeated alley experiments. Numerous experimenters (Hardy, Rand, & Ritter, 1951; Hardy, Rand, Rittler, Blank, & Boeder, 1953; Indow,

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 39
Inoue, & Matsushima, 1962b, 1963) have found that when subjects are asked to adjust lights to form alleys in the dark, there is a strong tendency for points nearest to the observer to be set closer together than those fartherest away, and for “parallel” alleys to fall inside “distance” alleys. How can one account for this data outside of Luneburg’s formulation?
A number of researchers (Baird, 1970; Battro, Reggini, & Karts, 1978; French, 1987; Wagner, 1982) have posited that the alley experiments are special cases of the more general phenomena of size constancy, which we will discuss in greater detail in Chapter 6. In the typical size constancy experiment, a near comparison stimulus is adjusted until it is perceptually equal to a standard stimulus placed at various distances from the observer. When such adjustments take place under reduced-cue conditions (such as a darkened room with most cues to distance eliminated), observers tend to perceive standard stimuli (as reflected by their adjustments of the comparison stimulus) as being progressively smaller as distance away from the observer increases—a phenomena known as underconstancy. Conversely, if observers were asked to adjust stimuli to be perceptually equal in length to a standard at one distance, they would be expected to adjust the length of more distant stimuli to be larger than they would for stimuli closer to the observer. This is exactly what is found in the alley experiments.
How can one explain the fact that the parallel alleys are often, but not always (Hardy, Rand, & Rittler, 1951; Higashiyama, Ishikawa, & Tanaka, 1990; Ishii, 1972) inside of the distance alleys? One possibility is that the data result from instruction effects. Underconstancy is least for objective instructions in which observers are asked make their adjustments reflect physical reality. Underconstancy is greater for apparent instructions in which observers are asked to make adjustments reflect their perceptions or how things “look.” Underconstancy is greatest under projective instructions in which the observer is asked to take an artist’s eye view of distance and have their adjustments reflect the amount of the visual field taken up by a stimulus. Projective instructions imply the observer should ignore depth cues. In terms of the alley experiments, this would mean that the alleys should be straightest (and on the outside) with objective instructions and most curved (and on the inside) for projective instructions.
In fact, Battro, Reggini, and Karts (1978) were able to precisely predict the shape of parallel alleys by assuming that judgments reflect a constant Thouless ratio (which will be defined and discussed in Chapter 6). Later experimental work confirmed that Thouless ratios were indeed constant for all distances when subject performed the alley task. According to their model, alleys should be straightest with Thouless ratios close to 1 (which generally occurs with objective instructions under full-cue conditions) and most curved and located inside with Thouless ratios close to 0 (which generally occurs with apparent or projective instructions under reduced cue conditions). Battro et al. found their size-constancy based model better fit the parallel alley data than Luneburg’s hyperbolic model.
A careful look at the instructions used to generate distance and parallel alleys shows they may not be exactly of the same type. For example, Indow and Watanabe (1984a) gave the clearest description of their instructions in their methods section of any study I’ve found. Here is what they told subjects:

40

CHAPT ER 3

Please keep in your mind the following points.

(1) In the case of the parallel series, you are not requested to place two series to be physically parallel like railway tracks. Perceptually they appear to converge at a certain distance because they are physically parallel. The series you construct have to be perceptually parallel. You need not care about what are physical positions of light points. Rely solely on your perceptual impression...

(2) In the case of equi-distance series, you have to equate lateral distances you perceive between two lights on the left and on the right. Farther pairs may look inside of nearer pairs though all pairs look equally separated, i.e., being spanned by the same invisible string if the string is moved back and forth between pairs appearing at different distances from you. (pp. 149, 151)

It would seem from the above description that great emphasis is placed on judgments not reflecting physical reality in the parallel alley instructions. They seem to fall somewhere between apparent and projective instructions. The distance alley instructions put much less emphasis on judgments not reflecting reality. In fact, the reference to the invisible string is similar to asking subjects to make their judgments according to a ruler. These instructions appear to fall closer to the objective type. Baird (1970) and French (1987) find that this difference in instructions is common for alley experiments. If true, this would account for parallel alleys being inside distance alleys as explained before.
Higashiyama, Ishikawa, and Tanaka (1990) directly tested the effects of instruction type on alley settings. They found that objective instructions consistently lead to straighter alleys that lie on the outside of alleys generated from apparent instructions. If the same type of instructions were used, parallel and distance alleys did not differ from each other under most conditions. Objective alleys were outside of apparent alleys, but parallel alleys did not differ from distance alleys.
Indeed, the classic Blumenfeld alley experiment has probably been misinterpreted. The defining feature of a hyperbolic geometry is the restatement of the Parallel Postulate in the form: through a point P not on a line L, there is more than one parallel to L. The Blumenfeld alleys might support the hyperbolic model if more than one parallel existed using the same instructions. As a matter of fact, only one parallel is produced for each instruction set. The reason different parallels exist across different instructions is that observers are performing different tasks. As evidence for this, I cite a phenomenon mentioned by Luneburg (1948) himself. Observers report that “parallel alleys” do indeed seem parallel; however, “distance alleys” do not seem parallel. In fact, distance alleys do not even appear to be composed of straight lines.
With this in mind a second alternative explanation of the alley results can be offered that does not rely on objective vs. apparent instruction effects: The distance alley and parallel alley data may result from very separate mechanisms. Let’s consider how each data set might arise in turn.

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 41
(1) Distance alleys: Distance alley instructions ask to observer to maintain a constant distance between pairs of luminous points. According to Holway and Boring (1941) frontal judgments under reduced-cue settings (such as being in the dark) tend to be nearly proportional to the visual angle of the stimulus. (Presumably this is because the visual angle is the only information that the observer has to determine size.) If observers are asked to adjust stimuli to be the same perceived size, then actual size must systematically increase with increasing distance in order to maintain the same visual angle. The function that describes the relationship between visual angle and distance from the observer will be discussed in Chapter 6. For now it is enough to say that this function is exponential like (it looks a bit like an exponential equation, but is not) and would produce a curved alley just as in the Blumenfeld experiment.
(2) Parallel alleys: Wagner (1982, 1985) and Wagner and Feldman (1990) have shown that angle perception is powerfully influenced by the orientation of an angle with respect to the observer. Angles facing either directly toward the observer or directly away from the observer are perceptually expanded while those viewed from the side are perceptually compressed. That is, when an angle is seen on its side, a line from the observer can cut through both legs of the angle. For an angle seen facing directly toward or away from the observer, the opposite would be true. A line from the observe could only cut through one leg of the angle.
As shown in Figure 3.2a, asking observers to make parallel alleys is equivalent to asking them to make the angle defined by a given pair of points and one more distant one equal to 90˚, a right angle. This angle is seen on its side. As such it should be seen as perceptually contracted. In order to achieve an angle that is perceptually equal to 90˚, the physical angle must be expanded slightly. If this expansion is applied to each pair of points (successively working inward) a curve like that seen in the parallel alleys will result. Because this curve arises from a separate mechanism than the distance alley, we would not necessarily expect the two curves generated to coincide.
By the way, this logic can generate an interesting prediction that (to my knowledge) no one has ever tested before. Let us say that instead of asking observers to make two straight lines, they are asked to create two outward bending arcs as seen in figure 3.2b (perhaps by showing the curve to the subjects on a card presented frontally near the observer). This time the angle (Ø1) will be seen frontally. This angle should seem perceptually expanded compared to physical reality. To adjust, the observer will need to set the points so that they curve less physically than the curve they were asked to produce. This is precisely the opposite of what occurred with the parallel alleys, and different from what a hyperbolic geometry would predict. (Note that the second curve is not really necessary.)

42

CHAPT ER 3

Figure 3.2. (a) The visual display observers are trying to produce in the parallel alley experiment. (b) The visual display observers are trying to produce in the experiment proposed in the text. (Observers are trying to produce two bending arcs.)
If we just look at the right hand curve in Figure 3.2b, we see that Ø2 is seen on its side. As such, it should seem perceptually contracted. To compensate, the observer will need to expand this angle in order to generate the curve they were asked to produce.

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 43
Although this prediction has not been tested empirically, I have confirmed it to my own satisfaction introspectively. When driving along the interstate, I often notice that moderately sharp curves seem sharper at a distance than when I am on the curve and thus seeing it in a more frontal orientation.
A few final comments on Luneburg’s theory. You might be surprised, given the foregoing analysis, to hear me express my admiration for Luneburg’s theory. His theory is eloquent and sophisticated. It represents a paradigm for model building. Given the right conditions, his theory may also represent a reasonable description of visual space. Yet, like the spherical model of Reid (1764/1813), Angell (1974), Daniels (1974), and French (1987), Luneburg’s model probably only works well under a fairly narrow set of conditions.
The Luneburg theory applies to stimuli that lie on the horizontal plane that passes through the eyes. Indow and Watanabe (1984a, 1988) show that the curvature of the fronto-parallel plane does not differ significantly from the Euclidean value of zero. It is only when stimuli are arrayed in depth that the hyperbolic model applies. Secondly, most of this work is done in the dark with luminous points under controlled conditions that eliminate all other cues to depth outside of binocular disparity and vergence. Such cues are limited in scope—primarily useful within two meters from the observer (Baird, 1970). As such, the most dramatic departures from what we might expect compared to a Euclidean model are for stimuli that are close at hand. Indow (1974) and Indow, Inoue, and Matsushima (1963) found that certain predictions of the hyperbolic model were somewhat “disappointing” when applied to a large-scale spacious field. In addition, the curvature of visual space seems to be attenuated under full-cue conditions, like a well lit field with plenty of monocular cues to depth (Battro et al., 1976; Hardy, Rand, & Rittler, 1951; Koenderink et al., 2002). While other experimenters sometimes find a general tendency toward negative curvature even under these conditions (Indow & Watanabe, 1984; Higashiyama, Ishikawa, & Tanaka, 1990), even these later researchers did not find negative curvature with all of their subjects under illuminated conditions, arguing for at least some attenuation in the effect.
In summary, within its restricted domain, Luneburg’s theory holds a place as one of the geometries of visual space. However, I do not believe that it is the only geometry that applies. As with the spherical model of the phenomenologists, the geometry of visual space varies with observer attitude and stimulus conditions.
Visual Space as a Euclidean Geometry
Why list Euclidean geometry second? You may wonder at why I chose to start this chapter on synthetic approaches to visual space by discussing the hyperbolic model instead of beginning with the traditional Euclidean approach. The reason for this organizational tack is that relatively few perception researchers have explicitly supported the Euclidean model for visual space. Compared to the hyperbolic model that has attracted some of the top minds in mathematical psychology, the Euclidean model seems to be something of a unwanted step child; per-

44

CHAPT ER 3

haps, because there is little excitement attached to proposing the traditional view.
Having said this, Euclidean geometry is not without its supporters. Fry (1950) constructed a synthetic model that describes visual space as Euclidean. Like Luneburg, Fry proposed a set of axioms that he thought were self-evident facts. For example, Fry asserts that physically straight lines are seen as straight, physically right angles are seen as right angles; physically parallel lines are seen as parallel. Such observations, if true, would indicate that visual space is more or less equivalent to physical space and this would support a Euclidean model (if one assumes that physical space is Euclidean). His primary empirical evidence to back up the claim that visual space is Euclidean is based on his observation that subjects (with fixation held constant) can reliably arrange eight points to form a square that has straight lines, right angle corners, and sides of equal length. The constructed object was said to not only be physically square, but it looked physically square to the observer. If true, this observation would be strong evidence for the Euclidean character of visual space. Unfortunately, Fry did not give any details for this experiment, and to my knowledge, no one has replicated his findings. Fry dismissed Blumenfeld and Helmholtz’s work as being artifacts of allowing free eye movement and the distortions that occur in visual perception due to asymmetrical convergence angles in the two eyes. Fry (1952) argued that some of the theoretical and empirical work on size constancy designed to support Luneburg’s theory (i.e., Gilinsky, 1951) actually are better explained via a Euclidean geometry.
A model for visual space that assumes constant fixation and symmetrical convergence angles would seem to be even more restrictive than the assumptions of Luneburg’s model which does allow free eye movement (but no monocular cues to depth). In addition, Fry’s model needs more empirical support.
More fundamentally, proponents of a Euclidean model for visual space face many of the same objections as discussed for the hyperbolic model. The Euclidean model also assumes that visual space is a metric space in which distance perceptions should display symmetry (the distance from point A to point B is the same as the distance from B to A) and the triangle inequality should hold. The Euclidean model also assumes that visual space is convex, compact, and Desarguesian. The Euclidean model requires free mobility to hold. All of these assumptions are necessary for visual space to be a Riemannian geometry of constant curvature of which Euclidean geometry is a special case. All of the empirical work that attacks these assumptions not only casts into doubt Luneburg’s hyperbolic model, but the Euclidean model of visual space as well.
Higashiyama (1981, 1984) and Ivry and Cohen’s (1987) determination that the curvature of visual space is not constant, but ranges between negative and positive values, is no more consistent with an Euclidean model than it is with a hyperbolic model because both require a constant curvature. Nor is the Euclidean model consistent with Moar and Bower (1983) who found that subjects tended to perceive the angles of a triangular structure to be greater than 180˚.

Modern Euclidean philosophers. A number of modern philosophers take a tack similar to Kant (1781/1923) to arrive at the conclusion that visual space is

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 45
Euclidean. For example, Ewing (1974) and Strawson (1976) believe that although we are able to conceive of non-Euclidean geometries in the abstract, the Euclidean conception of space is perceptually necessary in order to experience the world at all. Unlike the empirically derived observations we make about our world which often have exceptions, the structure of our phenomenal world is universal. For example, we cannot experience the world, except in terms of three dimensions, try as we might.
Ewing and Strawson do not distinguish between the physical space and visual space because both are aspects of phenomenal space. To them, the observations of physics are as much a part of this phenomenal space as our everyday experiences as individuals. All of these observations, of the scientist and of the average individual, reflect a coherent conception of space, not a randomly selected set of propositions that vary across time and space. The evidence from physical science is that visual space is Euclidean, at least at the terrestrial level. Where can this coherence arise from if it is not built into the universe or imposed as an organizing structure of the human mind?
One can make the same reply to these modern philosophers that Helmholtz (1869/1921) made to Kant. While it is certainly possible that Euclidean conceptions are built into our perceptual apparatus as part of the act of experiencing the world, this model can be subjected to empirical test. The axioms and metric relations found in perception can be tested through the judgments people make. It is also possible to empirically compare the judgments people make based on their perceptions to the measurement made by physical scientists to determine if indeed the phenomenal world is a coherent whole in which our perceptions match reality. The conception that our perceptions of space are universal can also be tested; in other words, we can determine if our perceptions of the world change under varying conditions and shifts in mental attitude. In truth, the data is in and few of these propositions hold. As we will see in subsequent chapters, visual space is clearly different from physical space and its nature depends on conditions. However logical the argument in favor of the necessity of Euclidean space, if the data contradict it, the theory of Euclidean necessity must be rejected.
J. J. Gibson’s model. Gibson (1950, 1959, 1966) also argues that visual space is Euclidean, although he comes to this conclusion from a very different direction. Gibson begins his development by attacking conventional views of space and vision. Gibson believes that visual space is neither an abstract structure as in mathematics nor a structure created by the human mind to organize experience as in Kant. To Gibson, the traditional viewpoint implies that the perception of space is totally divorced from the real world. Gibson feels that these abstract structures are empty vessels that evaporate into nothingness unless stimulation is presented to the observer. Rather than objects fitting into an abstract space, spatial perception arises from our perception of surfaces whose structure already exists in the real world and whose layout is specified by invariant patterns in stimulation (Cutting, 1993; Gibson, 1979; Turvey & Carello, 1986). The invariant patterns that specify the location of objects include occlusion, texture gradients, flow gradients, and motion parallax.

46

CHAPT ER 3

Furthermore, Gibson is a Naive Realist (Henle, 1974; Gibson, 1959, 1979). That is, he believes that the perceptual systems of people and animals have evolved to allow us to perceive the world veridically. In other words, our perceptions closely match physical reality under ordinary circumstances. If our perception of the world did not match reality, then we would be locking ourselves into a world of subjectivity. We would be making the error of “concluding that we can know nothing but our perceptions…. Once having made this argument, a theorist is trapped in a circle of subjectivism and is diverted into futile speculations about private worlds” (Gibson, 1959, pp. 462-463). In addition, if our perceptions did not match reality, then we would constantly be making perceptual errors that would handicap us in the struggle for survival.
Thus, Gibson submits that humans are capable of almost perfect perceptual constancy; that is, humans perceive the metric attributes of the world correctly under virtually all ordinary circumstances. Because an almost perfect relationship exists between physical space and visual space, and because physical space is Euclidean, visual space must also be Euclidean.
Ironically, Gibson’s assumption of perceptual constancy allies him with Luneburg, in that Luneburg’s free mobility axiom was motivated by this assumption of constancy. Gibson (1959) referred to his own early empirical work to bolster his claim of perfect constancy. Gibson (1933, 1947, 1950) had subjects estimate the size of objects at various distances from the subject under information rich natural conditions and in a cluttered office space, and he found that judgments matched physical reality to a high degree.
Of course, the problem here is that other researchers do not find perfect constancy. As summarized in Baird (1970), frontally oriented objects tend to exhibit overconstancy under full-cue conditions like Gibson employed; that is, distant objects tend to be seen as larger than near objects of the same physical size. Wagner, Kartzinel, and Baird (1988) found that objects lying on the ground oriented in-depth tend to exhibit strong underconstancy; that is, distant objects tend to be seen as smaller than near objects. Many other studies have also found a lack of perceptual constancy under a variety of conditions. Outside of constancy studies, illusions also represent occasions when constancy breaks down.
Gibson was aware that such contrary data existed. Yet, he swept this dat a aside as irrelevant. Gibson (1979) referred to data collected under controlled conditions as “aperture vision” or “bite-board vision” as opposed to the natural vision relevant to human and animal adaptation to the world in which he was interested. Gibson (1977) felt that perceptual error only arises under two conditions: when the perceptual system breaks down such as with eye injury or when the information necessary for accurate perception is denied to the perceiver. Thus, Gibson was convinced that laboratory studies controlled the phenomena of interest out of existence because they failed to provide the information necessary for veridical perception. In particular, Gibson stressed the importance of exploration and motion to provide the information necessary for veridical perception.
Following Gibson’s death, ecological psychologists have followed up on his theory along a number of different lines. Some, like Feldman (1985) and Turvey and Carello (1986), have attempted to quantify the invariant patterns in stimulation that specify location and guide motion. Others have shown the importance

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 47
of motion to perceiving perceptual constancy (Clocksin, 1980; Johansson, 1986; Johansson, von Hofsten, & Jansson, 1980). Others still have emphasized that perception is most accurate in information rich environments that provide plenty of redundant cues to depth (Bruno & Cutting, 1988). For example, Runeson (1988) explained how the environment normally provides enough redundant information to specify spatial layout even under static viewing conditions, and the distorted perceptions reported for the Ames’ Room are generated by eliminating much of this information.
Other researchers have presented evidence opposing the ecological viewpoint. Gehringer and Engel (1986) found that much of the Ames’ room illusion remained even after observers were allowed unrestricted head movement and binocular viewing, indicating that rich information conditions do not always produce perceptual constancy. Domini and Braunstein (1998) found that threedimensional layout produced by motion does not yeild perceptions with a Euclidean structure. Similarly, Loomis and Beall (1998) found that optic flow does not fully explain control of locomotion, and that other more cognitive information is necessary to guide action.
While I greatly admire the Gibsonian perspective (as any of my students would quickly attest to), I feel that it has three essential weaknesses in the present context. First, the assumption of perfect perceptual constancy, which lies at the heart of Gibson’s doctrine of Naive Realism, should not be taken on faith. This assumption can and has been tested empirically. Even under the most information rich conditions, perceptions do not always match physical reality as we will see in subsequent chapters. Second, cognitive factors such as the meaning the observer attaches to the concepts of size and distance will influence our judgments. Gibson too quickly derides the importance of such factors when he argues that perception is direct. Third, Handel (1988) points out that although perception can be thought of as exploration, we are often not explicitly interested in layout; so, we don’t always move around to explore the environment. Even under these static conditions, we still have an impression of the layout of a scene. Thus, static viewing is really no less natural than dynamic viewing. Static viewing conditions are a part of our ordinary life experience. Similarly, I would add that we experience information poor viewing conditions every day when the lights go out at night. We should not so narrowly define what is natural enough to be studied. The most complete understanding of visual space arises from looking at how the geometry of the space changes with conditions. Gibson’s desire to limit the domain of perceptual research would leave us with an incomplete understanding at best.
Crypto-Euclidean spaces. Although few theorists go out of their way to positively assert that visual space is Euclidean, the Euclidean perspective may still be the dominant perspective in space perception. This oxymoronic statement is possible because the Euclidean position pervades much of the theoretical work in perception without the assumption ever being acknowledged. For example, the trigonometry underlying the Size-Distance Invariance Hypothesis is Euclidean trigonometry. Many of the classic explanations for the moon illusion including the flattened-dome theory assume a Euclidean space in the process of their devel-

48

CHAPT ER 3

opment. Much of the logic behind the classical explanations for why cues to depth work have an Euclidean assumption at their root.
To give one example, Gogel has often used a head motion measure for perceived distance (Gogel, 1990, 1993, 1998; Gogel, Loomis, & Sharkey, 1985; Gogel & Tietz, 1973, 1980). This measure is based on the fact that the height of a triangle (perceived distance) can be calculated by knowing the length of the base of a triangle (the degree of head movement) and the angles that form the triangle (directions to fixated object) based on trigonometry. However, using this trigonometry presupposes that the space is Euclidean.
There is nothing wrong with assuming that visual space is Euclidean as part of the process of model building. This assumption should be made explicit however. In many cases, Euclidean mathematics is taken for granted without being acknowledged. In other words, Euclidean assumptions slip into the model building process, to give us a sort of crypto-Euclidean space.

Other Geometries Applied to Visual Space

Other geometries have been proposed to describe visual space. In most cases, these models are variations on the hyperbolic or Euclidean models. For example, Hoffman (1966) suggested that visual space displays the properties of a Lie transformation group. Hoffman uses his theory to explain size and shape constancy, motion perception, and rotational perception. Hoffman and Dodwell (1985) extend this theory to account for some of the Gestalt properties of visual perception. In the case of static perception, Hoffman’s model reduces to something similar to Luneburg’s theory of perception. I am not familiar with any independent empirical work that followed up on this theory to test its assumptions and predictions.
Drösler (1979, 1988, 1995) generalizes Luneburg’s theory. Drösler (1979) described visual space as a Cayley-Klein geometry. His more recent work attempts to tie space perception to more fundamental psychophysical invariance relations. Drösler (1988) assumes that the free mobility axiom holds, while Drösler (1995) attempts to tie space and color perception into a generalized version of Weber’s Law. In all cases, the metric of visual space is thought to be a variation on Luneburg’s model.
Yamazaki (1987) revisited the data used to support Luneburg’s theory. Yamazaki explained this data without assuming that visual space is Riemannian by thinking of visual space as being composed of a set of connected affine spaces. In this space, curvature need not be constant because visual space may be stretched slightly in one direction in a given location and stretched in a different direction in another location in the space. One consequence of this structure is that a line element that travels a complete circuit through these connected spaces need not end up in the same perceived location. Yamazaki applied this model to explain Blumenfeld’s visual alley data.
Dzhafarov and Colonius (1999) generalize the Fechnerian integration (that we will talk about in Chapter 5) to describe space perception. According to this approach, stimuli are associated with psychometric functions that determine the probability that they will be discriminated from other stimuli. This function is

SYNTHETIC APPROACHES TO VISUAL SPACE PERCEPTION 49
assumed to vary smoothly from one stimulus to adjacent stimuli and to be defined on an infitesimal level. The metric is determined by integrating along a path between two stimuli and using the minimum distance to define the metric. Of course, the assumptions of compactness and the ability to express JND’s as differentials are open to question, but the approach does integrate information from different psychophysical domains.
One may also doubt about whether Dzhafarov and Colonius’s approach really is a legitimate generalization of Fechner’s work. Dzhafarov and Colonius use a concept of similarity that Fechner avoided. See Link (1994) for a discussion of Fechner’s original approach.
Final Comments on the Synthetic Approach
Thus, a virtual plethora of geometries have been proposed for visual space. Given the appropriate axioms, each approach is internally consistent and undeniable. Unfortunately, the various proposed geometries are not mutually compatible. Visual space cannot be doubly elliptical, hyperbolic, Euclidean, a Lie group, and a non-Riemannian affinely connected space all at the same time. Perhaps one geometry may hold under one set of conditions and another may hold under another set of conditions, and we must accept that there is no single geometry that works under all conditions.
Yet, the difficulties for the synthetic approach run deeper than this. The various critical studies, such as the Blumenfeld alleys, that are meant to help us choose between models can be reinterpreted to support other geometrical formulations (c.f., Fry, 1952, Hoffman, 1966). It is not clear that a critical test exists that can differentiate between the models.
In addition, the validity of each synthetic model rests on the veracity of the axioms on which the theory is based. There are a host of studies reported earlier in this chapter that call into question the most fundamental of these axioms. It is not clear that any synthetic geometry except perhaps the most global topological systems (which would be able to make only the vaguest of predictions) can pass this rigorous test. In many ways, the synthetic approach to visual perception has failed.
I believe that an alternative approach is possible. Rather than indirectly attempt to specify the geometry of visual space through postulates and critical experiments, it may be possible to take a more direct approach. Rather than attempt to define the geometry of visual space synthetically, one can take an analytic approach. In this approach, observers are asked to judge the metric properties of visual space directly using the methods of psychophysics. In turn, these judgments can serve as the basis for deriving functions that relate the observer’s judgments to physical coordinates as a function of experimental conditions. These functions can be used to directly specify the metric of visual space, and the geometry of the space can be defined in terms of this metric. The next chapter will begin to develop this alternative.

4
An Analytic Approach to Space and Vision

In mathematics, there are two ways to approach the geometry of a space: synthetic geometry and analytic geometry. The previous chapter considered attempts to define visual space synthetically; that is, by listing a set of postulates meant to describe visual space and determining what geometry best fits the proposed postulates. While the synthetic approach has successfully accounted for a small set of classic experiments, none of the models presented can account for the effects of stimulus conditions and observer attitudes that are found in the literature. In addition, research does not appear to support the foundational axioms of the synthetic models.
I would like to propose a more direct way to define visual space by using the tools of analytic geometry. To apply this approach to visual space one must first assign coordinates to locations in visual space. Secondly, one must seek out equations based on these coordinates that describe our perceptions of distance and other metric properties such as angles, areas, volumes, etc.

Advantages of the Analytic Approach

Using an analytic approach has a number of advantages. First of all, the analytic approach is more general. While the synthetic approach has largely focused on the three geometries of constant curvature (hyperbolic, Euclidean, and spherical geometry), the analytic approach is under no similar restraint. An infinite variety of coordinate equations are potentially available to describe metric relationships. As an example of this flexibility, consider the Minkowski metric that we discussed in Chapter 2. According to this equation, distance between points is defined as

d(P1,P2) = [x1-x2 R + y1-y2 R]1/R

(4.1)

where the distance between points P1 and P2, d(P1,P2), is a function of the coordinates of the two points, (x1, y1) and (x2, y2), and the Minkowski parameter R. This single equation expresses an infinite number of geometries. If R is equal to 2, this metric equation specifies a Euclidean space. If R is equal to 1, the

50

AN ANALYTIC APPROACH TO SPACE AND VISION

51

equation specifies a city-block space. Yet, these are only two of an infinite number of values that the Minkowski parameter can assume.
As flexible as the Minkowski metric is, it is only one of an infinite number of possible metric equations that can be defined on a set of coordinates. Clearly, the analytic approach to defining a space is both general and powerful.
Second, the analytic approach is more direct than the synthetic approach and is more applicable to modeling the psychophysical literature as a whole. The synthetic approach relies on a small number of critical experiments that attempt to test the axioms of the synthetic models in order to specify the geometry of visual space. The vast majority of studies on space perception, however, are not directed at testing these axioms, but instead ask observers to judge various metric properties (particularly size and distance) as a function of conditions and instructions. From a synthetic viewpoint, these studies are essentially irrelevant.
The analytic approach, however, easily incorporates this corpus of the space perception literature. Indeed, according to the analytic approach, the judgments that observers give for size and distance directly define visual space. The theorist’s goal is to find coordinate equations that predict the judgments that observers generate. Thus, instead of being irrelevant, psychophysical studies of space perception provide the basic data that specify the geometry of visual space.
Third, the synthetic approach cannot easily incorporate the effects of stimulus conditions, stimulus layout, judgment methods, and instructions. As such, synthetic theorists tend to carefully limit the domain to which their theories apply. Luneburg (1947) and the other advocates for the hyperbolic model, for instance, make it clear that the theory applies only to binocular space perception in the dark with luminous points of light for stimuli and a stationary head position. Although it usually not explicitly stated, the modeling is also limited to dat a collected through various methods of adjustment, and numeric estimation methods are excluded. Similarly, Foley (1980) explicitly limits himself to binocular viewing conditions under which no other cues for depth are allowed except eye position and binocular disparity. He also rules out the use of verbal reports. This tendency to limit the domain to which a theory applies and to exclude large bodies of data seems to universally describe synthetic models for space perception. In fact, most of these models explicitly exclude from consideration the natural viewing conditions typically found in every day life.
The analytic approach does not need to be so exclusive. In fact, variations in method and conditions can be incorporated into the metric equations that define visual space. When quantifiable, these conditions can be included as parameters in the coordinate equations that predict the size and distance judgments people generate. Some conditions are not directly quantifiable, such as using different sorts of judgment methods or varying judgment instructions. In this case, one could develop separate metric equations for each condition to specify how the geometry of visual space changes from one condition to another or use “dummy coding” to incorporate the affects of these qualitative dimensions within a single equation. In short, the analytic approach need not exclude data, but can incorporate changes in stimulus conditions and experimental methods into the definition of the geometry of visual space.

52

CHAPT ER 4

Indeed, one should not be too quick to exclude data generated from any source. In general, most psychologists would feel more confidence in any conclusion if converging measures for the same concept yield similar results. Certainly, this also applies to space perception. When only a single method is applied, it is impossible to know whether the resulting function is due to perception or due to response processes inherent in the method. Both numeric estimation and adjustment methods are influenced by various response factors (Ono, Wagner, & Ono, 1995; Wagner, 1989; Wagner, Ono, & Ono, 1995). If several alternative psychophysical techniques are used to probe an observer’s perceptions, then commonalties resulting from these converging measures should represent aspects of visual space that are largely independent of the methods employed. Using a large number of methods also allows for a comparison of the methods themselves. In the analytic approach, judgment method can be thought of as another parameter to take into account when modeling spatial judgments.
Finally, before moving on, I must admit that the distinction I have drawn between the synthetic and analytic approaches to space perception is not as cut and dry as I may have made it seem. Most synthetic theorists do attempt to derive metric functions to describe distance perception. Similarly, the analytic approach has synthetic axioms buried implicitly within it. For example, the geometries of constant curvature can be approached analytically because their metric functions are well known. In fact, one can think of these synthetic geometries as simply being special cases of the more general analytic approach. The difference between the synthetic and analytic approaches really lies in a difference in emphasis. The synthetic approach to visual perception starts by listing the axioms of a limited set of synthetic geometries and attempts to validate those axioms and chose between the geometries considered. Metric functions are derived as an after thought. The analytic approach begins with metric judgments and attempts to model them directly, spending little thought on whether fundamental axioms hold.

Analytic Geometry and Visual Space

To implement the analytic approach to space perception, two questions must be answered. (1) How will the locations of stimulus points be specified? That is, what coordinates are most appropriate for visual space? (2) What function relates these coordinates to the judgments observers give for the metric properties of visual space? Let us deal with each of these questions in turn.

The coordinates of visual space. How should the locations of stimulus points be specified? In general, the location of a point in a plane may be specified by two coordinates, and the location of a point in three dimensions may be specified by three coordinates. Similarly, the locations of two points involve four coordinates in a plane and six coordinates in three dimensions. The exact nature of these coordinates is somewhat arbitrary. Points could be located by sets of Cartesian coordinates, (x1, y1), (x2, y2), etc. They could be located by sets of polar

AN ANALYTIC APPROACH TO SPACE AND VISION

53

coordinates, (R1, 1), (R2, 2), etc. They could also be located in many other ways.
I believe the way one specifies point locations should ultimately depend on four criteria: (1) The method must locate points using a disjoint set of coordinates. That is, the coordinate system must completely specify point locations while at the same time not over specify the points by including the same information more than once. (2) The method must be reasonably simple. The coordinate dimensions must be easy to understand and interpret. (3) Because one cannot directly examine conscious experience to determine coordinates, the only way to assign coordinates is by basing them on the physical position of stimuli. Some might object to this, because the goal is to describe visual space, not physical space, and it would seem more appropriate to assign coordinates to visual space directly. Unfortunately, one could never be certain that the set of coordinates assigned based on one person’s introspection are equivalent to the set derived from another’s. To operationalize our variables in a way that allows for scientific investigation requires us to tie the observer’s judgments to something concrete, such as physical position. In truth, little is lost here because the assignment of coordinates is somewhat arbitrary, and the differences between physical layout and experienced layout will find their expression in the metric functions that we derive from the observer’s judgments. (4) The method should be ecologically valid. That is, the method should naturally relate the person making the judgment to the points being judged. In so doing, an attempt should be made to specify point locations in a manner that a person might normally use to specify them. A coordinate system designed with this ecological criterion in mind is likely to be both easy to interpret and sensitive to systematic trends which might exist.
The classic Cartesian coordinate system certainly passes the first three of these tests, but I feel that it fails the fourth criterion. For example, if a person was facing north and looking at a point one meter away to the northeast, it is unlikely that the observer would conceptualize the point as being located 2/2 to the right and 2/2 forward. The polar coordinate system is a more natural way to locate a single point because people do think in terms of how far away something is and how much something deviates from straight ahead.
Another coordinate system that has been used extensively by Luneburg and others (Aczél et al., 1999; Blank, 1953, 1958, 1959; Foley, 1980, 1985; Heller, 1997a, 1997b; Indow, 1974, 1979, 1982, 1990; Luneburg, 1947, 1948) is bipolar coordinates. As shown in Figure 4.1, if the two eyes are fixated on a point in space, the bipolar coordinates ( , ) are determined by the angular deviation from straight ahead from the center of the two eyes.
This coordinate system makes if easy to work with the Vieth-Müller circles which in the idealized case represent those stimuli which fall on corresponding points of the two retina when one point on the circle is fixated; that is, the two eyes would need to change direction by the same amount to fixate on another point of the circle. To Luneburg, points lying along the same Vieth-Müller circle are perceived as being equidistant from the observer. Similarly, this coordinate system makes it easy to describe the hyperbolae of Hillebrand that form the

54

CHAPT ER 4

parallel lines in Luneburg’s hyperbolic space. If the observer fixates on a point along an Hillebrand hyperbola, other points along the hyperbola are projected onto retinal points that deviate the same angular extent from the fovea in both eyes but in opposite directions.
To Luneburg, the points along a Hyperbola of Hillebrand are perceived to lie in the same visual direction from the observer. Because both the Vieth-Müller circle and the hyperbola of Hillebrand can be described most simply in such angular terms, the bipolar coordinate system is probably the simplest and most elegant way to express them.

Figure 4.1. Bipolar coordinates for a point in space in terms of angular deviation from straight ahead for the right eye ( ) and the left eye ( ).

AN ANALYTIC APPROACH TO SPACE AND VISION

55

Unfortunately, when one leaves the world of Vieth-Müller circles and hyperbolae of Hillebrand and one attempts instead to model the size and distance judgments found in the bulk of the space perception literature, the bipolar coordinate system is awkward to use at best. Using the bipolar coordinate system also implies a number of implicit assumptions about the nature of space perception that I find questionable. First of all, most of the theorists who use this coordinate system act as though the only cues to depth that matter are binocular ones like disparity and convergence angle. In truth, there are a host of monocular cues to depth perception that also influence spatial perception (such as texture gradients, linear perspective, motion parallax, etc.), which have no place in a bipolar system. In addition, under most circumstances, our perception of space is a unitary phenomenon. We seem to be looking on the world from one place, one origin, not from two places at once. The origin of visual space is not in the eyes, but it is the mind, and it is located at a singular egocenter. Thus, in terms of the criterion mentioned earlier, bipolar coordinates pass the tests of being disjoint and defined physically, but fail the tests of simplicity and naturalness.

Implicit natural coordinates. A fourth coordinate system implicitly underlies the bulk of the space perception literature. In truth, most space perception studies don’t bother with specifying a coordinate system at all, but define stimuli in terms of egocentric and exocentric distance, stimulus orientation, and eccentricity (deviation from straight ahead of the observer). The following is an attempt to explicitly layout the coordinate system implicitly employed by the majority of space perception research. This “Natural Coordinate System” is not as simple as the previous systems, but it corresponds more closely to our ordinary, commonlanguage conceptions of space.
For the location of a single point, as in egocentric distance estimation tasks, the “natural” way to assign coordinates is with a polar coordinate system. Here the two coordinates are R and . To make this definition more concrete, let us define the radius (R) as the straight-line Euclidean distance from the observer to the point, and the polar angle ( ) as the counter-clockwise angle measured between an arbitrarily defined axis and the direction to the point. In my formulation, the arbitrarily defined axis extends from the observer directly to his or her right. Thus, a point directly to the observer’s right is at 0˚, a point located di rectly in front of the observer is at 90˚, and a point directly to the observer’s left is at 180˚. I use this convention to avoid needing to talk about negative angles or constantly having to specify on which side of the observer each stimulus lies, although I recognize that it might be even more natural to think of straight ahead as being 0˚, and other angles could be thought of as being so many degrees to the right and so many to the left. While most experiments present stimuli along a plane defined by the ground or a table, one could extend this coordinate system by including a third coordinate that specifies the elevation of the stimulus. This coordinate would also be expressed as a polar angle relative to some arbitrarily defined axis.
For size and exocentric-distance estimation, four Natural Coordinates are needed to capture the location of the two points that define the object being judged. Figure 4.2 displays these coordinates. Here, the four coordinates are: the

56

CHAPT ER 4

distance from the observer to the nearest point (R), the polar angle or the direction the nearest point lays relative to an arbitrarily defined axis to the observer’s right ( ), the orientation of the points with respect to the observer’s frontal plane ( ), and the Euclidean distance between the two points (D). This coordinate system for distance judgments can be easily extended to three-dimensional space by including polar and orientation coordinates that express elevation.
The coordinate system thus defined is disjoint. The location of the near point is defined by R and ; while and D define the location of the far point (with respect to the near one). The inclusion of D as one of the coordinates is particularly noteworthy. In this way, inter-point distance is factored out of the other coordinate dimensions. When determining judged distance as a function of actual distance and other factors about the location of the stimulus (as well as other non-spatial factors), factoring out distance prevents recursive effects of including

Observer
Figure 4.2. Natural coordinate system used to specify object locations for distance estimation. The four coordinates are: the distance from the observer to the nearest point (R), the polar angle or the direction the nearest point lays relative to an arbitrarily defined axis to the observer’s right ( ), the orientation of the points with respect to the observer’s frontal plane ( ), and the Euclidean distance between the two points (D).

AN ANALYTIC APPROACH TO SPACE AND VISION

57

distance twice in the formula, once as the distance itself and once hidden in the coordinates. It also allows one to directly address the effects of stimulus orientation, which turns out to be a critical factor in size judgments. Typically, stimuli are either presented frontally or in-depth relative to the observer (although seldom both at once). The natural coordinate system allows this experimental factor to be directly represented in a model.
This coordinate system is also easy to interpret, and it is defined in terms of the objectively observable physical layout of the stimuli. In addition, the coordinate system is relatively ecologically valid. The coordinate dimensions are defined with respect to the observer much as a naive observer might describe them. That is, an object could be described as far away (or near), to the right (or left), see straight on (or in-depth), and is so long. (At least, the specification is more ecologically valid than the obvious alternatives.)
On the other hand, Natural Coordinates fail the test of simplicity. Instead of each point being defined by two equivalent coordinates as in other systems, the four coordinates describe different aspects of the stimulus constellation as a whole. Despite this complexity, this coordinate system more directly relates to the size and distance estimation literature than other systems. Egocentric distance judgments involve estimating R; exocentric distance judgments (or size judgments) involve estimating D; and corresponds to variations in stimulus orientation found in different experiments.
One may also use a similar Natural Coordinate System to describe angle judgments. The smallest number of points that may define an angle is three. One point specifies the vertex, while two other points lay along the two legs. Because three points are involved a total of six coordinates will be needed to completely specify the position of the points. As seen in Figure 4.3, the six coordinates that define an angle are: the distance from the observer to the vertex of the angle (R), the polar angle between the vertex and an arbitrarily defined axis ( ), the orientation of the angle relative to the observers frontal plane ( ) where orientation is defined by the vector that bisects the angle, the physical size of the angle (A), and (to be complete) the length of the two legs of the angle (D1 and D2). Like the Natural Coordinate System for distance judgment, this one has the advantages of being disjoint, physically defined, and ecologically valid. It is also defined in terms of important experimental variables in a fashion that will make modeling easier. In this book, I will tend to prefer Cartesian coordinates when modeling metric functions for visual space as a whole, but Natural Coordinates will be used when reviewing the direct estimation and size constancy literature because this corresponds to the variables researchers typically emphasize.

The origin for visual space. Idhe (1986) points out that spatial localization is reciprocal. Looking out at the world, we are able to localize the position of objects relative to our own position and see each object from a certain point of view. This same information, however, also can serve to localize the point from which the observation occurs. Our point of view, the exact direction from which we look on an object, points back toward the direction that the observer him/herself lies. By a process of triangulation, the position of the self can be unambiguously identified, if the visual direction to more than one object is known. This

58

CHAPT ER 4

location from which observation occurs is known as the egocenter, and I believe that it is the logical origin of any coordinate system for visual space.
Where does the egocenter lie? One simple hypothesis might be that we see the world from the point of view of our dominant eye, and thus the origin of visual space lies there. This does not seem to be the case. Barbeito (1981), Ono (1979), and Ono and Barbieto (1982) have shown that our perception of visual direction does not originate in either eye, but from a point in between that they refer to as the cyclopean eye after Homer’s mythological Cyclopes.

A

Observer
Figure 4.3. Natural coordinate system used to specify object locations for angle estimation. The four coordinates are: the distance from the observer to the nearest point (R), the polar angle or the direction the nearest point lays relative to an arbitrarily defined axis to the observer’s right ( ), the orientation of the bisector of the angle with respect to the observer’s frontal plane ( ), the length of the two legs (D1 and D2), and the physical size of the angle (A).

AN ANALYTIC APPROACH TO SPACE AND VISION

59

If the egocenter is not coincident with either of the eyes, then where does it lie? Four methods have been proposed for locating the egocenter. Funaishi (1926) had subjects fixate a point straight ahead of them and equidistant between the two eyes (the median plane) and judge the direction to non-fixated targets in the same depth plane as the fixation point by pointing toward each target with their hands (which are out of sight). Judgments are made at two different fixation distances with stimuli placed at the same visual directions from the subject in each depth plane. Lines are drawn through corresponding judgments from the two depths and projected back toward the observer. The point of intersection for more than one of these lines is said to define the egocenter. One can criticize this method, because the experimenter has made a determination of what the visual direction is physically and not allowed subjects to make their own determination. In addition, pointing may be a very poor measure because the arm is in a different place than the egocenter and the mechanics of the arm might conceivably influence directional estimates.
Fry (1950) tried a more indirect method. He had subjects fixate the more distant of two stimuli along the median plane (straight ahead from the middle of the observers head). Subjects were asked to point at the locations of the two diplopic images produced by the non-fixated stimulus by pointing with their hands (which were out of sight). He derived the location of the egocenter from this information based on Hering’s (1879/1942) principles of visual direction. Once again, pointing may be a poor estimate of visual direction.
Roelofs (1959) had subjects look with one eye through a tube that is physically pointing toward the fovea of the eye, while the other eye is occluded. Subjects do not see the tube as pointing at the eye, but as pointing at their face in between the two eyes. Subjects indicate the point on the face where the tube appears to point, and a line is defined from the front of the tube through the point on the face. The point where the lines defined by each eye intersect is the egocenter.
Howard and Templeton (1966) developed the most direct method for measuring the egocenter, and one that does not require pointing. Subjects rotate a rod presented at eye level until it appears to be pointing directly at the self. That is, the front and back ends of the rods appear to lie in along the same visual direction. The rod is moved to different locations and the rod is once again rotated until it appears to point at the observer. The place where lines traced along the axis of the rod intersect is the egocenter.
Not only is this later method the most direct (and to me the most intuitively appealing), it also appears to work the best. While Mitson, Ono, and Barbeito (1976) found that all methods where highly reliable, producing consistent egocenter locations both within experimental sessions and between different experimental sessions, Barbeito and Ono (1979) found that the Howard and Templeton method had the highest levels of internal consistency and test-retest reliability. In addition, the Howard and Templeton method had the highest predictive validity for three experimental tasks involving locating the subjective median plane, judging the relative direction of three points, and an accommodative convergence task. The Howard and Templeton method also produced less variable estimates of egocenter location.

60

CHAPT ER 4

So, where is the egocenter located? According to Barbeito and Ono (1979), Funaishi’s (1926) method places the egocenter .28 cm to the right of the median plane and 2.69 cm behind the plane defined by the cornea of the eyes. Fry’s (1950) method places it .28 cm to the right of the median plane and 15.07 cm behind the corneal plane. Roelof’s (1959) method places it .29 cm to the left of the median plane and .99 cm in front of the corneal plane, a seemingly unlikely result. Finally, the Howard and Templeton (1966) method places it .28 cm to the right of the median plane and 1.16 cm behind the corneal plane. Given the greater reliability, predictive validity, and intuitive superiority of the Howard and Templeton method, I believe the later figure represents our best estimate for the location of the egocenter. Most recent research is consistent with the idea that the egocenter is located halfway between the two eyes and slightly behind the corneal plane (Mapp & Ono, 1999; Nakamizo, Shimono, Kondo, & Ono, 1994; Ono & Mapp, 1995; Shimono, Ono, Saida, & Mapp, 1998).
Blumenfeld (1936) believed that the egocenter might not be located at one fixed place, but it might shift location depending on attentional factors and the sense modality employed. A number of research studies appear to support Blumenfeld’s conclusion.
In an unpublished undergraduate project (Krynen & Wagner, 1983), we once attempted to replicate Howard and Templeton’s work and extend it to auditory space. In the auditory condition, two tiny speakers located at two different distances from the subject would alternately make a beeping sound. Subjects were asked to align the nearer of the two speakers so that the two appeared to lie in the same direct from the subject. In the visual condition, the subject aligned the speakers visually. The average location for the egocenter in the visual condition was .76 cm to the right of the median plane and 1.32 cm behind the corneal plane. For audition, the average location for the ego center was .71 cm to the right of the median plane and 10.11 cm behind the corneal plane. Thus, the auditory egocenter would appear to lie more or less between the two ears. The localization of the auditory egocenter was quite variable however. The standard deviation for both of the two coordinates was almost four times greater in the auditory condition than it was in the visual condition.
Shimono, Higashiyama, and Tam (2001) attempted to locate the egocenter for kinesthetic space in four experiments. Overall, they found that the kinesthetic egocenter is located in the middle of the body, on the surface of the skin or just below it. Thus, the location of the egocenter appears to change across sense modalities, although the location of the egocenter appears to be relatively fixed within a given sense modality.

The metrics of visual space. Although much will be said about metric functions that describe visual space, something should be said about the general form of these functions in terms of the coordinate systems I have just introduced.
Stevens (1975) demonstrated that judgments of unidimensional stimuli almost universally fit a power function. For distance estimation, Baird (1970) showed that the power function describes estimates of stimulus size both in the frontal plane and in depth. Thus, if we examine judged distance as a function of

AN ANALYTIC APPROACH TO SPACE AND VISION

61

actual distance for points in a given location and orientation in space, we would expect to obtain a power function of the form

J= D

(4.2)

where and are constants for that particular location and orientation and J and D are judged and actual distances, respectively. Stevens (1975) argues that all unidimensional estimates of the magnitude of stimuli (prothetic continuum) are best fit by a power function irrespective of the method employed (whether magnitude estimation, category estimation, etc.). Different methods, however, will give rise to different power function exponents. Following Stevens’s example, I will often use a power function to describe the metrics of visual space. In this way, cross-method consistency is gained and distortions in judgments due to the response method can be accounted for by alterations in the exponent.
The parameters and need not be the same for stimuli at all positions in space. More specifically, the values of these parameters may vary as a function of the Natural Coordinate dimensions developed in this chapter such that

J = (R, , ) D (R, , )

(4.3)

where R, , and are the coordinates mentioned above. As mentioned in Chapter 2, these functions need not depend on the coordinate
dimensions alone, but may also vary with stimulus conditions and instructions. Hence, a more complete form for this general metric function could be

J = (R, , , , , , . . .) D (R, , , , , , . . .)

(4.4)

where , , and represent varying experimental conditions. While the idea that exponents might be functions of stimulus conditions is
not entirely new (Stevens & Hall, 1966; Stevens & Rubin, 1970; Teghtsoonian & Teghtsoonian, 1978; Wagner, 1982, 1985, 1992), it is rarely done in practice. In fact, Lockhead (1992) criticized psychophysical work for not sufficiently taking into account context when generating psychophysical equations. Treating as a function of conditions is rare indeed.

Does the power function always work? While this formulation of the power function is very general, even this formula may not describe all metric judgments that observers give. The human mind is capable of conceptualizing distance in many ways. Different judgments are given in response to objective, apparent, and projective instructions that respectively ask people to objectively report distance, to say what things look like, or to take on a artist’s eye view and report how much of the visual field a stimulus takes up.
Yet, these common instructions only scratch the surface of the variation possible in the human conception of distance. People are able to view distance in terms of the shortest route between two places. In fact, peoples’ conception of

62

CHAPT ER 4

the “shortest” route between two places deviates greatly from their as-the-crowflies estimates and the nature of these judgments can be quite complex. Bailenson, Shum, & Uttal (1998) found that people tend to engage in “route climbing.” That is, they tend to begin their trip by selecting the longest and straightest route segment available heading in the direction of their goal even when another overall shorter path is available that is slightly less direct to begin with. This heuristic can lead to asymmetric path selection because the longest and straightest route segment heading out of point A in the direction of point B may place the subject on a different overall path than the longest and straightest route segment heading out of point B in the direction of point A.
Raghubir and Krishna (1996) found that paths with many sharp turns and switchbacks tend to be perceived as shorter than paths of the same length that move generally in the same direction along their whole route. Here, perceived path length may be biased toward the as-the-crow-flies distance traversed (without being identical to it).
Metric functions can also be influenced by cognitive factors and categorization effects. Howard and Kerst (1981) found that people tend to alter their distance judgments between locations on a rectangular map in a way that causes the map to “square up;” in other words, the left-right dimensions of the judged space are made to seem roughly equal to the up-down dimension even though this is physically untrue. In addition, near by objects tend to be perceived as being clustered together more tightly than they actually are, particularly in memory conditions.
In summary, although a power function appears to describe many distance judgments, people are mentally flexible enough to conceive of distance in very complex ways—particularly, when judgments are based on memory. One must be prepared to abandon the power function under these circumstances.

A Cautionary Note: Is Visual Space Metric?

It is tempting to think of the effects of instructions and the judgments of route length as being exceptions. According to this view, we know what we normally mean by perceived size and distance, and these other sorts of judgments are not what we mean by those words. Perhaps, if one eliminated these pesky exceptions, then a unitary view of visual space would be possible. Perhaps, if one eliminated the exceptions, the geometry of visual space would be singular.
I believe this viewpoint implicitly pervades much of the space perception literature. According to this standard view, people perceive the world in a single, internally consistent fashion, which they (more or less) accurately report on with the various judgment methods. Because all of the judgment methods are getting at the same underlying perceptual object, each of these methods should produce results that are largely consistent with each other. That is, they do not change the basic structure of visual space, but accurately reflect it.
What do I mean by our perceptions of visual space being internally consistent? In an internally consistent space, the judgments people give for each of the parts of visual space should fit together again to produce a coherent whole. If an object is broken into two parts, the sum of the perceived sizes of the parts

AN ANALYTIC APPROACH TO SPACE AND VISION

63

should equal the perceived size of the whole. At the very least, visual space should be internally consistent enough to qualify as a metric space.
Is the standard view right? Is there a single visual space? Do the various judgment methods produce results that are consistent with each other? Is visual space internally consistent? Is it even a metric space?
Baird, Wagner, and Noma (1982) explored these questions and concluded that visual space, as reflected by the judgments that observers give, appears to fail each of these criteria. This paper is not very well known among psychologists because it was published in a geography journal. I will lay out the basic argument here.

Does visual space satisfy the metric axioms? To be considered a metric space, the distance function on that space must satisfy the four axioms mentioned in Chapter 2 (Equations 2.1 to 2.4). Let us examine whether or not these axioms hold for visual space.
Let x, y, and z be elements of set X, then d(x,y) is a metric on X if

(1) Distance is always non-negative. That is,

d(x,y) 0

(4.5)

(2) Non-identical points have a positive distance. That is,

d(x,y) = 0 if and only if x = y

(4.6)

(3) Distance is symmetric. That is,

d(x,y) = d(y,x)

(4.7)

(4) The triangle inequality holds. In other words, a path between two points which is traced through a third point can never be shorter than the distance between the two points. That is,

d(x,y) d(x,z) + d(z,y)

(4.8)

No one quarrels with the first of these axioms, but one can dispute the other three. For example, all psychophysical modalities have a smallest stimulus that can be detected, an absolute threshold. This is also true of distance perception; non-identical points may seem to be in the same place if the distance between them is very small.
Similarly, there is evidence that distance perception is not always symmetric. This violation of symmetry is most often found in memory or cognitive mapping conditions (Burroughs & Sadalla, 1979; Cadwallader, 1979; Codol, 1985).
While neither of these violations of the metric axioms seems particularly serious, we will see that violations of the triangle inequality are common and that they have far reaching consequences for the internal structure of visual space.

64

CHAPT ER 4

The primary reason for the failure to satisfy the triangle inequality is that the power function exponent (seen in Equation 4.2) relating judged distance to perceived distance is almost never equal to precisely one. Wiest and Bell (1985) report that an average exponent for the direct perception of distance using magnitude estimation is 1.1 with wide variation around this number. In some cases, the exponent is much greater than one. On the other hand, the exponent is typically significantly less than one when other methods are used such as category estimation and mapping (Baird, Merrill, & Tannenbaum, 1979; Sherman, Croxton, & Giovanatto, 1979; Stevens, 1975; Wagner, 1985), under memory conditions (Weist & Bell, 1985), or under reduced-cue conditions (Baird, 1970). Chapter 5 will explore variations in the exponent as a function of stimulus conditions, instructions, and method in great detail.
Yet, we will see that the problems relating to the triangle inequality are even deeper than the power function, and apply to any concave or convex transformation of physical distance into perceived distance.

Concave and convex functions. A positive function, f, is said to be concave (downward) if it satisfies the following inequality

f(a) + f(b) > f(a+b)

(4.9)

and is said to be convex (downward) if it satisfies the inequality

f(a) + f(b) < f(a+b)

(4.10)

The power function is either a concave or convex transformation unless the exponent is precisely equal to 1.0. In fact, it is easy to show that the following theorem holds (For the proof, see Baird, Wagner, & Noma, 1982.):

a + b > (a+b) if a,b > 0, < 1 a + b < (a+b) if a,b > 0, > 1

(4.11) (4.12)

Thus, the power function is concave when the exponent is less than one and convex if the exponent is greater than one. To see why this is problematic, consider three points that lie along a line in physical space. Let us say the distance from the first point to the second is a, and the distance from the second point to the third is b, and the distance from the first to the third is a+b. Now let us say that the perceived distances (a', b', (a+b)') between the points is a power function of the physical distance; that is, the judged distances are

a' = a b' = b (a+b)' = (a+b)

(4.13)

AN ANALYTIC APPROACH TO SPACE AND VISION

65

Equations 4.11 and 4.12 imply that the straight line is only preserved as a unified whole when the exponent is equal to one. When the exponent is greater than one, the triangle inequality (Equation 4.8) does not hold because the total distance from the first point to the last (a+b) is greater than the sum of its parts. Because the exponent for size and distance judgments is often greater than one, this implies that the triangle inequality is violated on a regular basis.
Even if the exponent is less than one, the percei ved line can no longer be straight because the perceived distance between the first and last points is less than the sum of the two parts. In fact, this is a best case scenario, because any other path from the first point to the third that passes through a second point will produce parts whose perceived lengths will sum to an even larger number. The parts can never fit together to make the whole. In fact, contrary to the old Gestalt saying, the whole is always less than the sum of it parts. (Of course, one can make the parts fit together by moving into a higher dimension. The line segments could be plotted on a plane, although the line would not be straight. Conceptually, one might be able to keep a perception of straightness by moving this construction of pieces into an even higher dimension. Because visual space is three dimensional, the parts could fit together by moving to four dimensions, with the projection into three dimensions seeming straight. It’s not clear how to interpret the outcome of this operation that would be in the spirit of the classic book Flatland (Abbott, 1884).)
What is more, one can’t escape these difficulties by turning to one of the non-Euclidean geometries discussed in the previous chapter. Luneburg laid out the axioms that visual space must satisfy to be a geometry of constant curvature. As I said in the last chapter (Equation 3.1), one of these axioms (which Luneburg referred to as convexity—using the term in a different way than I am here) requires that for any two points (P1 and P3) on a line a third point (P2) must exist on the line between them such that

D(P1,P2) + D(P2, P3) = D(P1,P3)

(4.14)

where D is the metric for the space. Equations 4.11 and 4.12 flatly contradict this axiom and thus rule out any of the geometries of constant curvature.
By the way, Equations 4.11 and 4.12 can be extended to subtraction as well through a few simple substitutions to yield the following equations:

a - b < (a-b) if a > b > 0, < 1 a - b > (a-b) if a > b > 0, > 1

(4.15) (4.16)

(A result which I have used to explain several spatial illusions such as the Müller-Lyer Illusion, the Delbouef Illusion, and the moon illusion—although I have never published the theory.)

Spatial distortions implied by the power function. None of the forgoing has assumed that visual space had any particular geometry. Yet, to give you a idea of the sorts of distortions in visual space implied by the power law, let’s see what

66

CHAPT ER 4

would happen if we suppose that visual space were Euclidean. Let us say that we had three points on a plane in a triangular arrangement whose inter-point distances were a, b, and c as seen in Figure 4.4a.
Here, is the angle opposite side c. If perceived distance is related to physical distance by a power function, then

a' = a b' = b c' = c

(4.17)

If we try to map these perceived distances back onto a Euclidean plane, the resulting triangle is seen in Figure 4.4b. Let us see how , the perceived angle opposite c', changes as a function of the exponent.
By the Law of Cosines, c in Figure 4.4a can be reexpressed as

c2 = a2 + b2 - 2ab cos

(4.18)

Thus, c' in Equation 4.17 may be rewritten as

c' = ( a2 + b2 - 2ab cos ) / 2

(4.19)

(a)

(b)

a c

a’ c’

b

b’

Figure 4.4. Transformation of triangle 4.4a into triangle 4.4b after a power function is applied to distances between vertices. (Based on a figure from Baird, Wagner, and Noma, 1982. Copyright 1982 by the Ohio State University Press. Reprinted by permission.)

AN ANALYTIC APPROACH TO SPACE AND VISION In perceived space, the Law of Cosines gives us cos = a'2+b'2 - c'2 2a'b'

67 (4.20)

Substituting for a', b', and c' and solving for yields

= Arccos a2 +b2 - ( a2 + b2 - 2ab cos ) 2a b

(4.21)

Figure 4.5 shows the relationship between the original physical angle, , and the new angle in visual space, , based on Equation 4.21 for two different triangles: one where side a is equal in length to side b, and one where side a is ten times longer than b. Each curve displays this relationship based on a single exponent. Notice that the original angle equals the new angle only when the exponent is equal to one. The more the exponent deviates from one, the more the new angle is distorted from the original. The distortion is particularly extreme when a and b are very unequal in length.
The figure also shows that when the exponent is less than one, certain angles do not exist in the transformed space. For example, there are no straight lines through three points in the perceptual space because 180˚ angles are not represented in the space. If the exponent is greater than one, many larger physical angles cannot even be represented in the perceptual space.

Figure 4.5. Relationship between the Angle in a Triangle ( , Figure 4.4a) and the Angle ( , Figure 4.4b) after a power transformation of distances between vertices. Data are based on an evaluation of Equation 4.21 for the exponents ( ) listed on the graph (left, enclosing sides of are equal, a = b; right, a = 10b). From Baird, Wagner, and Noma, 1982. Copyright 1982 by the Ohio State University Press. Reprinted b y permission.

68

CHAPT ER 4

Not only do these findings cast doubt on the viability of geometries of con-
stant curvature to describe visual space, but they may be generalized in yet an-
other way. They also apply to any n-dimensional Minkowski space (as defined by Equation 2.7). More formally, if Rn is a complete n-dimensional Minkowski space (every n-tuple of real numbers is defined on the space), Sm is a complete
m-dimensional Minkowski space, and there is a concave or convex mapping of distances between Rn and Sm, then it can be shown that one or the other of
these spaces can no longer be complete if the mapping generally holds. For example, if the mapping is concave, then straight lines will not exist in Sm. If the mapping is convex, then a straight line in Rn cannot be mapped into Sm (Baird,
Wagner, & Noma, 1982).

Examples of impossible figures. While it is possible to systematically explore how power function transformations distort the distance and angular relationships between three points, when the layout of four or more points is examined, distance and angular relations can break down altogether. Let’s look at two examples of this that were presented in Baird, Wagner, and Noma (1982).
In Figure 4.6, two line segments intersect at right angles ( = 90˚) to form a cross where the end points of the cross (A, B, C, and D) are all the same distance (a) from the center point (O).
To find the angle in the perceptual space, we can substitute a = b and = 90˚ into Equation 4.21 to yield

= Arccos a2 + a2 - (a2+a2) 2(aa)
or

= Arccos 2a2 - (2a2) 2(a)2

= Arccos 2a2 - 2 a2 2(a)2

= Arccos 2 - 2 2

= Arccos 1 - 2 -1

(4.22)

Here, is only equal to 90˚ when = 1. Otherwise, the four angles that make up a complete circuit do not sum to 360˚, which would contradict the predictions of a Euclidean geometry. If the exponent is less than one (but still positive),

AN ANALYTIC APPROACH TO SPACE AND VISION

69

D

A

a

a

a

C

O

a

B
Figure 4.6. Diagram representing four points (A, B, C, and D) equidistant from a common origin (O). Based on a figure from Baird, Wagner, and Noma, 1982. Copyright 1982 by the Ohio State University Press. Reprinted by permission.
then the sum of the angles is less than 360˚. This result would be similar to the predictions of a hyperbolic geometry. If the exponent is slightly larger than one, then the sum of the four angles would be greater than 360˚. These results remind us of the predictions of a spherical geometry, although in both cases, we still have the problem we discussed earlier with Luneburg’s “convexity” axiom.
A second example. As a second example of the difficulties that arise from power function transformations, consider Figure 4.7. In this figure, four points (A, B, C, and D) define two equilateral triangles. All five line segments that make up this figure have the same length (a). Each of the angles in the triangle

70

CHAPT ER 4

( ), are equal to 60˚. Putting two of these angles together to make up ' should span a total of 120˚.
Now, let’s look at how each of these angles is affected by a power function transformation. Substituting = 60˚ and a = b into Equation 4.21 yields

= Arccos a2 + a2 - (a2 + a2 - 2a2(.5)) 2a2
or

= Arccos 2a2 - a2 2a2
= Arccos 1 = 60˚ 2

C

B

D

A
Figure 4.7. Diagram to represent four points (A, B, C, and D), where the lengths of line segments AB, BC, CD, and AD are all equal and = 60˚, = 120˚. Based on a figure from Baird, Wagner, and Noma, 1982. Copyright 1982 by the Ohio State University Press. Reprinted by permission.

AN ANALYTIC APPROACH TO SPACE AND VISION

71

Thus, the 60˚ angles that make up the two equilateral triangles remain the same after the power function transformation. The angle to the right of , CAD, will also transform to 60˚. The sum of these two angles should give us '. Now, let’s see what happens to the combined angle, ', when the formulas are applied to it directly. Substituting ' = 120˚ and a = b into Equation 4.21 yields

= Arccos a2 + a2 - (a2 + a2 - 2a2(-.5)) 2a2
or

= Arccos 2a2 - (3a2) 2a2

= Arccos 2a2 - 3 a2 2a2

= Arccos 2 - 3 2

(4.23)

The transformed angle found in Equation 4.23 is only equal to 120˚ when = 1. Thus, unless the exponent is one, the larger angle is no longer the sum of its two parts.

Why visual space is not a Banach space. In Chapter 2, we mention another space that is important in mathematics, the Banach space. In a Banach space, spatial coordinates are thought of as vectors whose lengths are called norms (||x||). In terms of space perception, these norms may be thought of as the percei ved distance from the self, or the origin of visual space, to objects. The above analysis indicates that visual space also fails to live up to the axioms of a Banach space.
First, a Banach space assumes that the norm is always non-negative and only zero if the vector has no length. Once again, the existence of thresholds for the detection of distance makes this unlikely. Second, a Banach space assumes that the triangle inequality holds, and we have seen that this is often untrue for egocentric distance judgments. Finally, a Banach space assumes that multiplying the coordinates by a scalar should increase the norm in a like manner. That is,

||cx|| = |c| ||x||

(4.24)

A power function transformation of physical distance into perceptual distance does not satisfy this final axiom either because if D' = c D in physical space and judgments are transformed by the equation

72

CHAPT ER 4

J= D then substituting the value for D' into the equation yields

J' = (c D) = c D c J if 1

(4.25)

A Few Final Comments

A few general conclusions can be drawn from the preceding analysis. First of all, one may wonder if space perception data gathered using different methods are consistent with each other. If magnitude estimation gives rise to exponents greater than one, category estimation gives rise to exponents less than one, and mapping techniques force subjects judgments to fit onto a plane, it would seem that the methods are giving very incompatible results. Statements about the geometry of visual space may be conditioned on the method employed. One may need to derive different, incompatible models for visual space for data generated by each method.
This conclusion that the methods for exploring visual space lead to mutually inconsistent results led MacLeod and Willen (1995) to conclude that no unitary visual space exists. They used experiments involving sinusoidal stimuli and the classic Zollner and Müller-Lyer illusions to show that judgments of orientation and location do not always agree.
Second, if numeric estimation judgments are used as the basis for defining visual space, then we know quite a bit about what visual space can not be. It is not really a geometry of constant curvature because the judgments observers give are des cri bed by a power function which does not in general satisfy Luneburg’s “convexity” axiom. Visual space is not satisfactorily des cri bed by any Minkowski metric (including the Euclidean metric) because the power function transformation would imply that either visual space or physical space is not complete. Under many circumstances, visual space is does not even satisfy the axioms of a metric space. At least three of the axioms are open to question, and whenever the exponent is greater than one, the triangle inequality is flatly violated. Visual space also does not satisfy the axioms of a Banach space.
This is one of the reasons for introducing the concept of a quasimetric in Chapter 2. Although visual space may not in general be a metric space, people do make metric-like judgments. That is, while an observer’s judgments of size and distance may not always satisfy all of the axioms of a metric space, they may still reflect the observer’s perceptions of these quantities.
One might wish that the data generated by subjects would fit a predetermined model more closely than it does. Yet, I believe that one should not reject dat a just because it does not fit our preconceptions. Data must be primary, and theory must follow to describe it. I feel that the only way to approach the problem of space perception is to rely on the judgment’s generated by subjects, even though those judgments may lead one to conclude that visual space has a distorted structure.

AN ANALYTIC APPROACH TO SPACE AND VISION

73

The analytic approach is better able to handle this difficult data rather than the synthetic approach. The same logic that leads us to question the metric nature of visual space also leads us to question the synthetic models proposed by others. If visual space is not convex (in Luneburg’s terms), then neither is it hyperbolic, spherical, or Euclidean.
On this sobering note, it is now time to examine the data in more detail. Subsequent chapters will look at the size and distance judgments subjects make under a variety of conditions. In particular, the next chapter will look at unidimensional judgments; that is, size and distance judgments when stimuli are all presented in a single orientation (e.g.., frontally or in-depth). Later chapters will look at the more complicated case of multidimensional stimuli.

5
Effects of Context on Judgments of Distance, Area,
Volume, and Angle

In Chapter 4, I argued that judgments of the metric properties of space (distance, area, volume, etc.) can be related to their corresponding physical dimensions using a power function of the form

J= S

(5.1)

where J represents the subject’s judgments of distance, area, or volume, is a scaling constant, S is the physical distance, area, or volume and is an exponent. In addition, I argued that the parameters of this equation, and , are not constants, but rather vary as a function of stimulus location and experimental conditions.
This chapter presents the results of a meta-analysis on the space perception literature. Conditions that significantly influence spatial judgments and the degree to which they alter the parameters of the power function are examined. I also look at how well the power function fits spatial data and under which circumstances the fits are particularly good or particularly poor.

Previous Reviews

Prior to Stevens (1957) and Stevens and Galanter (1957) researchers seldom reported power function parameters. However, after Stevens convinced the majority of the psychophysical community that the power function was the best description of judgments for a wide variety of perceptual modalities, particularly for judgments arising from magnitude estimation techniques, researchers began to regularly report the power function parameters for experiments involving spatial judgments.
Eventually, researchers generated enough data that meta-analytic summaries began to appear. Perhaps the first review of the spatial literature may be attributed to Baird (1970). Baird reported 69 exponents from 14 studies on judgments of length, area, and volume. Baird found that exponents for length are largest when the standard is in the middle of the range. He also discussed the effects of average stimulus size, stimulus range, and instructions, but believed that these
74

EFFECTS OF CONTEXT ON JUDGMENTS

75

variables had little influence of length judgments. For area, Baird indicated that objective instructions typically produced higher exponents than apparent instructions, but other contextual variables had little effect. In addition, he suggested that fits to the power function were poor for area judgments of complex shapes. Overall, Baird believed the average exponents for length, area, and volume averaged about 1.0, .8, and .6, respectively.
Two meta-analytic reviews of distance perception appeared in 1985. DaSilva (1985) reported 76 exponents from 32 studies on egocentric distance estimation, where egocentric distance refers to the distance from the observer to an object. DaSilva excluded exocentric distance judgments, which refers to the length or distance between two endpoints that are both located away from the observer. DaSilva indicated that a number of variables might potentially influence exponents including judgment method, whether the data is collected indoors or outdoors, instructions, size of the standard, size of the number assigned to the standard, range of stimuli, and stimulus cue conditions, although he did not produce any statistics from the 32 studies to examine these claims. Based on a number of his own experiments, DaSilva concluded that the typical exponent for egocentric distance estimation was about .90. He also found that magnitude estimation produces lower exponents that ratio estimation or fractionation, that increasing stimulus range produces lower exponents, and that the exponents for individual subjects are quite variable but less than 1.0 about 78% of the time.
Wiest and Bell (1985) analyzed 70 exponents taken from 25 studies on distance estimation. Their main finding concerned performance across perceptual, memory, and inference conditions. To Wiest and Bell, judgments are perceptual when the judged stimuli are available to the subject throughout the judgment process; memory judgments occur when stimuli are presented perceptually to the subject at one time but judgments are made at a later time when the stimuli are no longer available, and inference judgments occur when knowledge about stimulus layout is acquired across time such as in cognitive mapping. Wiest and Bell found that the average exponents were 1.08, .91, and .7 for the perceptual, memory, and inference conditions respectively. They also found that larger stimulus ranges are associated with smaller exponents and that judgments collected outdoors tend to produce smaller exponents than those collected indoors.
A great deal of spatial perception research has been reported in the decades following these prior reviews. The present review updates and greatly expands on them, because it is based on over seven times as much information as the most extensive previous work.

The Scaling Constant

A complete description of spatial judgments should include a discussion of variations in the scaling constant, , as a function of stimulus conditions. In truth, most all of the foregoing analysis will focus on the exponent and measures of goodness of fit, because the scaling constant is typically only meaningful within a single procedural context due to variations in the numerical scale used for judgments in different experiments.

76

CHAPTER 5

In fact, Borg and Marks (1983) mention 12 factors that can influence the scaling constant. These factors include the units of measurement for the physical stimulus, the units of measurement for the psychological scale, the psychophysical method, sensory and processing variations across conditions, and individual differences. In addition, if the exponent is determined in the usual fashion by fitting a straight line through a log-log plot of the data and subjects accurately judge the size of the standard, then the scaling constant (which is the yintercept) will vary along with the slope of the line and therefore along with changes in the exponent. Only when the exponent and experimental conditions are relatively constant will variations in the scaling constant be meaningful.
Given these conditions, however, variations in the scaling constant can be quite important because they indicate an across the board tendency for judgments under one set of conditions to be greater than under another. In fact, a number of experiments have uncovered interesting variations in the scaling constant.
Teghtsoonian (1980) had children of various ages engage in cross modality matching of length and loudness. She found that the scaling constant increased significantly with increasing age although the exponent did not differ significantly.
Butler (1983a) had subjects estimate the lengths of horizontal and vertical lines and found that the scaling constant for a vertical standard was about 12% greater than for a horizontal standard while exponents did not differ significantly across conditions. Butler thought of this variation in the scaling constant as a direct measure of the magnitude of the horizontal-vertical illusion. Butler (1983b) also found that the scaling constants associated with judgments of line length were significantly influenced by whether the line appeared in the context of the edges of a box or by itself. The scaling constant was significantly larger for lines viewed in the box context. Butler interpreted this difference as being an alternate measure of the constant error that is traditionally measured with discrimination techniques such as the method of constant stimuli.
Wagner (1985) and Wagner and Feldman (1989) had subjects estimate distances at various orientations with respect to the observer using four psychophysical methods. Wagner found that the scaling constant for stimuli oriented indepth with respect to the observer tended to be about half as large as for stimuli oriented frontally, while exponents did not differ across conditions. Wagner felt this variation indicated a general tendency for in-depth stimuli to be seen as smaller than frontally oriented stimuli of the same physical length. In other words, Wagner viewed variations in the scaling constant to be indicative of the presence of a perceptual illusion.

Methodology

The remainder of this chapter will focus on variables that influence the exponent and the coefficient of determination (R2) of the power function. This work is meta-analytic in that it statistically combines data gleaned from many sources to estimate the overall effects of different contextual variables on the exponent. However, I see the basic purpose of this investigation as being primarily descriptive; the amount of data collected and the number of variables involved are so

EFFECTS OF CONTEXT ON JUDGMENTS

77

large that the main points could easily be lost in a sea of convoluted statistical analysis. I aim to keep things as simple as possible.
The initial list of works to include in the analysis came from a computer search for articles on length, distance, size, area, volume, and space perception that included mention of power functions or exponents. I then eliminated nonEnglish articles. This eliminated eight articles mentioned in the initial computer search. The list was then supplemented by adding references from the previous meta-analytic reviews (Baird, 1970; DaSilva, 1985; Wiest & Bell, 1985) and other articles.
In the end, the analysis includes 104 space perception articles and a total of 530 power function exponents. The later number is greater than the former because most articles reported exponents for multiple experimental conditions. In a few cases, the experimenters broke down the data into so many conditions that very few subjects participated in a given combination. In these cases, I would average exponents across variables that were not at issue in the present study. (Otherwise, some studies would carry more weight in the final analysis simply by virtue of multiple exponents being reported in the same study with no other useful information being added to the analysis.)1
For each study, 22 variables were recorded (when available). The effects of 13 of these variables on the power function exponent and coefficient of determination are listed in Table 5.1 and will be discussed in this chapter. The other 9 variables are not reported either because they are redundant re-expressions of reported variables or because they lacked sufficient variation to make analysis valid.
The results for distance, area, volume, and angle judgments will be reported separately, starting with distance estimates. For size and distance judgments, I will discuss the effects of each variable four ways. First, an analysis of the entire data set will be discussed. Second, the analysis will be repeated for perceptual data alone, excluding inference and memory conditions. Third, the analysis will be refined even further to focus on perceptual data collected using magnitude estimation or ratio estimation. (Two methods which Wiest and Bell (1985) and others argue are equivalent.) Finally, I will discuss the results of articles that specifically examine a given variable (if such articles exist).
The data are presented in a variety of formats in recognition of the dangers involved in meta-analytic research. Data collected under memory and inference conditions may not be equivalent to perceptual data. Similarly, certain judgment methods such as mapping, fractionation, or triangulation place physical constraints on the judgment process that may render them non-equivalent to numeric techniques like magnitude and ratio estimation.

1A complete copy of the Excel database which was used in the following analyses can be obtained from the author by sending a diskette (being sure to indicate if your computer is IBM compatible or Macintosh) to the following address: Mark Wagner, Psychology Department, Wagner College, 1 Campus Road, Staten Island, NY 10301.

78

CHAPTER 5

Table 5.1

Mean Exponent and Coefficient of Determination (R2 ) as a Function

of Recorded Variables for Judgments of Distance Calculated from

All Data, Perception Data, and Magnitude or Ratio Estimation

Perception Data

________________________________________________________________

Value of

Exponent

Coefficient of Determination

Variable

Variable All Per. Mag. Est. All Per. Mag.Est.

________________________________________________________________

Overall

0.96 1.02 1.04

0.91 0.95 0.96

Age

Pre Under. 0.93 0.92 1.05

0.97 0.97 0.99

Undergrad. 0.95 1.02 1.02

0.91 0.97 0.96

Post Under. 1.04 1.04 1.05

0.91 0.91 0.95

Cue Conditions Full

1.00 1.02 1.04

0.92 0.94 0.95

Reduced

0.99 0.99 1.01

0.98 0.98 0.99

Ego vs. Exocentric Egocentric 0.99 1.01 1.06

0.96 0.98 0.99

Exocentric 0.95 1.03 1.01

0.88 0.91 0.92

Immeadiacy

Perception 1.02

1.04

0.96

0.96

Memory 0.87

0.90

0.87

0.85

Inference 0.77

0.76

0.82

0.81

Inside or Outside Inside

1.02 1.05 1.04

0.91 0.96 0.95

Outside

0.89 0.97 1.03

0.91 0.95 0.97

Instructions

Objective 0.94 1.03 1.08

0.90 0.97 0.96

Neutral

0.93 1.00 1.03

0.90 0.94 0.92

Apparent 1.02 1.03 1.04

0.96 0.95 0.97

Road Path 0.70

0.82

Method

Fractionation 0.89 0.90

0.97 0.97

Ratio Est. 0.82 1.02 1.02

Mag. Est. 0.98 1.04 1.04

0.91 0.96 0.96

Mapping 0.76 0.96

0.88

Production 1.02 1.06

0.92 0.99

Triangulation 0.96 0.96

Number of Subjects 15

1.00 1.03 1.03

0.92 0.94 0.95

> 15

0.90 1.01 1.04

0.91 0.97 0.97

Standard Used

No

0.96 1.03 1.05

0.93 0.98 0.98

Yes

0.96 1.00 1.00

0.88 0.92 0.93

Standard Size

Small

0.94 1.02 1.02

0.84 0.82 0.86

Midrange 0.96 0.94 0.96

0.93 0.92 0.93

Large

1.04 1.03 1.03

0.98 0.98 0.98

Stimulus Orientation Horizontal 1.02 1.01 1.02

0.95 0.93 0.86

Vertical

1.04 1.04 1.04

0.91 0.91 0.91

In-depth 1.01 1.04 1.09

0.95 0.98 0.98

Log Stimulus Range < 1

1.06 1.06 1.10

0.97 0.97 0.96

1 x 1.5 1.06 1.06 1.08

0.94 0.94 0.96

> 1.5

0.86 0.89 0.91

0.98 0.97 0.98

Stimulus Size

< 1 m

1.04 1.04 1.02

0.95 0.95 0.94

1m x 10m 1.10 1.09 1.11

0.99 0.99 0.99

> 10m

0.88 0.93 0.99

0.90 0.94 0.96

________________________________________________________________

EFFECTS OF CONTEXT ON JUDGMENTS

79

More researchers focus on perception and most of those use magnitude estimation techniques, and so the more refined analyses reported here will conform to this emphasis. In addition, because conditions are not randomly assigned to researchers and some combinations of conditions may occur more frequently than others, it is possible for variables to be correlated with one another. Thus, the apparent influence of one variable on the exponent might really be due to it being correlated with another recorded variable. Examining studies that explicitly look at the effects of a given variable within a single experimental context can serve as a check to confirm or disconfirm meta-analytic conclusions.

Size and Distance Estimates

There were a total of 362 exponents for distance estimates that entered into the analysis. The perceptual data set had 257 exponents, while the magnitude/ratio estimation data set had 182 exponents. Table 5.1 summarizes the effects of different contextual variables on judgments of size and distance. I will discuss the effects of each variable in turn.

Overall. In general, exponents for distance estimation are very close to 1.0. For both perceptual data sets, it appears that the average exponent is slightly larger greater than 1.0, and the data are well-fit by a power function with coefficients of determination averaging about .95. The total data set shows a lower average exponent and a lower coefficient of determination due to the influence of memory and inference conditions.

Age. Only a small number of studies reported the average age of the subjects involved. Typically, studies either reported an age range or described the population from which subjects were drawn—such as undergraduates. I used this information to estimate the average age of the subjects. In truth, there was little variation in age because 84% of all studies were based on undergraduate students.
Based on the total data set, exponents appeared to significantly increase with age, r = .15, p < .05, but this trend is not significant for the other two data sets. Coefficients of determination appear to decline with age for the total data set, r = -.19, p < .05, and for perceptual judgments, r = -.28, p < .05.

Cue conditions. Some studies limit perceptual exploration and/or cues to distance more than others. Full-cue conditions allow relatively complete layout information, while reduced-cue conditions limit perceptual information in some way. In the present analysis, most studies (81.2%) were classified as full cue.
Exponents were significantly greater under full-cue conditions for all dat a combined and for perceptual magnitude estimates. In truth, the effects of cue conditions may be greater than the summary data would make it seem because 60.4% of the studies classified as “reduced-cue” only limited information by fixing head position with a bite bar. This attempt at increased experimental control may also explain why the coefficient of determination was significantly greater under reduced conditions.

80

CHAPTER 5

One study that looked at cue conditions (Hagen & Teghtsoonian, 1981) found higher exponents for egocentric distance estimates under binocular conditions than monocular ones. Similarly, Kunnapas (1968) found a low exponent (.82) for monocular conditions, an intermediate exponent (.93) for binocular conditions with a fixed head position, and a high exponent (.97) for full cue conditions. Also consistent with this general tendency, Wagner and Feldman (1989) found much lower exponents for distance judgments collected under dark conditions (.70) than under full lighting (.99). Predebon (1992), however, found few consistent effects of head position or binocularity.

Egocentric vs. exocentric. There were no significant differences in the exponent between egocentric and exocentric distance judgments. Coefficients of determination, however, were significantly higher for egocentric judgments for all three data sets.

Immediacy. For want of a better word, immediacy here refers to the effects of perceptual, memory, and inference conditions as defined by Wiest and Bell (1985). The stimulus is most immediately available in perceptual conditions and never directly available under inference conditions.
The exponent is highest for perceptual conditions and lowest in the inference condition. In addition, coefficients of determination were highest for perceptual conditions and lowest for inference conditions. These differences in exponents and coefficients of determination between all three conditions are statistically significant at the .001 level according to Duncan follow up tests.
Wiest & Bell (1985) restricted their analysis to magnitude and ratio estimates. To make the present analysis equivalent, Table 5.1 also reports mean exponents for perceptual, memory, and inference conditions based exclusively on magnitude and ratio estimates. (This is the only time the third column of the table includes memory and inference conditions.) The exponents reported here are similar to those reported by Wiest and Bell; however, they deviate a bit less from 1.0 than in their report. Once again, coefficients of determination are highest for perceptual conditions and lowest for inference conditions.
The present results are also consistent with many individual articles examining the effects of memory on psychophysical judgment. Bradley and Vido (1984), Kerst, Howard, and Gugerty (1987), Moyer et al. (1978), Radvansky and Carlson-Radvansky (1995) all found higher exponents in perceptual than in memory conditions. DaSilva, Ruiz, and Marques (1987) found high exponents for perception, generally lower exponents for memory, and generally lowest exponents for inference conditions. However, there are two exceptions to this general rule. DaSilva and Fukusima (1986) and Kerst and Howard (1978) found perceptual exponents to be slightly greater than one and memory exponents to be even greater. One might be tempted to assume that memory exponents will be less than perceptual exponents if the perceptual exponent is less than one, and greater than perceptual exponents when the perceptual exponent is greater than one consistent with Kerst and Howard’s reperception hypothesis, but three of the studies who found perceptual exponents to be greater than memory ones also had mean perceptual exponents greater than one.

EFFECTS OF CONTEXT ON JUDGMENTS

81

For memory conditions, the amount of time that passed between stimulus presentation and the judgment process was recorded. One might expect than exponents would decline with increasing retention intervals, but the data did not support this. There was no consistent pattern in the size of exponents as a function of judgment interval and the correlation between retention interval and the exponent was not significant. Past studies that specifically looked at this variable are inconsistent regarding the effects of retention interval on the exponent (DaSilva & Fukusima, 1986; DaSilva, Ruiz, & Marques, 1987; Kerst, Howard, & Gugerty, 1987).
For inference conditions, Foley and Cohen (1984a, 1984b) and Wagner and Feldman (1990) found a tendency for the exponent to increase (and approach 1.0) the longer subjects reside in a location, but there was not enough data to test this hypothesis in the current meta-analysis.

Inside vs. outside. Experiments conducted indoors produce significantly higher exponents than those conducted outside for the two larger data sets. However, this difference is not significant for perceptual data collected using only magnitude/ratio estimation. Coefficients of determination do not differ significantly between inside and outside studies.
Teghtsoonian and Teghtsoonian (1970a) were the first to find higher exponents for distance judgments collected indoors. More recently, one condition of Higashiyama and Shimono (1994) directly tested this effect and found results consistent with the Teghtsoonians and those reported here.

Instructions. Studies differ in the way they describe the judgment task to subjects. In the present analysis, instructions were classified into four types. Apparent size instructions were the most commonly employed (39.2% of studies). These instructions ask subjects to judged how the distance “looks”, “appears”, or “seems to be” subjectively. Objective instructions, which explicitly emphasize physical accuracy, are less commonly employed (24.3% of studies). A third categorization used here was a neutral category, that neither emphasized physical accuracy nor how things appear subjectively, but rather simply asked subjects to judge the distance between two points. This categorization describes 34.3% of the studies. Finally, a relatively small number of studies (2.2%) asked subjects to judge the distance from one place to another, not as the crow flies, but according to the length of the route one would need to take to drive from one place to another.
Instructions ap peared to have inconsistent effects on the exponent. For the complete data set, apparent instructions produced significantly higher exponents than objective or neutral instructions. For perceptual magnitude/ratio estimates, objective instructions tended to result in higher exponents than either apparent or neutral instructions although this trend is not significant. Road-path length produced consistently and significantly lower exponents and coefficients of determination than other instruction types.
The studies that directly looked at this issue generally either found little difference in exponents due to instructions or found objective instructions resulted in slightly higher exponents. Two experiments by DaSilva and DosSantos

82

CHAPTER 5

(1984) found objective instructions produced slightly higher exponents (.88) than apparent instructions (.86). Similarly, Gogel and DaSilva (1987) found very slightly higher exponents with objective instructions (1.02) than apparent instructions (1.00). Teghtsoonian (1965) also found slightly higher exponents for objective instructions (1.00) than apparent instructions (.98). None of these differences were statistically significant, and they also do not seem to be “significant” in the common English meaning of that word.

Method of judgment. Analysis of variance indicates that the exponent differs significantly as a function of method used to collect judgments for the total dat a set at the .001 level. Duncan follow up tests indicate that mapping, complete ratio estimation, and fractionation exponents are significantly smaller than exponents based on magnitude estimation and production. Exponents also differ significantly (at the .001 level) as a function of method for the perceptual data set. Here, Duncan follow up tests reveal that the exponents for magnitude estimation, complete ratio estimation, and production are all significantly greater than those produced by fractionation. The magnitude/ratio estimation data set found no significant difference between magnitude and ratio estimation. There was too little data collected employing category estimation or absolute judgment to make meaningful statistical statements, but such as there is indicates that category estimation exponents seem to be low (.90) while absolute judgment exponents seem to be very high (2.44). Coefficients of determination did not differ significantly as a function of method.
Many studies have explicitly looked at how method influences exponents. Often, these studies have produced contradictory results. Baird, Merrill, and Tannenbaum (1979) and Bradley and Vido (1984) both found magnitude estimation produced higher exponents than mapping when testing knowledge of the spatial layout of a familiar environment. Wagner (1985) found that magnitude estimation exponents were higher than mapping exponents (and category estimates were the lowest) in a perceptual task. However, Kerst, Howard, and Grugerty (1987) found magnitude estimation produced lower exponents than mapping in a memory task. Baird, Romer, and Stein (1970) found that magnitude estimation led to much lower exponents than absolute judgment. Bratfisch and Lundberg noted that magnitude estimation exponents tended to be slightly larger than those associated with complete ratio estimation. In MacMillan et al. (1974) and Mershon et al. (1977), magnitude estimation exponents tended to be slightly higher than magnitude production exponents under similar conditions; however, Masin (1980) and Teghtsoonian and Teghtsoonian (1978) showed no consistent difference between magnitude estimation and magnitude production exponents and Pitz (1965) actually found higher exponents with magnitude production.

Number of subjects. The number of subjects in the experiment proved to be significantly negatively correlated with the size of the exponent for the total dat a set, r = -.30, p < .001. The most probable explanation of this relationship is that memory studies typically had many more subjects than perceptual ones. Because memory exponents are much lower than perceptual exponents, this would explain why exponents went down with the number of subjects under

EFFECTS OF CONTEXT ON JUDGMENTS

83

memory conditions. When memory is factored out as in the two perceptual dat a sets, the correlation between the number of subjects and the exponent disappears.

Standard. Although no difference is found with the total data set, studies using a standard had significantly lower exponents and coefficients of determination than studies that used no standard for perceptual data collected using magnitude or ratio estimation. Using a standard gave rise to more accurate judgments with the exponent averaging precisely 1.0. It would appear that putting the standard in the middle of the stimulus range gives rise to lower perceptual exponents than placing it at either extreme, but this trend is not significant.
Past research is equivocal. DaSilva and DaSilva (1983) and Kowal (1993) found the presence of a standard had little consistent influence on the exponent. On the other hand, Pitz (1965) found consistently lower exponents when the standard was present.

Stimulus orientation. The orientation of the stimulus did not significantly influence the exponent for any of the data sets. However, for both of the perceptual data sets, orientation significant influenced the coefficient of determination. In both data sets, Duncan follow up tests show that in-depth oriented stimuli gave rise to significantly higher R2 values than frontally presented stimuli in a vertical orientation. For the magnitude/ratio estimates, in-depth stimuli were also associated with higher R2 values than frontally presented stimuli oriented horizontally.
Butler (1983b), Hartley (1977), and Künnapas (1958) all found little di fference in exponents between vertical and horizontal frontal stimuli. Baird and Biersdorf (1967) found higher exponents for frontally oriented stimuli than for those oriented in-depth. Unlike the analysis reported above, Teghtsoonian (1973) conducted four studies and also found a marked tendency toward higher exponents for frontally oriented stimuli than those oriented in-depth.
As mentioned before, even though exponents may not differ much due to stimulus orientation, orientation may still have profound effects on perception of distance. Butler (1983b) found the scaling constant differed significantly between horizontal and vertical stimuli. Similarly, Wagner (1985) and Wagner and Feldman (1989) found very different scaling constants between in-depth and frontally oriented stimuli. These studies indicate a general tendency for frontally oriented stimuli to seem larger than those oriented in-depth.

Stimulus range. Teghtsoonian (1971, 1973) proposed that the exponent was closely related to the range of stimuli presented to the subject. Large stimulus ranges should theoretically produce consistently smaller exponents. In the present study, the stimulus range was determined dividing the maximum stimulus presented to the subject by the smallest. To be consistent with Teghtsoonian’s work, I then took the logarithm of this ratio.
Stimulus range proved to be one of the most powerful predictors of the exponent. The exponent was negatively correlated with log stimulus range at the .001 level for all three data sets (r = -.45 for all data, r = -.38 for perception data,

84

CHAPTER 5

r = -.44 for perceptual magnitude estimation data). The coefficient of determination was not significantly correlated with stimulus range for any of the data sets.
Past research is largely, but not entirely, consistent with Teghtsoonian’s theory as well. Da Silva (1983b), DaSilva and DaSilva (1983), Gibson and Bergman (1954), Künnapas (1958), and Markley (1971) all found lower exponents for larger stimulus ranges than for smaller ones. A few other studies found no consistent effect of stimulus range (Galanter & Galanter, 1973; Kowal, 1993; Teghtsoonian & Teghtsoonian, 1970, 1978).

Stimulus size. For the total data set, stimulus size—defined as the midpoint of the stimuli presented to the subject—was significantly negatively correlated with the size of the exponent, r = -.16, p < .01. Once again, this correlation probably arises because inference conditions, which are associated with lower exponents, often use large-scale environments to test cognitive mapping knowledge. When memory and inference conditions are factored out, as in the two perceptual data sets, the correlation between stimulus size and the exponent is no longer significant.

Multivariate analyses. Of course, there are certain statistical dangers associated with conducting a large series of univariate significance tests. First of all, the more tests one conducts, the higher the likelihood that some of the significant findings arise by chance (Type I error). Secondly, it is possible that real trends in the data may be produced (or obscured) by the effects of secondary factors that are accidentally associated with the variable due to the non-random nature of the data collection process. Because some combinations of conditions may occur more frequently than others, it is possible for variables to be correlated with one another. Thus, the apparent influence (or lack of influence) of one variable on the exponent might really be due to it being correlated with another recorded variable.
To overcome these deficiencies, I performed a series of linear multivariate regression analyses. The exponent was the outcome variable and all variables that displayed any significant univariate associations with the exponent served as predictor variables. These predictor variables included age, cue conditions, immediacy, inside vs. outside location, instructions, method, number of subjects, standard presence, and log stimulus range. Mean stimulus size was not included as a predictor because its association with the exponent was non-linear. Variables with no significant univariate association with the exponent were excluded to limit the multivariate model to a reasonable size.
Categorical variables were recoded as dummy variables where “1” represented the presence of a factor and “0” represented the absence of a factor. If more than two levels existed for a categorical factor, a series of dummy variables were used to represent the information. For example, instructions were broken down into two variables, one for the presence of objective instructions and one for the presence of apparent instructions.
For the total data set, four factors proved to be significantly associated with the exponent in the multivariate analysis: inference conditions (t = 3.70, p < .001), the presence of a standard (t = 2.64, p < .01), judgment method (t = 2.84,

EFFECTS OF CONTEXT ON JUDGMENTS

85

p < .005), and log stimulus range (t = 9.54, p < .0001). Once the significant variables were determined, a second regression analysis was performed employing only these four significant variables to determine the best fitting equation to predict the exponent. This equation accounted for 42.03% of the variance in the exponent.
The results of the second regression equation can be substituted into equation 5.1 to yield the following general equation to predict distance judgments:

J = D .083(mag) - .058(stan) - .210(inf) - .167(log(max/min)) + 1.172 (5.2)

where J is the subject’s judgment for distance, is a scaling constant, D is the physical distance, “mag” is code “1” if the method is magnitude or ratio estimation and “0” otherwise, “stan” is coded “1” if a standard is used and “0” otherwise, “inf” is code “1” for inference stimuli and “0” for memory and perceptual stimuli, and log(max/min) refers to the base 10 logarithm of the ratio of the largest stimulus used in the experiment to the smallest.
Because there was a linear dependency between the way magnitude/ratio judgments and the various production methods were coded (by knowing that the method was not any of the production methods automatically implied that the method was magnitude or ratio estimation), a second equation could be generated based on the various production methods (fractionation, triangulation, mapping, and magnitude production) where magnitude estimation was coded as “0.” Only one of these methods significantly predicted the exponent, fractionation. The linear regression including the influence of fractionation accounted for 47.76% of the variance in the exponent. When this regression equation was substituted into the power function it yields the following general equation to predict distance judgments:

J = D -.264(frac) - .067(stan) - .249(inf) - .143(log(max/min)) + 1.232 (5.3)

where “frac” is coded “1” when the fractionation method was employed and “0” otherwise.
For the perceptual data set, three (of the eight remaining) variables proved to be significantly related to the exponent in the multivariate regression analysis. Consistent with the total data set, the three variables that were significantly related to the exponent were method (t = 2.75, p < .01), presence of a standard (t = 2.40, p < .05), log stimulus ratio (t = 8.17, p < .0001). (Of course, the inference condition could not be an element in the perceptual equation because inference data was factored out of this data set.) When a regression equation was generated based on these three factors it accounted for 29.46% of the data. Substituting this linear equation into the power function yields the following general equation to predict perceptual distance judgments:

J = D .084(mag) - .055(stan) - .160(log(max/min)) + 1.161 (5.4)

86

CHAPTER 5

Once again, if production methods are focused on, fractionation was the only method significantly associated with the exponent. The regression equation including fractionation accounts for 36.87% of the variance. The power function based on this regression equation is:

J = D -.268(frac) - .065(stan) - .134(log(max/min)) + 1.219 (5.5)

Finally, for the perceptual data set focusing on magnitude and ratio judgments, only two factors were significantly related to the exponent in the multiple regression equation, presence of a standard (t = 3.02, p < .01) and log stimulus ratio (t = 6.25, p < .0001). (Method is no longer a factor because it has been factored out of the data set.) This regression equation accounted for 23.48 % of the variance in the exponent. Substituting this equation into the power function yields the following equation to predict distance judgments:

J = D - .071(stan) - .136(log(max/min)) + 1.223

(5.6)

Area Estimates

A similar meta-analysis was performed for the effects of contextual variables on judgments of area. In this case, only two data sets were examined; one based on all area judgments and another solely based on perceptual judgments that excluded memory and inference conditions. There was no point to separately analyze perceptual judgments using magnitude or ratio estimation alone because almost all (95%) area judgments employed either magnitude or ratio estimation. There were 117 exponents in the total data set, and 91 exponents in the perceptual data set. Table 5.2 shows a summary of the effects of various contextual variables on area estimation exponents and power function coefficients of determination (R2). The following looks at each of these variables in more detail:

Overall. In general, area exponents are smaller than those reported for distance judgments and are in line with Baird’s (1970) estimate of .8. For perceptual judgments, area exponents averaged .84, and these judgments followed a power function quite well as indicated by the high average coefficient of determination of .95. The total data set showed somewhat lower exponents and coefficients of determination due to the influence of memory and inference conditions.

Age. There was a small, but significant, positive correlation between the age of subjects and the size of the exponent for the total data set (r = .21, p < .05). A similar correlation between age and the exponent is seen in the perceptual dat a set, but it is not significant (r = .21, p > .05). The coefficient of determination declines sharply with age for the perceptual data set (r = -.58, p < .001), but this trend is not significant for the total data set (r = -.13, p >. 05). These data are somewhat consistent with Borg and Borg (1990) who generally found higher area estimation exponents with older, more educated subject populations.

EFFECTS OF CONTEXT ON JUDGMENTS

87

Table 5.2

Mean Exponent and Coefficient of Determination (R2 ) as a Function

of Recorded Variables for Judgments of Area Calculated from All

Data and Perception Data

________________________________________________________________

Value of

Exponent

Coefficient of Determination

Variable

Variable All Data Perception All Data Perception

________________________________________________________________

Overall

0.78 (.20) 0.84 (.18) 0.90 (.16) 0.95 (.11)

Age

Pre Under. 0.81 (.06) 0.81 (.06) 0.99 (.01) 0.99 (.01)

Undergrad. 0.76 (.23) 0.83 (.21) 0.89 (.23) 0.97 (.03)

Post Under. 0.80 (.18) 0.87 (.16) 0.87 (.18) 0.76 (.25)

Cue Conditions

Full

0.75 (.14) 0.80 (.11) 0.89 (.17) 0.94 (.13)

Reduced

0.97 (.36) 0.97 (.36) 0.95 (.03) 0.95 (.03)

Immeadiacy

Perception 0.84 (.18)

0.95 (.11)

Memory 0.67 (.10)

0.81 (.15)

Inference 0.50 (.12)

0.81 (.24)

Inside or Outside Inside

0.79 (.20) 0.85 (.18) 0.92 (.14) 0.97 (.03)

Outside

0.63 (.16) 0.63 (.16) 0.54 (.10) 0.54 (.10)

Instructions

Objective 0.96 (.47) 1.01 (.49) 0.99 (.01) 0.99 (.01)

Neutral

0.79 (.18) 0.87 (.13) 0.90 (.16) 0.97 (.03)

Apparent 0.75 (.13) 0.78 (.11) 0.90 (.16) 0.94 (.13)

Method

Ratio Est. 0.97 (.13) 0.97 (.14)

Mag. Est. 0.76 (.15) 0.82 (.11) 0.91 (.15) 0.96 (.07)

Number of Subjects 15

0.83 (.23) 0.86 (.23) 0.93 (.13) 0.92 (.14)

> 15

0.72 (.14) 0.79 (.06) 0.87 (.18) 0.98 (.03)

Standard Used

No

0.77 (.16) 0.80 (.14) 0.97 (.01) 0.99 (.02)

Yes

0.77 (.16) 0.83 (.12) 0.86 (.18) 0.92 (.14)

Standard Size

Small

0.82 (.21) 0.82 (.21)

Midrange 0.81 (.12) 0.81 (.12)

Large

0.64 (.13) 0.76 (.08)

Stimulus Orientation Frontal

0.80 (.19) 0.84 (.18) 0.93 (.13) 0.97 (.03)

Flat

0.63 (.16) 0.63 (.16) 0.54 (.09) 0.54 (.09)

Log Stimulus Range 1.5

0.98 (.14) 0.98 (.14) 0.98

0.98

1.5 x 2.0 0.84 (.25) 0.84 (.25) 0.99 (.01) 0.99 (.01)

> 2.0

0.73 (.10) 0.73 (.10) 0.88 (.19) 0.88 (.19)

Stimulus Size

< 100 cm2 0.87 (.14) 0.87 (.14) 0.98 (.01) 0.98 (.01)

in between 0.85 (.37) 0.85 (.37) 0.98 (.00) 0.98 (.00)

> 1000cm2 0.75 (.11) 0.74 (.11) 0.89 (.21) 0.89 (.21)

Cue conditions. There appears to be a tendency for the exponent to be larger under reduced cue conditions than it is under full cue conditions, although this result is not statistically significant for either data set. The total data set shows a significantly higher coefficient of determination for reduced cue conditions, perhaps due to the greater experimental control often employed in reduced settings. In truth, there are too few experiments that have employed reduced cue settings to allow for any firm conclusions. Given only 12 reduced cue data points, one