The Global Structure Of Visual Space

227 Pages·2004·6.456 MB·English

by  Tarow Indow

Advanced Series on Mathematical Psychology VOl
Global Structure
Tarow Indow
•The
Global Structure
Visual Space
ADVANCED SERIES ON MATHEMATICAL PSYCHOLOGY
Series Editors: H. Colonius (University of Oldenburg, Germany)
E. N. Dzhafarov (Purdue University, USA)
Vol. 1: The Global Structure of Visual Space
by T. Indow
Advanced Series on Mathematical Psychology
The
Global Structure
of
Visual Space
Tarow Indow
University of California, Irvine, USA
Y|g5 World Scientific
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THE GLOBAL STRUCTURE OF VISUAL SPACE
Advanced Series on Mathematical Psychology — Vol. 1
Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd.
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Foreword
In moving around, most animals guide the body in accordance with the
information obtainable through vision. Humans are not exceptional. For
humans, however, perceived space has played a more important role.
Our ability to see the highly structured surrounds as they are has
provided us with another ability to grasp the geometrical structure
within a figure or between figures. This is a starting point of logical
thinking in humans. As will be discussed in this book, "null or void"
and "infinity" are beyond our direct perceptual experience. This may
have something to do with the fact that humans have had difficulty in
achieving these concepts. The use of the positional notation
corresponding to 0 was an invention in India, perhaps around 6th
century. Medieval abacus had the column corresponding to 0 but
mathematicians had difficulty to conceptualize it. "The zero is
something that must be there in order to say that nothing is there"
(Menninger, 1969, p.400). Conceptualizing infinity is even subtler.
Often people imagine a condition of "countless" as the image for
infinity because this condition can occur as a phenomenon of our visual
experience. The space we see extends in three directions. Gardner
(1990) vividly describes how difficult it is to visualize something that is
not in our perceived space. According to him, until the 19th century,
mathematicians did not realize the possibility of extending Euclidean
geometry to dimensions higher than three.
Investigating visual perception has been a main subject of
experimental psychology. However, most studies have been concerned
with local phenomena in the space we see. It is especially true since
modern technology was introduced in experiments of visual perception.
Generating visual patterns on a CRT by computer and the availability of
fMRI (functional magnetic resonance imaging) have revolutionized the
experimentation, and most investigators are more and more interested in
detailed study of phenomena in a small area in the visual space.
Consequently, studying the global structure inherent in what we see
escapes the attention it deserves. This is the problem to be taken up in
this book.
The publication of experimental results referred to in this book began
in 1962. Many of my students at Keio University in Tokyo made
contributions. The following names deserve explicit mention: Emiko
v
vi Global Structure of Visual Space
Inoue, Nobuko Momomi, Mitsuho Ohta, Mitsuko Shimada, Keiko
Matsushima, Yasuo Nishikawa, Hisako Miyauchi, Noriko Yamashita,
Akira Nakada, and Toshio Watanabe. I taught at Keio University until
1979, then moved to the University of California Irvine (UCI).
Watanabe was accepted to UCI as a graduate student and received
a Ph.D. with the study cited in Sec.5.1.1. Experiments numbered as 5 in
Table 2.2 were performed by a student Kevin Wright.
While I was in Japan, I was given an opportunity to work as
a research fellow at the Laboratory of Psychophysics in Harvard, 1963 -
1966. S.S. Stevens was the director. During that period, I met R.D.
Luce, a professor of University of Pennsylvania at that time. Later, he
invited me to the Institute for Advanced Study in Princeton as a visiting
member, 1971-1972, and recommended me to UCI when he moved from
UCI to Harvard. I am particularly indebted to these two distinguished
scholars in the United States.
My study on visual space at UCI was partially supported by the
National Science Foundation Grant, IST80-23893. As to the global
structure of visual space, I have exchanged opinions with many
scientists all over the world. Discussions with Patrick Suppes of
Stanford University and Jan Drosler of University of Regensburg in
Germany were especially helpful to me. The writing of this book is due
to strong recommendation by two colleagues in the Society for
Mathematical Psychology, E.N. Dzhafarov, a professor at Purdue
University, and H. Colonius, a professor at Universitat Oldenburg,
Germany.
Irvine, California Tarow Indow
December, 2003
Menninger, K Number words and number symbols:A cultural history of
numbers (MIT Press, Cambridge, 1958).
Gardner, M The new ambidextrous universe: Symmetry and asymmetry
from mirror reflections to superstrings (W.F.Freeman, New York,
1990).
CONTENTS
Foreword v
Abbreviations and Symbols ix
1. Visual Space 1
1.1 Global Structure of Visual Space 1
1.1.1 Features of VS 2
1.2 Binocular Vision 8
1.2.1 Cyclopean Vision in the Horizontal Plane of Eye-level 9
1.2.2 3-D Cyclopean Vision 11
1.2.3 Spatial Behavior 14
2. Luneburg Model 17
2.1 P- and D-alleys, H-curves in the Horizontal Plane 17
2.1.1 Experiments with Stationary Points 17
2.1.2 Discrepancy between {Q/}p and {Qj} 20
D
2.2 VS as a Riemannian Space of Constant Curvature 22
2.2.1 Riemannian Space of Constant Curvature 22
2.2.2 Eudlidean Map (EM) 26
2.2.3 Equations of P-and D-alleys, H-curves in EM2 29
2.3 Theoretical Curves in X2 33
2.3.1 Luneburg's Mapping Functions 33
2.3.2 Equations of P-and D-alleys and H-curves in X2 37
2.3.3 Comments on Results of Alley Experiments 39
2.3.4 Comments on Values of-K and a 44
2.4 Derivations and Explanations 47
2.4.1 Supplementary Explanations to Sec.2.2.1 47
2.4.2 Derivations of Equations in Secs.2.2.2 and 2.2.3 51
3. Two Extensions of Luneburg Model 67
3.1 Alleys on a Frontoparallel Plane 68
3.1.1 Theoretical Equations 68
3.1.2 Experimental Results 75
3.2 Direct Mapping according to Riemannian Metric 79
3.2.1 Multidimensional Mapping according to Riemannian
Metric 80
3.2.2 Experimental Results 91
vii
viii Global Structure of Visual Space
3.2.3 Concluding Remarks to Sec.3.2 97
4. Visual Space under Natural Conditions 103
4.1 The Perceived Sky and Ground 103
4.1.1 Bisection of the Sky 104
4.1.2 The Moon Illusion 109
4.1.3 Multidimensional Construction of the Night Sky 113
4.1.4 Horizon 117
4.2 Scaling of Radial Distance 8 122
0
4.2.1 Scaling based on Difference Judgment 122
4.2.2 Scaling based on Ratio Judgment 129
4.2.3 Discussion on the Form of d(x) 133
4.3 Perceived Spatial Layouts under Full Cue Conditions 137
4.3.1 Three Experiments 137
4.3.2 General Discussion 143
4.3.3 Regular Triangles with S at the Barycenter 145
5. Related Experiments and Theoretical Considerations 149
5.1 Spatial Layouts in Frameless Visual Space 149
5.1.1 An Experiment with Circles 149
5.1.2 Experiments using Triangles 152
5.1.3 General Comments and Derivations 159
5.2 Mapping Functions 163
5.2.1 Experimental Data 163
5.2.2 Theoretical Considerations 167
5.2.3 Roles of Mapping Functions 173
5.3 Experimental Tests of Properties of VS as an R 175
5.3.1 Two Experiments of Foley 175
5.3.2 Bottom-up Experimental Approaches 180
5.4 Discussion on the Postulate that VS is anR 186
5.4.1 Helmholtz-Lie Problem 186
5.4.2 Congruence and Similarity in VS 189
5.4.3 Linear Perspective 193
5.4.4 Concluding Remarks 197
References 201
Author and Subject Indexes 211
Credits 216
Abbreviations and Symbols
S subject (observer) in experiment
VS visual space
X physical space
E Euclidean space
R Riemannian space of constant curvature
K Gaussian (totoal) curvature
r curvature radius
EM Euclidean map to represent R in E
(Poncare)
BS, BC basic sphere, basic circle in EM
2 inSec.2.4.2 Klein's model for R2 of K<0
Q in Sec.2.4.2 sphere representing R2 of K > 0
Q, Q(x,y^\ Q(y,<|>,9) stimulus point in X, see Fig.2.3
y, <>| , 9 convergence, lateral, elevation angles
Pv(£v,Tlv£v)> Pv($o, <Pv, $v) perceived point in VS, see Fig.2.3
P(£,TI,Q, P(p0, (p, $) point representing Pv in EM,
see Fig.2.3
O or 0 origin of coordinates, body in X, self in
VS,EM
DP(8) X2 extending in depth along x with an
elevation angle 9 or its conterpart in VS
and EM
horizontal DP DP with 9 = 0
HP frontoparallel plane in VS or its
conterpart in X, EM
e, e distance, radial distance (Euclidean)
0
in "X"
111 _/V
P, Po distance, radial distance (Euclidean)
in EM
8, 8 distance, radial distace (perceptual)
0
in VS, latent variable
d,d scaled value of 8, 8
0 0
IX