Fechner, information, and shape perception
Joseph S. Lappin & J. Farley Norman & Flip Phillips
# Psychonomic Society, Inc. 2011
Abstract How do retinal images lead to perceived envi-
ronmental objects? Vision involves a series of spatial and
material transformations—from environmental objects to
retinal images, to neurophysiological patterns, and finally to
perceptual experience and action. A rationale for under-
standing functional relations among these physically differ-
ent systems occurred to Gustav Fechner: Differences in
sensation correspond to differences in physical stimulation.
The concept of information is similar: Relationships in one
system may correspond to, and thus represent, those in
another. Criteria for identifying and evaluating information
include (a)resolution, or the precision of correspondence;
(b)uncertainty about which input (output) produced a given
output (input); and (c) invariance, or the preservation of
correspondence under transformations of input and output.
We apply this framework to psychophysical evidence to
identify visual information for perceiving surfaces. The
elementary spatial structure shared by objects and images is
the second-order differential structure of local surface
shape. Experiments have shown that human vision is
directly sensitive to this higher-order spatial information
from interimage disparities (stereopsis and motion paral-
lax), boundary contours, texture, shading, and combined
variables. Psychophysical evidence contradicts other com-
mon ideas about retinal information for spatial vision and
object perception.
Keywords Fechner. Information . Perception . Shape .
Surfaces . Space . Depth . 3-D perception . Stereo .Motion .
Contours . Texture . Shading . Cue combination . Spatial
vision
Part 1. Information and the representation
of structure
Mind from matter
The fundamental problem of vision is to understand
how retinal images lead to perceived environmental objects
Spatial vision is so marvelously effective that usually it is
taken for granted. We perceive environmental objects but
not the visual instrument by which they are revealed.
Spatial vision pervades our conscious experience, but our
understanding of the underlying visual mechanisms has
substantial gaps.
Gaps in understanding occur especially at the interfaces
between visual subsystems, where information is trans-
ferred from one physical and spatial format to another. How
does vision transcend vast changes in format—involving
3-D objects and spaces, 2-D optical images, neural spike
trains and synaptic interactions in multiple neural areas,
conscious perceptions, and behavioral actions? How do
immaterial knowledge and experience arise from material
objects and events?
J. S. Lappin
Vanderbilt University,
Nashville, TN, USA
J. F. Norman
Western Kentucky University,
Bowling Green, KY, USA
F. Phillips
Skidmore College,
Saratoga Springs, NY, USA
J. S. Lappin (*)
5083 Hanging Moss Lane,
Sarasota, FL 34238, USA
e-mail: joe.lappin@vanderbilt.edu
Atten Percept Psychophys
DOI 10.3758/s13414-011-0197-4

The present article offers a theoretical framework for
addressing these issues—based on the concept of informa-
tion, involving corresponding relational structures in phys-
ically separate domains. We describe criteria for identifying
and evaluating structural correspondence and then apply
these criteria to research on shape perception.
Fechner’s insight
In the 19th century, Gustav Fechner struggled for years to
find a scientific rationale for linking the material and mental
worlds. Material objects have mass and spatial extent, but
mental experience has neither. Some have concluded that
mental events belong to a spiritual world. Fechner,
however, believed that mind and body are inseparable and
that mental events are emergent properties of complex
material systems, such as brains (see Heidelberger, 2004,
pp.118–119). His insight was that sensory discrimination
experiments offer meaningful quantitative bridges from
physics to psychology.
According to Fechner, the idea that came to him on
“October 22, 1850 at dawn in bed” was to make “the
proportionate increase in living energy .. . be the measure of
the increase of pertinent mental intensity” (Vol.2: Fechner,
1860, 2004, p.56; translated from Fechner, 1860, p.554).
The idea was expressed mathematically in his “fundamental
formula”:
dg ¼ K db=b; ð1Þ
where dg is a change in sensation magnitude, db is a
change in physical stimulus magnitude, b is a positive ratio-
scaled physical magnitude, and K is an arbitrary constant.
As Fechner said, “the fundamental formula .. . simply
expresses the relation holding between small relative
stimulus increments and sensation increments. In short, it
is nothing more than Weber’s law...” (Vol.2: Fechner, 1860;
1965, p.70; translated from Fechner, 1860, p.10).
Discussions of Fechner’s psychophysics usually focus
on his logarithmic scale of sensory magnitude, but his
fundamental formula is more important. Integrating the
fundamental formula yields the logarithmic scale.
Fechner’s insight entails two main ideas: First, differ-
ences in sensation can meaningfully correspond to differ-
ences in physical energy. Physical quantities do not
meaningfully correspond to experiential quantities, but
relations among physical quantities may correspond to
relations among sensations.
A second aspect of Fechner’s insight is “Weber’s law”—
that a detectable change in physical intensity is (approxi-
mately) proportional to that intensity. In Fechner’s words,
“Weber’s Law ... should, because of its wide generality and
the extent of the boundaries within which it is strictly or
approximately valid, be looked upon as fundamental to
psychic measurement theory” (Vol.1: 1860, 1966, pp.54–55).
The validity of the hypothesized invariance of the Weber
fraction is not important here. The important idea is that
physical and mental variations share a common structure.
The Weber fraction is but one possibility.
Information, communication, and structural representation
About a century after Fechner’s insight, related ideas
emerged in theories of information, communication, and
control—generalizing the idea that relationships in one
domain can correspond to and represent those in another.
Information, as the term itself suggests, is “in the
formation”—based on organization rather than energy
(Wiener, 1954, 1961).1 Information is transmitted by
corresponding variations in separate systems. Information,
communication, and control involve changes, and changes
involve energy, but the quantity of information is unrelated
to the quantity of energy. Physical events involve transfers
of energy, but communication and perception involve trans-
mission of information. Energy is “conserved” (constant), but
1 Information is often understood as a statistical concept, based on the
probabilities of alternative states of a system, as in Shannon’s (1948/
1949) theory. Applications of this statistical theory to experimental
psychology have been reviewed by Garner (1962) and Laming (2010).
Shannon’s goal was to quantify transmitted information—when the
sets of messages and physical signals are already specified.
The present goal is to define information when the relevant
variables are not known prior to investigation. Specifically, we seek to
clarify the representation of variations in one system by those in
another. The problem of representation logically precedes a statistical
description.
Shannon (1948/1949) sought a general quantitative method
applicable to any variable, regardless of its physical dimensions or
scale, applicable to symbolic as well as physical variations. Accord-
ingly, variations were treated simply as categories, where the relations
among alternative values are simply same or different, members of the
same set or different sets.
An expanded set of relationships is required to understand
representation more generally, especially those involving variations
found in nature. Optical and spatial variations relevant to visual shape
perception, for example, involve ordinal and topological structures in
multiple dimensions of space and time and involve differentials of at
least second order.
Shannon’s generalized representation of categorical structures
involved no assumptions about physical form or relations among
categories. Categorical structures are readily represented by sets of
symbols, but symbols are less useful for many other relational
structures. A common misunderstanding is that “information” is
fundamentally symbolic, without physical or geometric or meaningful
structure. This unfortunate misconception fails to recognize the great
variety of forms of information in physical and biological systems. It
is also insufficient for characterizing representation in general.
The informal concept of ‘information’ used in both science and
everyday language implicitly encompasses a wide range of physical
forms and relationships. The present aim is to develop a generalized
concept of information relevant to visual perception and other
applications.
Atten Percept Psychophys

information is not conserved and not governed by natural laws
of space, time, mass, and energy.
To understand vision, one must identify the information
it uses. Retinal images and physiological responses can lead
to perceived environmental objects if and only if these
physically different variables have a corresponding struc-
ture. As Marr (1982) pointed out, “a process may be
thought of as a mapping from one representation to
another” (p.31). Thus, representations of input and output
information shape inferences about visual or computational
processes for mapping one to the other. Identifying what
information is processed is a prerequisite for determining
how it is processed.
Marr (1982, p.31) represented optical input as a scalar
field of intensity values, spatially structured by retinal
anatomy. Marr and many others have regarded this
representation as obvious, but it is a hypothesis, one of
several empirically testable alternatives. Abundant experi-
mental evidence that will be reviewed in Part 2 of this
article shows that retinal information has topological
structure—involving relations among image points.
Many of the present ideas resemble those of James
Gibson. Gibson (1950) concluded that visual information is
based on invariant higher-order spatiotemporal structure,
and we seek to identify that structure. Surface structure had
a basic role in Gibson’s conception of sensory information,
and it is also basic here. Gibson (1966, 1979) also
emphasized that (a)the structure of sensory stimulation
reflects the structure of the environment, and (b)relevant
environmental structure covaries with the perceiver’s
perspective, actions, and attention. Those ideas are also
reflected here. Not surprisingly, rationale for these ideas
may be found outside Gibson’s writings.
The present approach generalizes concepts of information
beyond symbolic and statistical structures (e.g., Laming,
2010; Shannon, 1948/1949) to encompass geometric organi-
zation. This generalization is not new. Applications of
cybernetics to biology (Ashby, 1963; Meyer-Eppler, 1969;
Wiener, 1961) led naturally to descriptions of sensory
information as spatiotemporal patterns structured by inter-
actions with the environment.
Psychophysics and representation
The principal goal of psychophysics is sometimes regarded
as the numerical representation of sensations; that was a
primary aim of Fechner, and has been of many others since.
Luce (2004) and Steingrimsson (2009) are contemporary
examples. The present application, however, entails nonnu-
merical representation.
The first problem of psychophysics is what to measure.
As S.S. Stevens put it, “In a sense, there is only one
problem of psychophysics, namely the definition of the
stimulus. In this sense, there is only one problem in all of
psychology—and it is the same problem” (1951, p.31). If
“stimulus” is replaced by “information,” then Stevens states
the present problem: What is the information for perceiving
environmental objects?
In general, representation involves a correspondence
between two relational structures.2 The theoretical rationale
for representation has been studied in the theory of
measurement, concerned with the problem of representing
an empirical structure of qualitative relations by a structure
of numerical relations (e.g., Krantz, Luce, Suppes, &
Tversky, 1971; Roberts, 1979; Suppes & Zinnes, 1963).
Two subtasks in measurement are known as the “represen-
tation” and “uniqueness” problems.
Both problems have analogues in studying biological
systems and in identifying optical and visual representa-
tions of environmental objects. Methods for identifying
such representations are developed in the section below on
“Evaluating structural correspondence”; the methods are
then illustrated by application to the specific problem of
shape perception in Part 2 of this article.3
Basic concepts and terminology
Many different terms might be used for concepts of
structure and information. We will rely on the following
terminology:
Forms are variable local elements that carry informa-
tion. For shape perception, the local forms are spatial
variables that occur in smoothly varying arrays on
surfaces and their images.
“Form” is used here as a generic term for both spatial
and nonspatial elements. Symbolic forms—for instance,
letters, words—label discrete categories, where spatial
shape is arbitrary except to distinguish among alternatives.
Temporal forms occur in music, speech, and other acoustic
events. In spatial vision, the elementary forms of moving
objects deform smoothly over time in moving images and
change discretely between different viewpoints and differ-
ent objects.
Structure refers here to relations among forms—
synonymous with the concept of “relational structure”
in measurement theory.
2 A relational structure is a set of elements, relations among the
elements, and (often) operations that yield other elements in the set.
3 Representation refers here to a correspondence between physically
separate relational structures—where variations in one system corre-
spond to, and thus represent, those in another. The same term is
applied to many different types of representation, including spatio-
temporal, numeric, and symbolic structures, natural or biological
structures, as well as artificial structures of human design. The term
implies neither mental nor symbolic representation.
Atten Percept Psychophys

Corresponding structures do not imply corresponding
forms. The elementary forms typically differ between
domains—as in linguistic and numerical representations—
and in visual correspondences between objects, images,
physiological responses, and perceived objects. Correspon-
dence involves the relational structure, not the forms
themselves.
Here, information is said to exist if and only if two or
more physically distinct structures correspond to one
another. “Information” is distinguished from the concrete
forms or signals that carry it. Thus, “information” is neither
objectively defined nor concrete, because (a)it involves
relationships among forms rather than forms per se, (b)
relational structure may be represented in multiple ways,
and (c)it requires corresponding representations by both
receiver and sender. Despite its nonobjectivity, this
approach resolves basic ambiguities in identifying
information, because arbitrary physical variables seldom
correspond across systems. In Shannon’s (1948/1949)
statistical theory of communication, the structures were
defined a priori, but in sensory systems, the information-
carrying forms and structures are generally not known
prior to investigation.
Evaluating structural correspondence
Spatial correspondences of objects, images,
and perceptions
Structural organization can be represented in many ways.
No structure can be singled out, especially beforehand, as
objectively correct or optimum. How, then, can relevant,
information-carrying structure be identified?
Three quantifiable criteria for evaluating structural corre-
spondence are resolution, uncertainty, and invariance. The
aim is to identify corresponding relational structures that
maximize resolution, minimize uncertainty, and maintain
correspondence under transformations of observational con-
ditions. Correspondence of particular relational structures may
be quantified by Shannon’s (1948/1949) information theory.
Three physical domains are relevant: environmental
objects, optical images, and physiological patterns—say
O, I, and <. The relevant structures in these domains are
inferred (by a scientist) from correspondences across
domains. Structures in O and I are scientifically observable
but permit multiple representations. Structures in < are not
directly observable by scientists, and can only be inferred.4
The maps between structures in these domains may be
labeled f, g, and h, along with their inverses:
f : O ! I f 1 : I ! O0;
g : I ! < g1 : < ! I 0; and
h : < ! O00:
Three different sets of the environmental objects—O, O',
and O''—are distinguished because these are defined
differently. The set of environmental objects, O, may be
described by a scientist without reference to retinal or
photographic images, but another set of objects, O',
indirectly defined by their images, is typically smaller,
including multiple environmental objects with the same or
indistinguishable images. Similarly, a third set of perceived
objects, O'', is operationally defined by behavioral discrim-
inations. A distinction between the image structures I and I'
is also needed, since I' is defined by behavioral discrim-
inations. The inverse map h–1 is not meaningful in this
context and is safely ignored. Combining functions, the
map from physical to perceived objects is
h g f : O ! O00:
These are generic maps that might not satisfy correspon-
dence criteria and might differ in dimensionality. Structures
and forms that carry information are subsets of many
possible representations. Identifying information requires
investigation. Figure 1 illustrates these maps.5
Resolution
Resolution refers to the precision of representation.
Bandwidth and capacity are related concepts involving
the complexity of a representation. In psychophysical
experiments, resolution is often evaluated by discrimina-
tions among neighboring signals, but dynamic range is
also relevant.
Shannon’s (1948/1949) measure of information trans-
mission offers a general method of quantifying resolution.
This measure assumes nothing about the structure of
relations among the alternative forms. In Shannon’s theory,
4 The structure of < correlates with patterns of neural spike trains and
synaptic interactions. Presently available scientific observations do not
reveal the information-carrying organization of these physiological
patterns, however. Instead, physiological structure must be inferred
from correspondences among objects, images, and behavioral
discriminations.
5 These functions map variations—structures of possible elementary
forms—not individual forms, symbols, or objects. Transfers of
information involve sets of potential events, not particular causal
events or processes operating on individual stimuli.
The sets of objects, images, and perceived objects in Figure 1
involve an investigator’s representations of forms and structures of
variation among these forms. Thus, the representations may be
adjusted to maximize correspondence across the domains of environ-
ment, optical images, and behavioral discriminations. The quantitative
evaluations of uncertainty and information transmission—for instance,
H(X) and T(X:Y) as defined in the next section—are contingent on the
representations of these sets of objects, images, and perceived objects.
Atten Percept Psychophys

the structure is simply a set of categories, with relations
same versus different. Thus, the variability of a set X of
categories, xi∈ X, is given as
HðX Þ ¼ 
X
p xið Þlog2 p xið Þ½ f g; ð2Þ
where p(xi) is the probability of occurrence of category xi,
with ∑ [p(xi)] = 1.0. The probability distribution is not
important in most of the present geometric analysis, and in
this article, the categories usually are treated as equally
likely, p(xi) = p(xj).
The information transferred by corresponding variations
of two different sets X and Y is given by
T X :Yð Þ ¼ HðY Þ  H Y jXð Þ; ð3Þ
where H(Y|X) is the conditional uncertainty of Y given X.
Thus, if H(Y|X) = 0.0, then T(X:Y) = H(Y); if H(Y|X) = H
(Y), then X and Yare statistically independent, and T(X:Y) =
0.0. This correspondence measure is symmetric, T(X:Y) = T
(Y:X). The maximum value of T(X:Y) is the smaller
variability of the two sets: T(X:Y) ≤ min{H(X), H(Y)}.
Psychophysical experiments cannot directly access
forms or structures in <, so we rely instead on observable
correspondences such as T(O:O''). “Low-level” visual
resolution of spatial structure may be evaluated as T(I:I'),
and perceptual discriminations of environmental objects
may be evaluated as T(O:O''). One aim of psychophysical
experiments is to discover what vision sees best—that is,
what spatial structures maximize T(I:I') and T(O:O'').
Uncertainty
Uncertainty refers in this article to the set of forms
corresponding to a given form in another structure. When
a map from one structure to another is many-to-one, the
inverse map is uncertain. Uncertainty implies ambiguity. In
information theory terms, the present use of “uncertainty”
refers to conditional uncertainty—for instance, H(O'|I), the
set of objects that potentially correspond to a given image.
Image ambiguity is basic in analyzing spatial vision.
Geometric uncertainties are especially relevant. Pro-
jected images obviously provide information about 3-D
spatial structure, but images have inherent ambiguities.
Perceptual uncertainties, H(O''|<), often arise from image
uncertainties, H(O'|I). Environmental and perceived objects
necessarily correspond less than objects and images, T(O:
O'') < T(O:O'), but the differences can be small.
Invariance
Invariance refers to the permissible transformations of a
structure that do not alter its correspondence with another
structure. Indeed, “structure” is defined by the groups of
transformations that leave it invariant. Thus, invariance is a
fundamental aspect of the concept of information, and it is
fundamental in identifying information in natural systems.
The set of potential information-carrying structures is
sharply reduced by the requirement that visual resolution
is maintained under a particular group of transformations.
Perceptual constancies of an object’s shape under
observational transformations—such as changes in view-
point, illumination, and so forth—are not just puzzles
awaiting explanation; they define perceived “shape.” The
invariance criterion is essential in experimental research on
shape perception. The experimental aim is to test whether
discriminations between objects are robust under changes in
viewpoint, for example.6
In theoretical physics and mathematics, invariance is
often called symmetry—equivalence of structure after
Perceptions
Objects Images
g-1
g
f
f -1
h
Fig. 1 Visual information involves structural correspondences be-
tween multiple physical domains: environmental objects, retinal
images, and perceptions. Objects defined by the inverse map from
images to objects, f–1: I → O', differ from the input environmental
objects, O ≠ O', and a third set of objects, O'', is defined by behavioral
discriminations, h: <→ O''. Similarly, behaviorally discriminated
images, g–1: <→ I', also differ from the input retinal images, I ≠ I'.
Psychophysical analyses of visual information transmission are based
on empirical correspondences such as T(O:O")
6 Consider the map m=hgf from environmental objects to perceived
objects, m: O→O''. Suppose that t: O→O is a group of trans-
formations that map objects in O onto other objects in O. If T[t(O):
O''] ≈ T(O:O''), then we say that the correspondence between O and
O'' is invariant under the transformation group t. This is an
experimentally testable criterion. The method is especially powerful
if the transformations are randomly varied, where H[t(O)] > > H(O).
Atten Percept Psychophys

specific groups of transformations. All physical laws may
be expressed as symmetries—as structural relations that are
conserved under specific groups of transformations of
observational parameters (Lederman & Hill, 2004). Con-
servation of energy, for example, is equivalent to the
invariance of physical interactions under shifts in time of
occurrence. Symmetry has several (related) meanings and
applications in mathematics, which generally involve
equivalent structures of different mathematical objects.
Symmetries between different mathematical systems, such
as algebra and geometry, offer methods for revealing
patterns in one structure through investigations of another
(Descartes, 1637/1886; Stewart, 2007).
Invariance (symmetry) is a powerful research tool,
because a structure may be deduced from its invariance.
Rather than inducing invariance from empirical observa-
tions, one can postulate invariance under a specific
transformation group and then search, theoretically and
experimentally, for a structure that satisfies the invariance.
Symmetry is used in this way in physics and mathematics.
In measurement theory, “uniqueness” of the numerical
representation specifies the scale of measurement (Luce,
Krantz, Suppes, & Tversky, 1990, chaps. 20 & 22; S.S.
Stevens, 1951; Suppes & Zinnes, 1963). In psychology, the
methods of converging operations (Garner, Hake, & Eriksen,
1956) and the receiver operating characteristic (e.g., Swets,
1996) exemplify use of the invariance criterion, but other
uses have been limited. Importantly, visual information about
the shapes of environmental objects is identified by its
invariance under changes in viewpoint, illumination, and
context.
Measuring resolution
Psychophysical resolution may be measured in many ways.
Shannon’s statistical measure of transmitted information is
general: It involves merely discrimination among alternatives
and is applicable to any relational structure—symbolic,
spatiotemporal, numeric, and so forth. Information often
involves relations of order, geometry, or so on, and resolution
measures may usefully reflect such structure. In fact,
information theory measures such as T(X:Y) are rarely used
in contemporary psychophysics, and were not used in the
experiments reviewed in this article.
Spatial vision often involves ratio-scaled dimensions
such as length. The Weber fraction, Δs/s, is often useful in
such cases—where Δs is a discriminable difference in, say,
spatial position, and s is a spatial length. The Weber fraction
is a dimensionless measure applicable to many physical
variables. The comparative visual resolution of various
variables (length, depth, slant, curvature, etc.) may be
evaluated by their Weber fractions. Large Weber fractions
(e.g., 15% or greater) indicate poor resolution.
A similar measure is the coefficient of variation, SD/M,
where SD and M are the standard deviation and mean of a
positive ratio-scale variable. If one uses the standard
deviation as the numerator of the Weber fraction, Δs, then
the Weber fraction and the coefficient of variation are
equivalent measures applicable to different experimental
procedures.
A related approach is used in signal detection theory,
where resolution is scaled in units of standard deviations.
The Weber fraction and the coefficient of variation can be
computed from a pair of physical values for which d' = 1.0.
Another version of the Weber fraction is the familiar
contrast ratio, C = (x1 – x2)/(x1 + x2), where x1 and x2 are
the values of two stimuli on a positive-valued ratio-scaled
dimension. If the difference in stimuli is such that
discrimination accuracy, d'(x1, x2) = 2.0, then each of the
two stimuli differs from their mean, (x1 + x2) / 2, by one
standard deviation.
Part 2. Visible information about surface structure
Image information
Image resolution of surfaces
Given the vast spatial and material differences between
objects and their optical images, what spatial commonalities
support the perceptual effectiveness of images? A basic
answer is that the spatial structure of images corresponds to
the spatial structure of environmental surfaces.
Koenderink and van Doorn contributed this fundamental
idea (Koenderink, 1984a, 1990; Koenderink & van Doorn
1975, 1976a, 1976b, 1976c, 1980, 1992a, 1997; Koenderink,
van Doorn, Kappers, & Todd, 2001; Lappin & Craft, 2000).
Key aspects include:
1. Environmental surfaces and their images are both 2-D
manifolds. Both are differentiable, and both can be
described by spatial derivatives.
2. Spatial relations on smoothly connected regions of a
surface and on generic images of those regions are
related to one another by local linear coordinate
transformations. These coordinate transformations vary
smoothly over the surface, with the relative orientation,
curvature, and shape of the surface.7
7
“Smooth” surfaces are differentiable almost everywhere. Polyhedral
objects in which planar faces join at a sharp edge may be treated as
smooth, since they are differentiable up to an arbitrarily small and
spatially isolated region at the edge. This description of smoothly
curved surfaces includes polyhedra, such as those analyzed by Pizlo
(2008). Pizlo’s analysis concerned mainly global shape properties,
whereas the present analysis is primarily local.
Atten Percept Psychophys

Thus, the differential structures of smooth regions of
a surface and its image are approximately isomorphic
(“diffeomorphic”). There is a smooth map from one
manifold onto the other with a smooth inverse—say,
f: O → I, and f–1: I → O, where f and f–1 are both
smooth. See Fig. 2. Violations of this isomorphism can
result from several causes, including random noise and
“accidental” violations of spatial continuity.
Nevertheless, a given image corresponds to a group of
permissible surfaces, which share a common spatial
structure but differ in relative depths (as discussed below).
3. The structure of a given surface may be defined by (a)
multiple image variables—motion, binocular disparity,
texture, shading, boundary contours—and (b) multiple
images—from different viewpoints, focal lengths, illu-
minations, contexts, observer movements, and so on.
Importantly, multiple images of the same surface are
linearly related to the same underlying structure. Thus,
adding variables and images usually reduces uncertainty
about the surface (Koenderink & van Doorn, 1997).
Spatial uncertainties about objects given their images
Images in our eyes and in pictures yield compelling
impressions of spatial structure, and images provide real-
time guides for our actions. Nevertheless, images have
inherent ambiguities (see Koenderink, 2001). Spatial
uncertainties, H(O'|I), arise from several sources:
Optical noise, resolution, and scale space: At the photonic
level, optical information is statistical. Light reflected from
a given point on a surface is spatially distributed in the
image by scattering, diffraction, and refraction.
Information about the spatial structure of images is also
affected by the sampling density of photoreceptors and by
neural encoding of “local signs.” These factors influence most
aspects of spatial vision—for instance, acuity, peripheral
vision, low vision, amblyopia, “tarachopia” (Hess, 1982),
and the apparent visual field (Koenderink, van Doorn, &
Todd, 2009). Image motions add information about spatial
order that partially compensates for neural limitations
(Lappin, Tadin, Nyquist, & Corn, 2009).
An important literature on “scale space” deals with the
effect of image resolution on representations of surface
structure (Florack, 1997; Koenderink, 1990). The basic idea
will suffice here: Complexity of surface topography in
images should be monotonically related to image resolution
(Koenderink, 1990, chap.9). Discrete pixels can introduce
high-frequency artifacts (jagged curves), but local smoothing
operators can remove these artifacts.
Spatial continuity: Spatial continuity is potentially ambiguous
in three situations.
1. Disconnected surface regions may appear connected in an
image, when boundary contours of separate surface
regions happen to coincide. The “Ames chair” is an
illustration (Ittelson, 1952). Such “accidental views” are
rare, however, and disappear with shifts in viewpoint.
2. More commonly, physically connected surface regions
may be disconnected in an image, due to occlusion by a
nearer surface. The relative image locations of these
discontinuities change with the viewer’s position, however.
3. In experiments and occasionally in natural images,
surfaces may be represented only as interpolated relations
among spatially separate texture elements. Psychophysi-
cal experiments with “point-light walkers” (e.g., Blake,
1993; Fox & McDaniel, 1982; Johansson, 1973) and
random-dot stereograms (Julesz, Papathomas, & Phillips,
2006) and movies (e.g., Braunstein, 1962; Lappin &
Bell, 1976; Lappin & Craft, 2000; Rogers & Graham,
1979; Turner, Braunstein, & Andersen, 1995) have
demonstrated that coherent image motion facilitates
perception of connected structure.
Image shading: Changes in surface orientation cause
changes in image illuminance, but changes in image
Fig. 2 Diffeomorphic spaces.
The rectilinear coordinate space
on the left may be transformed
into the space on the right by
a smoothly varying linear
coordinate transformation. As
should be evident, these
coordinate transformations
describe local surface shape
Atten Percept Psychophys

intensity are also caused by the field of illumination,
viewpoint, and surface scattering. The surface locations of
image highlights are not fixed but vary with the directions of
illumination (a hemispherical field) and viewpoint. Binocular
highlights have different locations in each eye, and the
highlights change with movements of observer and object.
Only in rare cases, with Lambertian scattering, are variations
in relative intensity independent of the viewpoint.
Nevertheless, the second-order differential structure of
the image (associated with surface curvature in two
dimensions, as described in the next section on the
invariance of surface shape) tends toward invariance with
illumination and viewpoint—as illumination directions are
dispersed, as surface scattering increases, as viewing
distance increases, and at discontinuities between surfaces
(Koenderink & van Doorn, 1980). The extrema of image
shading tend to occur near changes in surface shape (at
parabolic lines—see Koenderink & van Doorn, 1980).
Shading in a given image region may be affected by
multiple sources and surfaces—indirect reflections, shadow-
ing, and intervening media such as glass and fog. Uncertain-
ties about local surface structure are reduced by additional
image data—wider field of view, added viewpoints, additional
objects with the same illumination, surface texture, and so
forth. Distinguishing the multifactored influences of the
illumination field, surface reflectance, and viewpoint is akin
to solving simultaneous linear equations (Anderson &
Winawer, 2008; Koenderink, Pont, van Doorn, Kappers, &
Todd, 2007; Koenderink & van Doorn, 1997; Pont, 2011).
Image shading does not determine the depth scale (bas-
relief) of the surface (Belhumeur, Kriegman, & Yuille, 1999;
Koenderink et al. 2001; Todd, 2004). Stretching an object in
depth does not necessarily alter its shaded image.
Linear perspective: The depth scale of photographs and
other images depends on parameters such as focal length
and the distance of the object. Without prior knowledge of
global scene and object structure, image structure alone
does not determine 3-D lengths, directions, angles between
lines, or slants of planes. This basic image ambiguity is
readily illustrated by comparing photographs of the same
scene using lenses of different focal lengths: The photos are
clearly different, but usually none appears either distorted
or uniquely “correct.” Imaging parameters such as focal
length and vantage point are ambiguous in pictures and
even in movies, even though they appear subjectively
unambiguous in specific cases (Cutting, 1987).
Linear perspective and depth scale are also uncertain in
retinal images, despite the limited range of focal length.
Many experiments have found that depth judgments are
inconsistent between observers, objects, viewpoints, and
sometimes even repeated observations by the same observer
(e.g., Koenderink, van Doorn, & Kappers, 1996b; Lappin,
Shelton, & Rieser, 2006; Norman, Crabtree, Clayton, &
Norman, 2005; Norman, Norman, Lee, Stockton, &
Lappin, 2004; Norman & Todd, 1996, 1998; Norman,
Todd, Perotti, & Tittle, 1996; Todd, Koenderink, van Doorn,
& Kappers, 1996; Todd & Norman, 2003; Wagemans, van
Doorn, & Koenderink, 2011).
Summary: Key aspects of image uncertainties include the
following: (1)Image structure does not specify metric
structure in the environment; spatial scale in depth
relative to the frontal plane is indeterminate. (2)Image
structure has a robust correspondence with the connect-
edness and topography of object surfaces. (3)Image
uncertainties about environmental surface structure de-
crease with added image data, especially from moving
observers and objects.
Invariance of surface shape
What specific aspects of surface structure reliably corre-
spond to image structure, at least approximately invariant
with changes in viewpoint and focal length? The answer:
Local surface shape.
Simpler spatial features of surfaces, associated with
relative depth (zero-order structure) and slant (first-order
structure), do not satisfy the invariance criterion for
information. The critical importance of local surface shape
is revealed by its invariance. These ideas about invariance
are illustrated in Fig. 3.
Surface shape is studied in the classical theory of differential
geometry, and more recently in the psychophysical literature
(see Gray, 1993; Koenderink, 1990; Koenderink & van Doorn,
1992b; Lappin & Craft, 2000; Phillips & Todd, 1996; Phillips,
Todd, Koenderink, & Kappers, 2003). Here we mention only
a few basic facts relevant to visual psychophysics.
At each point on a smooth (differentiable) surface,
local shape is defined by the two principal curvatures—
the maximum and minimum curvatures, κmax and κmin,
which are orthogonal on the surface. Surface curvature in
a given direction is given by a second-order spatial
derivative, measured as κ = 1/r, where r is the radius of a
circular arc that coincides with the surface at the given
point.
Curvature in any given direction is the rate of change
in direction of the surface normal relative to changes in
position along the surface. This curvature is a 3-D
property, not directly observable in 2-D images of the
surface. Nevertheless, the ratio of the principal curvatures
is intrinsic to the surface (i.e., is independent of a 3-D
reference frame), is defined in images of the surface, and
specifies the local surface shape.
Certain basic properties of surface shape are also given
by the Gaussian curvature, Κ = κmax κmin. Surface points at
Atten Percept Psychophys

which Κ = 0, Κ > 0, or Κ < 0 have qualitatively different
shapes; they are intrinsically different and have qualita-
tively different images. If Κ > 0, where κmax and κmin
have the same sign, then the surface is elliptic—concave
or convex, valley or hill. If Κ < 0, where κmax and κmin
have opposite signs, then the surface is hyperbolic—
saddle-shaped. If Κ = 0, then the surface is either planar or
parabolic; if κmax = κmin = 0, then the surface is planar (flat);
if Κ = 0 but κmax ≠ 0, then the surface is parabolic—
cylindrical, or flat in one direction but curved in others.
Parabolic points occur in continuous nested curves at
boundaries between elliptic and hyperbolic regions. Parabolic
lines describe the surface topography. The shape at every point
on a smooth surface is one of these four qualitative types
(elliptic, hyperbolic, parabolic, or planar). Figure 4
illustrates the surface topography of a randomly shaped
smooth solid object.
Among the possible numerical scales of shape is Koenderink
and van Doorn’s (1992c; Koenderink, 1990) shape index,
S ¼ 2=pð Þ arctan kmax þ kminð Þ= kmax  kminð Þ½ : ð6Þ
The shape index, S, scales shape variations by numbers in
the interval (−1,+1), as illustrated in Fig. 5. As the difference
in principal curvatures, kmax– kmin, approaches zero, then S
approaches +1 or −1, where curvature is equal in all
directions. This index scales qualitative surface variations
that are independent of size, depth, and orientation.
The “curvedness” at each surface location is given by the
combined values of the principal curvatures. Koenderink
and van Doorn’s (1992c) “curvedness index” is a useful
ratio scale of surface curvature:
C ¼ k2max þ k
2
min
 
=2
 1=2
: ð7Þ
Fig. 3 Local surface shape is
invariant with changes in
viewpoint. Relative depths
and 3-D orientation vary
with the viewpoint
Atten Percept Psychophys

Variations in shape and curvedness, S and C, can be
pictured as polar coordinates within a Cartesian coordinate
frame given by the two principal curvatures, as shown in
Fig. 6. Curvedness is a 3-D property, not intrinsic to the
surface, and not generally specified in images.
Summary: Image information about local surface shape
involves the second-order differential structure of
images. Local surface shape is an intrinsic surface
property, independent of a 3-D coordinate frame. Image
information about shape is invariant with the object’s
distance and orientation relative to the image. Depth
relief, curvedness, and slant, however, require a 3-D
reference frame. Images offer unreliable information
about these 3-D properties.
Perception and psychophysics of surface shape
What image information supports perception of surface
shape? This question may be answered by psychophys-
ical experiments, by evaluating the resolution, uncer-
tainty, and invariance of human discriminations. The
following psychophysical results converge with analyses
of image structure to indicate that vision is sensitive to
image information about local surface shape—involving
the second-order differential structure of images of
surfaces. Extensive psychophysical evidence is avail-
able about the roles of binocular disparity and motion
parallax in spatial vision, so we focus first on this
evidence.
Binocular disparity and motion parallax
Stereoscopic vision derives from spatial differences (“dis-
parities”) between two simultaneous images of an object
from different viewpoints. Similar interimage disparities
occur sequentially when an object and observer rotate in
depth relative to one another. In both cases, the spatial
disparities vary with relative depth, and both are visually
effective information about surface structure. To simplify,
Fig. 4 An illustration of the surface topography described by
Gaussian curvature. (Right)A randomly shaped smooth solid object,
which resembles one often used in studies by Norman and Todd (e.g.,
Norman et al., 2006). This shape is visible by virtue of its boundary
contours, texture, and shading. (Left)Red areas are elliptic (bumps or
dimples); green areas are hyperbolic (saddle-shaped); and the black
areas separating the elliptic and hyperbolic are parabolic, where a
principal curvature vanishes in one direction
Fig. 5 Local surface shapes can be described as a one-dimensional
variable, scaled here by Koenderink and van Doorn’s (1992c) shape
index (see the text). [The illustration is from Phillips & Todd, 1996,
Fig. 3, p.932. Copyright 1996 by the American Psychological
Association. Reprinted with permission.]
Atten Percept Psychophys

we focus on binocular disparity, but similar comments
apply to motion parallax.
What exactly is binocular disparity? The answer is less
obvious than one might expect. Differences in the spatial
properties of two optical images can be described in several
ways, but these possibilities are seldom considered in the
literature on binocular vision. The issue of defining
binocular disparity exemplifies the representation problem
in spatial vision research.
A common assumption is that binocular disparity is a
difference between the two retinal positions of a given
image feature. That is, the elementary spatial form is
assumed to be a single point (e.g., an individual texture
element), and its spatial positions in the two eyes are
regarded as given by the anatomy of the two retinas.
An alternative class of spatial representations involves
topological relations between a given point and its surround-
ing neighborhood. One may recognize at the outset that (a)
relative spatial positions might be represented in multiple
ways, involving relationships with various numbers of
surrounding points, and (b)we do not know beforehand,
without investigation, which of these possibilities best
represents the effective optical input for binocular vision.
The experimental evidence reviewed here shows that
binocular vision exhibits (a)very high resolution for
detecting disparities (ignoring for the moment exactly how
these are defined), (b)uncertainties about the depth scale for
disparities, with corresponding uncertainties about surface
slant and curvedness, and (c)invariance of perceived local
surface shape under image transformations associated with
3-D shifts in viewpoint. This evidence implies that
binocular vision is directly and especially sensitive to
second-order differential structure of disparities associated
with local surface shape.8
Resolution: Vision is said to have “hyperacuity” for
interimage disparities in spatial position (Westheimer,
1975, 1977, 1979). Hyperacuity is an apt term: Human
discrimination thresholds for binocular disparity and rela-
tive motion are well below seeming physical limits of about
30 arcsec, based on optical diffraction, the eye’s line-spread
function, and photoreceptor spacing.9
Monocular acuity for relative position is good, sometimes
as low as 30 arcsec. For example, without disparity, a 2° space
between two parallel vertical lines can be bisected with
precision (SD) near 30 arcsec, and the Weber fraction (SD/
separation) is about 1% over varied line separations and
configurations (De Valois, Lakshminarayanan, Nygaard,
Schlussel, & Sladky, 1990; Lappin & Craft, 2000).
Acuity for binocular disparity is much better, however.
Using the same three-line configuration, thresholds (SD) for
binocular disparity are about 10 arcsec for 2° reference-line
separations, and Weber fractions are below 0.3% (e.g.,
Lappin & Craft, 1997, 2000). Thus, the resolution of
binocular differences in spatial position is substantially
better than spatial resolution in either of the monocular
images alone! Another example: The disparity threshold is
below 5 arcsec for detecting a depth difference between two
adjacent planes of a 60-Hz dynamic random-dot stereogram
(Cormack, Stevenson, & Schor, 1991).
Hyperacuities for relative motion are quantitatively and
functionally similar to those for binocular disparity (Lappin
& Craft, 2000; Rogers & Graham, 1983). These hyper-
acuities are robust and well documented.
Uncertainty, Despite hyperacuity for detecting binocular
disparity, vision does not reliably represent magnitudes of
8 The definition of a relationship does not require definition of the
elementary components of the relationship. This is true for both
mathematical and sensory representations. The Weber fraction—a
dimensionless ratio—is a good example. Definitions and evaluations
of derivatives are not necessarily derived from the specific values
involved in a function or change. Visual representations of differential
structures of optical patterns do not require elementary retinal
positions of component points. Higher-order differential structure of
images may be measured directly by local operators that are higher-
order derivatives of Gaussians, without subtracting differences
between lower-order components (see Koenderink, 1990). Indeed,
measuring differences by subtracting measures of lower-order values
is vulnerable to noise: The variance of a difference between two
independent variables equals the sum of their two variances.
9 The mechanisms underlying visual hyperacuities for binocular
disparity and relative motion do not supersede physics, of course,
but explanations in the literature are often unclear. Evidently, the
mechanism involves spatial phase interference between coherent
physiological signals from the two eyes (Lappin & Craft, 1997, 2000).
Fig. 6 Koenderink and van Doorn’s (1992c) shape index, S, and
curvedness index, C, describe a two-dimensional space. [The
illustration is from Phillips & Todd, 1996, Fig. 5, p.933. Copyright
1996 by the American Psychological Association. Reprinted with
permission.]
Atten Percept Psychophys

disparity or depth. Indeed, perceived depth changes were
completely abolished when the binocular vergence and global
disparity of a 30° × 30° random-dot pattern oscillated over the
range of binocular fusion (Erkelens & Collewijn, 1985a,
1985b; Regan, Erkelens, & Collewijn, 1986). Perceiving
motion in depth requires relative disparity.
How reliable is the relation between perceived depth and
disparity? Two studies have shown that discriminations of
depth from disparity are imprecise, and worse than those
for monocular relative positions. McKee, Levi, and Bowne
(1990) found that increment thresholds for depth separa-
tions were much larger than similar monocular thresholds
for differences in width separation. Norman, Norman, Craft,
Walton, Bartholomew, Burton, Wiesemann, and Crabtree
(2008) tested depth-order discriminations for two targets at
varied base disparities. Thresholds were about 22% of the
base disparity.
How visible are temporal variations in disparity? Tyler
(1971) found that even with a stationary reference line,
disparity thresholds for motion in depth were two to four
times larger than those for monocular motion at
oscillation rates of 0.1–5 Hz. Monocular thresholds were
below 30 arcsec for speeds of 0.5 Hz and above, but
binocular disparity did not achieve hyperacuity at any
oscillation rate.
How visible are gradients of disparity and depth for
slanted surfaces? Stereoscopic slant perception is compli-
cated by the fact that many factors influence relationships
between surface slant and disparity and between disparity
and perceived slant—especially viewing distance, but also
stimulus size, vergence, cyclovergence, and direction of
slant (see Howard & Rogers, 2002, chap. 21). For example,
vertical surface slant (around the horizontal axis) is much
more visible than horizontal slant (e.g., Gillam & Rogers,
1991; Gillam & Ryan, 1992; Howard & Kaneko, 1994;
Rogers & Graham, 1983).
How does vision resolve ambiguities in the many-to-
many correspondence between surface slant and disparity
gradients? Some researchers have proposed that slant is
perceived by averaging depth estimates from individual
cues (e.g., viewing distance, horizontal shear, and vertical
shear), weighting each estimate by the cue’s reliability
(Backus & Banks, 1999; Banks, Hooge, & Backus, 2001).
Another possibility is that binocular vision is simply
insensitive to spatial gradients of disparity—consistent with
the inherent image ambiguity.
Indeed, stereopsis seems to add little to the limited
precision of slant discriminations without disparity. Norman,
Crabtree, Bartholomew, and Ferrell (2009) tested slant
discriminations, with large (22° diameter) random-dot stereo-
grams at a constant viewing distance. They found that stereo
slant thresholds were similar to those for texture and motion
gradients without disparity. Weber fractions averaged
(root-mean squared [r.m.s.]) 14% for 10 observers.10 Slant
discriminations for these planar surfaces were more precise
than those obtained when well-trained observers compared
the relative slants of two regions on randomly shaped solids
defined by binocular disparity, texture, and shading (Norman,
Todd, Norman, Clayton, & McBride, 2006). Even with the
added texture and shading cues, discrimination thresholds (at
d′ = 1.0) were about 5°–10°. Stereo slant discriminations
were modest in these studies but would have been worse if
viewing distances were randomly varied.
Invariance: What is the reference frame for visual infor-
mation about spatial position, binocular disparity, and
motion? The issue is empirical: What groups of trans-
formations leave perceived relative positions, depths, and
motions invariant? Hyperacuities for detecting interimage
disparities offer a sensitive experimental method for
identifying the spatial reference frame.
Ifmonocular images and binocular disparities were spatially
defined by retinal anatomy (e.g., Marr, 1982), then stereoa-
cuity and detection of differences in depth would vary with
the fixation point, viewing distance, ocular vergence, and
image motion. Experiments have clearly shown, however,
that large variations in retinal disparities are (a)not perceived
when applied globally over a large visual area and (b)have
little to no effect on stereoscopic acuity for detecting a small
local depth difference (Erkelens & Collewijn, 1985a, 1985b;
Lappin & Craft, 1997, 2000; Regan et al., 1986; Steinman,
Levinson, Collewijn, & van der Steen, 1985; van Ee &
Erkelens, 1996; Westheimer & McKee, 1978).
Lappin and Craft (2000), for example, independently
jittered each monocular image, with 10-Hz random hori-
zontal and vertical shifts; the horizontal image shifts
averaged (r.m.s.) 340 arcsec in each eye (= 455 arcsec
horizontal disparity). In a stereoacuity task, observers tried
to eliminate relative depth by adjusting a central vertical
line to be coplanar with two parallel outer reference lines
separated by 1°–8°. In a similar motion task (with no
binocular disparity), observers tried to eliminate additional
independent horizontal motions of the target line relative to
the outer reference lines. Random image jitter had almost
no effect on acuity for relative position in either task. When
the two monocular images were independently jittered in
the stereoacuity task, the average disparity threshold was
0.29% of the separation between the reference lines, as
compared to 0.22% for stationary patterns. (With reference
lines separated by 2°, for example, disparity thresholds were
13.2 and 12.4 arcsec for the jittered and stationary patterns,
10 This experiment tested both young and older observers, but here we
report performance of only the young observers. Weber fractions for
slant estimations by a palm board are defined here as SD/M of the
estimated slants, using the SD of slant estimations by individual
observers at a given slant, data not reported in the published article.
Atten Percept Psychophys

respectively.) Similarly, when the images were globally
jittered in the motion task, acuities for relative motion
averaged 0.18% of the reference line separation, as compared
to a Weber fraction of 0.06% for stationary patterns.
Thus, spatial information for binocular disparity and
relative motion is topological—defined by reference to the
surrounding image, not by anatomical retinal positions. The
simplest spatial relation involves pairs of points. Pair-wise
(two-point) structure may be described by first-order spatial
derivatives, Fourier power spectrum, and auto- or cross-
correlation. Pair-wise spatial relations are invariant with
global image translations but are perturbed by transformations
such as global dilations (expansion/contraction) or 2-D
rotation of the images. Lappin and Craft (1997, 2000) found
that acuities for detecting local depth and relative motion
were maintained when uncorrelated random dilations and
image rotations were added to the random translations
(Lappin & Craft, 1997, 2000). Thus, the topology of image
information for stereoscopic vision and motion perception
involves more than two-point relations.
Other possibilities include three-point forms, described by
second-order spatial derivatives and the Fourier phase spec-
trum. Triangular (2-D) three-point forms are invariant under
first-order image transformations—dilations and rotations—
but are perturbed by changes in surface slant. Thus, three-point
forms cannot provide direct information about surface shape.
As shown earlier, local surface shape is described by the
second-order 2-D differential structure of surfaces and
images. This second-order 2-D spatial form involves five-
point relations—describing the 2-D neighborhood around
any given point, as illustrated in Fig. 7.11 The following
experiments found that human vision is directly sensitive to
this higher-order spatial information about surface shape.
Perotti, Todd, Lappin, and Phillips (1998) found that
precise visual resolution of local surface shape was invariant
with image transformations produced by movements in 3-D
space. Motion parallax fields described smooth surfaces
rotating back and forth around a vertical axis. The surface
shapes were randomly varied: z = (κ1 x
2 + κ2 y
2)/2, where z
was the depth axis, κ1 and κ2 were principal curvatures,
randomly varied between patterns, and x and y were the
image coordinates. Schematic illustrations are shown in
Fig. 8. Observers estimated the shape and curvedness of the
moving surface by adjusting the curvatures (κs1 and κs2) of a
stationary stereoscopic surface.
Shape indices (Eq. 6) of the observers’ adjusted
stereoscopic surfaces corresponded to those of the moving
standard surfaces (average R2 = .995), but judgments of
curvedness (Eq. 7) were both imprecise (average R2 = .415)
and inaccurate. Representative results for 1 observer are
shown in Fig. 9. The precision of perceived shape and the
imprecision of perceived curvedness are consistent with the
image information.
Importantly, Perotti et al. (1998) tested the invariance of
perceived shape by three additional conditions, with added
transformations of first-order image structure—curl (2-D
rotation), divergence (expansion/contraction), and shear
(expansion and contraction in perpendicular directions,
associated with slant). Curl and divergence alter the first-
order spatial structure, and shear also deforms the triangular
three-point forms. None of these added image transforma-
tions reduced the visual resolution of shape: Correlations
between simulated and adjusted shapes in the curl,
divergence, and shear conditions, respectively, averaged
R2 = .997, .993, and .997. Visual demonstrations of this
invariance appear unremarkable: The same shape simply
has added rigid motions—rotating in the plane, moving
closer or farther, or slanting in depth. The phenomenon is
illustrated in Fig. 10.
11 The complexity of this elementary form is fourth-order. The “order”
of complexity is one less than the number of defining points—a scalar
array is zeroth-order, two-point structures are first-order, and so forth.
The shape-related forms of surfaces and their images are often referred
to as “second-order,” but there are two independent dimensions,
involving the relative magnitudes of two second-order relations
around a given point.
Fig. 7 Deformations of second-order 2-D differential structure at a
given image point produced by each of four different local surface
shapes when the object rotates horizontally relative to the observer.
Before the rotation, the initial pattern was circular, and the local image
deformations produced by the rotation are shown above. Each pattern
is centered on the axis of rotation; the center is a reference point that
does not move. Binocular disparities between the two monocular parts
of stereoscopic images involve the same image deformations. The
second-order 2-D differential structure of these patterns is invariant
under translation, 2-D rotation in the image plane, expansion or
contraction, and slant of the surface relative to the image. Such image
transformations produced by movements in 3-D space affect all lower-
order structure, however. As can be readily seen, the five shapes, from
left to right, are a plane, horizontal cylinder (parabolic), vertical
cylinder (parabolic), sphere (elliptic), and saddle (hyperbolic). [The
illustration is from Lappin & Craft, 2000, Fig. 3, p.14. Copyright
2000 by the American Psychological Association. Reprinted with
permission.]
Atten Percept Psychophys

Lappin and Craft (2000) tested the invariance of stereo
and motion hyperacuities for surface shape. Observers
adjusted 3-D positions of target points onto planar and
spherical surfaces. The patterns were perturbed by random
translations, dilations, and rotations, and the surfaces were
slanted in depth. The gradient for a slanted plane involves
first-order spatial relations, but the doubly curved sphere
involves a 2-D relationship among second-order relations.
Hyperacuities were obtained for the spherical surface as well
as for the plane. Stereo thresholds (SDs) averaged 12 arcsec for
the plane and 15 arcsec for the sphere; motion thresholds
averaged 19 arcsec for the plane and 20 arcsec for the sphere.
These results coincide with those of Perotti et al. (1998).
Summary: Visual resolution, uncertainty, and invariance of
interimage disparities in stereopsis and motion perception
converge with our previous analysis of image information:
Vision is directly sensitive to local surface shape, involving
second-order 2-D differential structure of the image dispar-
ities. The invariance tests demonstrate that this information is
not derived from lower-order image properties. Indeed, vision
is not very sensitive to the lower-order properties, especially
not the zero-order property of retinal position.
Boundary contours
Image contours correspond to surface points where the
viewing direction is tangent to the surface, perpendicular to
the surface normal. Thus, contour shapes are informative
about surface shapes. The sign of contour curvature is the
sign of the surface Gaussian curvature: Convex contours
occur at convex elliptic patches (bumps, hills); concave
contours, at hyperbolic (saddle-shaped) patches; inflections
in contour curvature, at parabolic surface curves; and
noncurved straight contours appear along edges of para-
bolic (cylindrical) shapes (Koenderink, 1984b, 1987; 1990,
pp.431–439). A famous drawing by Picasso, in Fig. 11, is
an elegant illustration. The same relationships are also
shown in the left panel of Fig. 4.
Studies suggest that contours are an especially effective
form of visual information about solid shape. Koenderink,
van Doorn, Christou, and Lappin (1996) and Norman,
Bartholomew, and Burton (2008) found that adding surface-
related information such as shading, texture, and motion
yielded only small improvements in shape judgments
beyond those obtained with boundary contours alone.
Wagemans, van Doorn, and Koenderink (2010) found that
contour shape strongly influenced perception of shape from
shading.
Silhouettes are devoid of all information about surface
structure except for the outside boundary contours, yet
they are often sufficient for perception. Attneave (1954)
suggested that visual information is concentrated at
contours’ curvature extrema, at both convexities and
concavities. Hoffman and Richards (1984) also showed
that the maxima of negative curvature (concavities) of
contours mark junctions of component parts. Norman,
Phillips, and Ross (2001) tested these hypotheses with
silhouettes of similarly shaped natural objects (potatoes),
which observers tried to represent as dotted figures of 10
points. As predicted, these descriptive points were located
mainly at the curvature extrema.
By definition, contour curvature involves a second-order
spatial derivative. Many scientists have suggested that
vision measures contour curvature by changes in (first-
order) orientation (e.g., Watt & Andrews, 1982; Wilson,
1985). Koenderink and Richards (1988) and Dobbins,
Zucker, and Cynader (1989), however, showed that
Fig. 8 Three stereoscopic illustrations of surface shapes used by Perotti et
al. (1998) to evaluate the perception of shape frommotion. (Top)Ellipsoid.
(Center)Saddle (hyperbolic). (Bottom)Ridge (almost parabolic, but with
slight curvature in the direction from lower right to upper left, making it
hyperbolic). [From Perotti et al., 1998, Fig. 3., p.381. Copyright
1998 by the Psychonomic Society. Reprinted with permission.]
Atten Percept Psychophys

second-order operators are biologically feasible and com-
putationally efficient.
Boundary contours depend on the viewpoint; they are
not fixed on the surface. When the observer or object
moves, the resulting image motions are quite different from
those of surface texture. Nevertheless, moving contours and
silhouettes carry visible information about solid shape, as
found in the experiments described below.
Resolution: Human vision is remarkably sensitive to
contour curvature and connectedness. Experiments typical-
ly have investigated 2-D contour shapes rather than surface
shapes, but the results probably generalize to surface shape.
Vision has hyperacuity for variations in contour curva-
ture. Wilkinson, Wilson, and Habak (1998) evaluated
thresholds for detecting sinusoidal modulations of the
radius of circular contours. For radial frequencies (cycles
per circumference) greater than 2 cycles, amplitude thresh-
olds were a constant fraction of the radius, with Weber
fractions averaging about 0.35%. Detection thresholds were
invariant with contour width, and robust under reduced
contrast. Modulation thresholds for circular contours were
similar to those for straight lines (Tyler, 1973).
Perceptual continuity.—Humans are sensitive to the
implied continuity of smooth contours defined by spatially
separate image segments and features. Field, Hayes, and
Hess (1993) demonstrated that smooth curves of separate
Gabor patches could be accurately detected in dense
random patterns of such patches. Perturbations of local
orientations rapidly reduced detectability of the target
contours. Even collinear dots will visually “pop out” of
dense random-dot backgrounds (Beck, Rosenfeld, & Ivry,
1989; Uttal, 1975). Global patterns of multiple curved
contours are readily visible in random-dot patterns when
neighboring pairs of dots specify smoothly varied local
orientations (Glass, 1969; Wilson & Wilkinson, 1998).
Perceptual closure.—Human vision is also sensitive to
the global property of contour closure: Closed contours can
be detected more rapidly than open contours (Elder &
Zucker, 1993, 1994) and have lower contrast detection
thresholds (Kovács & Julesz, 1994). The shapes of closed
contours often constitute visually effective information
about global shapes, especially those that are bilaterally
symmetric or have distinctive appendages.
Closed contours permit a global shape description in
terms of the medial axis of the outside contours. The medial
axis is a type of second-order differential property—a set of
points centered in a diffusion field bounded by the outside
contours. Theoretical and experimental results support the
visual utility of this description (Blum, 1973; Burbeck &
Pizer, 1995; Elder & Zucker, 1994; Pizlo, 2008). Boundary
contours and medial axis both change with the observer’s
viewpoint, but these deformations are informative about
solid shape.
Uncertainty, Because contours are visually important for
detecting and recognizing solid objects, contour ambigui-
ties are important in both biology and warfare. Literatures
on camouflage are relevant to the study of shape percep-
tion. A recent issue of the Philosophical Transactions of the
Royal Society B (vol. 364, 27 February 2009) has provided
helpful reviews of the biological literature. Articles by
Hanlon, Chiao, Mathger, Barbosa, Buresch, and Chubb
(2009), M. Stevens and Merilaita (2009a, 2009b), Tankus
and Yeshurun (2009), and Troscianko, Benton, Lovell,
Tolhurst, and Pizlo (2009) examined contour camouflage.
Two camouflage mechanisms that conceal boundaries
are background matching and disruptive coloration. Dis-
Fig. 9 Representative results for 1 observer in the study of Perotti et
al. (1998). The left graph shows the observer’s adjusted shape
characteristics as a function of the simulated (randomly generated
and displayed by the computer) shape characteristic of the displayed
surface pattern. The right graph shows the adjusted versus simulated
curvedness values for the same data points in the left-hand graph.
Clearly, the observer accurately and reliably matched the shape but not
the curvedness of these surfaces. [From Perotti et al., 1998, Fig. 4,
p.382. Copyright 1998 by the Psychonomic Society. Reprinted with
permission.]
Atten Percept Psychophys

ruptive coloration adds misleading contours unrelated to an
object’s shape and boundary contours. Disruptive coloration
can be effective even when it increases contrast with the
background (Hanlon et al., 2009).
Another potential ambiguity of contours involves the
similarity between smoothly curved surface boundaries and
sharp edges of planar “cutout” figures. The surface
locations of image contours usually change with the
observer’s viewpoint, but the edges of planar figures remain
in essentially the same surface location. Thus, the image
uncertainty usually disappears with changes in viewpoint.
From a single viewpoint, however, the ambiguity may be
important. Indeed, flat decoys resembling planes and
buildings on runways when seen from above were used in
WWII to misdirect or delay attacking pilots.
Invariance: Boundary contours and silhouettes change with
the relative positions of a solid object and its image. Many
experiments have found that human observers can identify
and discriminate solid shapes seen only as silhouettes, even
when viewpoints are varied (Newell & Findlay, 1997;
Norman, Bartholomew, & Burton, 2008; Tjan, Braje,
Legge, & Kersten, 1995; Wagemans et al., 2008).
The silhouettes of solid objects change when the objects
rotate in depth. These contour motions are very different
from those of surface texture, because they correspond to
changing surface locations as the object rotates. Neverthe-
less, observers typically perceive the underlying object
shape (Norman, Bartholomew, & Burton 2008; Norman &
Raines, 2002). Norman, Lee, Phillips, Norman, Jennings,
and McBride (2009) demonstrated this visual capacity by
Fig. 10 Stereoscopic illustration
of the invariance of perceived
shape under added image
transformations produced by
2-D rotation (“curl”) and shear
(“def”). Surface shape and
grayscale shading were random
and mutually independent. (Top)
Undistorted stereo, with right
image rotated in depth around the
vertical axis by about 5°. (Center)
Right image rotated 7°. (Bottom)
Right image horizontally
expanded and vertically
compressed about 7% in each axis
Atten Percept Psychophys

showing that observers accurately discriminated among
similarly shaped natural objects (bell peppers) rotating in
depth by seeing their moving shadows.12 Observers evidently
perceived invariant solids rather than changing contours as
such — because their discriminations were accurate even
when the shadows were projected onto curved background
surfaces that altered the shapes and motions of the shadows.
Textured images
When surfaces are covered with dense, isotropic, and
homogeneous texture, image variations in the orientations,
shapes, and spacing of texture depict the coordinate
transformations that map surface space to image space.
This image information resembles that in stereopsis and
relative motion, but is also influenced by the isotropy,
shapes, and contour directions of texture elements.
A visually important role of texture may be simply to
mark fixed surface positions that are invariant with changes
in viewpoint. Unlike many human-made objects, most
natural scenes are cluttered with organic and inorganic
natural materials. Textures may be inherent in the material
structure of an object, or the elements may be physically
distinct objects scattered over the surface; may be random
or regular, contoured, polygonal, or blob-like; may be
exposed by fracturing or carving solid objects that contain
other shapes; and so forth. The variety of natural textures
seems endless. The visual validity of assumptions about the
spatial homogeneity and isotropy of surface texture seems
dubious. Figure 12 shows a few examples, but these are
hardly special. Importantly, perceived surface shape seems
to be robust over wide variations in texture characteristics
that violate theoretical assumptions such as homogeneity
and isotropy (Todd & Oomes, 2002; Todd, Oomes,
Koenderink, & Kappers, 2004).
Surface textures may also add image contours, and the
directions of these contours may carry visual information
about surface shape (e.g., Knill, 2001; Todd & Oomes,
2002; Todd & Reichel, 1990; Zaidi & Li, 2002). Contour
orientation is often considered important visual informa-
tion, but information about surface curvature is probably
associated with contour curvature (a second-order space
differential property, as illustrated in Figs. 7 and 11) rather
than the first-order orientation as such. Todd and colleagues
(Todd & Oomes, 2002; Todd & Reichel, 1990) have found
that perceived shape from texture contours is quite robust.
Resolution: Todd et al. (2004) found that even inhomoge-
neous and anisotropic textures can provide reliable judgments
of surface structure. Six different textures were applied to four
randomly shaped solid objects. (See Fig. 13.) In one task,
observers moved colored dots horizontally along a given
latitudinal scan line to identify the local near and far points at
that latitude. In another task, dots were equally spaced
horizontally over the textured image, and observers adjusted
their vertical positions on a second blank monitor to scale the
perceived depths at the given positions.
Spatial resolution was not directly reported, but the
reported correlations indicate that all six surface textures
yielded reliable judgments of surface structure. The average
judged locations of near and far points were highly correlated
with the correct locations, r2 = .985. Pair-wise correlations
between different textures for the same object averaged r2 =
.940, and pair-wise correlations between observers for given
stimuli averaged r2 = .949. The average judged depths also
correlated with the correct values (r2 = .902), but the depths
were underestimated, averaging 45% of the correct values.
The depth judgments of different observers ranged from
25%–62% of the correct values, although correlations
between observers were high, r2 = .865. Different textures
on the same object yielded similar and correlated depth
judgments (r2 = .909).
Uncertainties: A principal uncertainty of texture information
is the depth scale. James Gibson (1950) and many others
since have suggested that gradients (first-order spatial
12 Cast shadows and boundary contours are slightly different, though
both are effective sources of information about solid shape. Both
derive from space curves on the surface of solid objects, but the space
curves are different because they involve tangents originating from
different directions in 3-D space (Norman, Lee, et al., 2009).
Fig. 11 A famous drawing by Pablo Picasso, Fragment de corps de
femme, elegantly illustrates the correspondence between the curvature
of 2-D boundary contours and 3-D surface curvature. The drawing
depicts primarily elliptic (ovoid) surface regions, but also the
beginning of a hyperbolic (saddle-shaped) region at the upper right,
and a parabolic line at the inflection between hip and waist. Parabolic
lines would also occur at the ends of the contours on the left side of
the right buttock (Koenderink & van Doorn, 1982)
Atten Percept Psychophys

derivatives) of texture density constitute information about
surface slant. Slanted planes are usually clear in illustrations,
but slant discriminations usually are not reliable. Norman et
al. (2006), found that Weber fractions (SD/M) for judging
differences in local slant (in stereo images of textured
randomly shaped solids) averaged only about 40%.
One limitation is that image texture gradients are often too
small to resolve local slant. Another limitation examined by
Todd and colleagues (Todd, Thaler, & Dijkstra, 2005; Todd,
Thaler, Dijkstra, Koenderink, & Kappers, 2007) is that the
equations relating surface slant to variations in texture
density are expressed in units of visual angle. Accordingly,
Fig. 12 Textures provide visible markings of positions on surfaces
and evidently carry information about the spatial structure of surfaces.
Textures in cluttered natural scenes occur in an endless variety of
configurations and scales—for instance, contours, scattered debris,
volumetric textures exposed by cutting or fracturing, and pits and
holes. Uniform distributions of texture elements with the same size
and shape are rare in most biological and geological structures, though
more common in human-made structures. Correlations between
texture patterns and surface structure are not immediately obvious in
many natural scenes
Fig. 13 Six different textures,
both dotted and contoured,
applied to one of the solid
objects in the study of Todd
et al. (2004). [From Todd et al.,
2004, Fig. 1, p.41. Copyright
2004 by the Association for
Psychological Science.
Reprinted by Permission of
SAGE Publications.]
Atten Percept Psychophys

the map from image texture to surface slant varies with the
combined perspectives (focal length) of both a photograph
and of an observer of the photograph; and these two
perspective transformations are rarely the same.
Image size is (approximately) inversely proportional to
object distance. Therefore, contrasts in image size corre-
spond to contrasts in distance,
wN  wFð Þ= wN þ wFð Þ  dF  dNð Þ= dF þ dNð Þ ð8Þ
where wN and wF are the angular image widths (perpen-
dicular to the viewing direction) of the same texture
element at near and far depths, respectively, and dN and
dF are corresponding distances from the eye (Gårding,
1992; Purdy, 1958; Todd et al., 2007). As described by
Equation (8), the depth contrast for a given pair of surface
elements decreases as viewing distance increases. There-
fore, uncertainty about slant increases with uncertainty
about viewing distance.
In recent experiments, Todd et al. (2007) found that
observers’ judgments of slant in computer-generated images
of hyperbolic cylinders were accurately predicted by Equation
(8). Slant judgments were reliable—with an average correla-
tion of .95 between judgments of the same stimuli in different
experimental sessions—but judged slants deviated systemat-
ically from the correct values. Slant judgments varied with
perspective and with convexity/concavity; and depths of
concave surfaces were greatly underestimated.
Invariance: Insofar as image information for perceiving
shape from texture may involve the deformations of second-
order structure described in Fig. 7, this information should be
invariant under the image transformations produced by
perspective changes associated with distance, direction, and
focal length. The studies of Todd and colleagues (Todd &
Oomes, 2002; Todd et al., 2004; Todd & Reichel, 1990; Todd
et al., 2007) have suggested that perceived shape from
texture may satisfy invariance under these image trans-
formations, but such invariance has not been tested directly.
Image shading
Surface curvature causes variations in image shading, but
two other global factors also influence shading: the
irradiating field of light and the scattering of light by the
surface material. Unlike shape information from stereopsis,
motion parallax, or boundary contours, shading information
is global, not local! Koenderink and van Doorn (2004) have
reviewed the fundamentals of image shading. Here, we
highlight selected basic aspects.
Irradiance converges at a given surface location from a
hemisphere of directions. This irradiance field is composed
of (a)direct illumination, (b)indirect reflection from other
surfaces, and (c)shadows and partial occlusion. Direct
illumination can be approximately unidirectional (e.g.,
direct sunlight), from multiple local or extended sources
(e.g., rooms with multiple lamps), from a diffuse hemi-
sphere (e.g., overcast sky), or countless combinations.
Additionally, the indirect contributions of reflections,
shadows, and partial occlusion by neighboring surfaces
may be as great as the direct illumination. The irradiation at
a location in a natural scene is a flow field structured in part
by the surrounding scene. Image shading is an ecological
phenomenon (Gibson, 1950, 1979; Koenderink et al., 2007;
van Doorn, Koenderink, & Wagemans, 2011).
Reflection and scattering from a given surface material
are described by its bidirectional reflectance distribution
function (BRDF). The BRDF is the ratio of radiance
scattered in a given direction relative to the incident
irradiance from a different given direction. Thus, for a
given surface location, the BRDF ratio (output radiance/input
irradiance) depends on four independent parameters—the
azimuth and elevation angles of both the incident and
reflected rays. The directions of the incident and reflected
rays correspond to points on a hemisphere in which the
central pole is the surface normal, and BRDF ratios are
defined on the four-dimensional product of two such
hemispheres of directions. The BRDF generally also
depends on wavelength, and it is an aspect of color vision,
but we ignore that aspect here. The BRDF is the visual
signature of a surface material (Koenderink & van Doorn,
1998; Koenderink, van Doorn, Dana, & Nayar, 1999; Oren
& Nayar, 1995), but the complexity of the BRDF makes it
difficult to measure.
An idealized model known as a Lambertian surface
scatters light uniformly in all directions, independent of the
directions of reflection and incidence. The reflected radiance
(photons/time/area) is then proportional to the incident
irradiance, and the BRDF is a constant. In this special case,
image intensity decreases with the angle between the surface
normal and the direction of irradiation, providing informa-
tion about variations in surface orientation.13 This model is
conceptually simple but physically unusual.
Surface scattering increases with the microscopic rough-
ness of a surface (Tadin, Haglund, Lappin, & Peters, 2001).
Very smooth surfaces (metals, glass, water) are shiny,
reflecting light like a mirror in a tightly constrained bundle
of rays. Such shiny surfaces are called specular. Surfaces
are sometimes modeled in computer graphics and psycho-
physical experiments as the sum of two components,
13 The surface area irradiated by a source in a given direction
increases proportionally with the cosine of the angle of incidence.
Thus, the radiant intensity reflected from a given surface location
decreases inversely with the angle of incidence.
Atten Percept Psychophys

Lambertian plus specular, but this is an oversimplification for
most surfaces. Scattering usually is centered on the specular
direction, with greater dispersion for rougher surfaces.
Specular surfaces produce highlights—images of incident
light from an angle of incidence equal to the angle of
reflection. Thus, highlights carry information about surface
orientation. Highlights aid perception of surface shape
(Norman, Todd, & Orban, 2004).
Variations in shading carry image information about
microscopic roughness, macroscopic texture, surface color,
and boundary contours. Figure 14 offers a few illustrations.
Shading is not an independent physical cue. Roughness,
texture, and boundary contours may be considered aspects
of image shading information. Experimental separation of
shading from other surface properties may be misleading.
Resolution: The current experimental literature does not
establish the visual resolution of surface shape from
shading. Koenderink and van Doorn (2004) concluded,
“There is remarkably little psychophysical material on
shape from shading that might be considered in any way
definitive” (p.1100). “Perhaps the majority of the literature
is irrelevant due to incomplete description of the stimuli, . . .
extreme stimulus reduction, . . . or invalid paradigms due to
incomplete understanding of the relevant geometry and
photometry” (pp.1101–1102).
When variations in local shape are experimentally
isolated from global information about the irradiance
distribution and surrounding surface structure, local shape
is very poorly resolved (Erens, Kappers, & Koenderink,
1993a, 1993b). In the Erens et al. (1993b) study, observers’
categorical shape judgments (eight regions of the shape
index—Eq. 6, Fig. 5) were essentially random. When a cast
shadow revealed the illumination direction, then concave
and convex shapes were discriminated, but elliptic and
hyperbolic shapes were still confused.
Perceiving shape from shading involves simultaneous
perception of both the surface and the surrounding light
field. A recent study by Koenderink et al. (2007) found that
vision is quite sensitive to the structure of light fields in
natural scenes. Observers adjusted the direction, diffuse-
ness, and intensity of illumination of a gauge object
(Lambertian sphere) at various positions in a multiobject
scene. They easily and accurately matched the illumination
of the gauge object to the local physical parameters of the
light field. Correlations (r2) between the physically correct
and adjusted shading values averaged (over conditions and
observers) .77 for the direction and diffuseness parameters,
and .42 for the intensity. Quantitative details suggested that
shaded images of natural scenes support perception of local
surface shape, but shape discriminations were not evaluated
in this experiment.
Fig. 14 Images are composed of light reflected and scattered from
surfaces. Image shading depends on the surface orientation but also on
many other factors, including the spatial distribution of illumination,
the direction of view, shadows, reflections from other surfaces,
occluding transparent materials, and the reflectance characteristics of
the surface. Sand, water, wood, flowers, and feathers have very
different reflectance characteristics. Roughly textured sand and wood
scatter light broadly; the smooth surface of water reflects light in a
narrow distribution of directions; and scattering by flowers and
feathers is intermediate between these extremes. The flower at the
lower left is illuminated from behind, thus appearing luminous. [The
two bottom photos were contributed by Dominic Ali.]
Atten Percept Psychophys

Uncertainties: Varied combinations of surface shape, BRDF,
and illumination may produce the same patterns of image
shading. Like most forms of image information about surface
structure, both the absolute distance and relative depth
variations across surface regions are ambiguous in shaded
images.
Belhumeur et al. (1999) presented a clear analysis and
demonstration of this “bas-relief ambiguity” of depth from
shading. This ambiguity includes convexity versus concavity
and, in certain cases, even connections in depth (van Doorn et
al., 2011).
Perceived convexities and concavities of surface
regions are known to vary with perceived directions of
illumination (e.g., Kleffner & Ramachandran, 1992;
Ramachandran, 1988; van Doorn et al., 2011). A popular
idea is that perceived convexity versus concavity involves
an automatic visual “assumption” or Bayesian “prior” that
illumination is from above. The important point, however,
is that the image information is inherently ambiguous.
Moreover, experiments have found significant variability
between individuals, within individuals, strong effects of
the outer contour shape, and both interactions and
inconsistencies among spatially separate surface regions
(Kleffner & Ramachandran, 1992; van Doorn et al., 2011;
Wagemans, van Doorn, & Koenderink, 2010).
Invariance: In general, perceived surface shape is not
invariant with changes in shading (Koenderink & van
Doorn, 2004). Changing the illumination of a given surface
may produce locally varied changes in relative depth and in
convexity versus concavity. Often, the changes are subtle,
with the topographical complexity and global shape
appearing similar after a change of illumination, but
changes in local details may become visible under closer
scrutiny. Careful control of illumination is important to
professional photographers, and commercial success in the
cosmetics industry involves controlling skin reflectance to
produce perceived changes in facial structure.
Locations of highlights vary with viewpoint. When a
shiny surface is viewed stereoscopically, the highlight can
sometimes be seen in depth off the surface. This stereo
effect can alter the perceived shape of a shiny surface
(Muryy, van Mierlo, Fleming, & Welchman, 2011; Nefs,
2008), though vision is usually robust to such effects.
Surface texture, scattering, and diffuse illumination tend to
counteract the potential uncertainties caused by the
viewpoint-dependent effects of highlights (Nefs, 2008;
Todd, Norman, Koenderink, & Kappers, 1997). Norman,
Todd, and Phillips (1995) found that image transformations
produced by binocular vision and observer/object motions
aided in recovering invariant surface structure from shaded
images. Perceived shapes of shaded surfaces are usually
stable under relative movements of observer and object and
under changes in illumination (Koenderink & van Doorn,
1980), but this tendency is not a generalized invariance.
Combining multiple forms of image information
Information about surface structure is carried by multiple
image variables—binocular disparity, motion parallax,
boundary contours, texture, shading, and so forth. These
variables carry amounts of information that vary both
within and between scenes, and between sensory channels
that detect this information. How, then, does vision
combine multiple sources of information? This is a basic
issue in sensory and perceptual research.
To resolve this issue, one must first identify the
information to be combined, but this first step has received
insufficient attention. Different assumptions about the
visual input and perceptual output yield differing concep-
tions of information integration.
Two contrasting approaches are statistical and intrinsic
constraint models. The extensive theoretical and experi-
mental literature on this topic is beyond the scope of this
article, but the underlying conceptions of image informa-
tion are quite relevant.
The statistical approach includes a variety of models that
aim to identify quantitative rules for combining multiple
forms of sensory information to infer surface structure—
typically slant or depth. These rules usually use Bayesian
statistical analyses of correlations between image cues and
environmental properties. A simple but frequently used
version assumes that independent estimates of relative
depth are derived from separate sensory channels that
detect different image cues (e.g., motion parallax, binocular
disparity, or texture foreshortening). Typically, these cues
involve zero- or first-order image properties, which are
used to estimate surface depth or slant. In a simple linear
version, evidence from multiple cues is linearly combined,
weighting each cue by its statistical reliability and by other
nonvisual information. Because these low-order image and
surface properties are not reliably correlated across varia-
tions in viewpoint, estimates are improved by (a)combining
evidence from multiple cues, (b)statistics of environmental
and image variables, and (c)adding evidence from prior
probabilities of certain environmental properties. Among
many relevant studies are Landy, Maloney, Johnston, and
Young (1995); Jacobs (1999); Mamassian and Landy
(2001); Ernst and Banks (2002); Hillis, Watt, Landy, and
Banks (2004); van Ee, Adams, and Mamassian (2003);
Knill (2003); Knill and Saunders (2003); and Adams and
Mamassian (2004).
An intrinsic constraint model developed by Domini and
colleagues (e.g., Caudek, Fantoni, & Domini, 2011; Di
Luca, Domini, & Caudek, 2007; Domini & Caudek, 2009,
Atten Percept Psychophys

2010; Domini, Caudek, & Tassinari, 2006; Vuong, Domini,
& Caudek, 2006) is based on a different representation of
image information and perception. Here, multiple image
variables are constrained by the same underlying surface
structure and converge on the same ordinal relations in
depth, despite possible differences in resolution, uncertainty,
or sensory modality. Image information in this model is
coherent (correlated) among neighboring image points and
surface points, and multiple image cues refer to the same
ordinal relations among points on a given surface.
The rationale and evidence reviewed here are consistent
with the intrinsic constraint model but not with the statistical
models. The statistical models are limited by the image and
surface information they use, not by the statistical processes
that combine this information. Information may indeed be
statistical, but the fruitfulness of statistical analysis depends on
the variables to which it is applied. The lower-order properties
used by most current statistical models are not invariant with
changes in viewpoint, and statistical analyses will not
overcome this limitation.
The intrinsic constraint model derives its power from
information about surface shape—where multiple image
cues describe the same structure, independent of the scale
of resolution. Linear integration of these multiple structures
converges on a common structure (Koenderink & van
Doorn, 1997).
Resolution: Suppose that information from an unreliable
depth cue is added to that from a cue with greater reliability
and greater visual resolution. To what extent does the added
information increase the perceptual resolution of relative
depths and surface structure?
According to many statistical models (e.g., the “modified
weak fusion” model of Landy et al., 1995), depth maps are
estimated independently from each cue. When depth
estimates from the individual cues are combined as a
weighted average, the variance of the combined-cue
estimate will be slightly lower even when the second cue
has low resolution (Jacobs, 1999).
In contrast, the intrinsic constraint model can predict
significantly better depth discrimination with added low-
resolution information, because two different variables can
provide complementary constraints on the surface shape
(Vuong et al., 2006). Surface shape may be described
independently of the image resolution, structurally compatible
across resolution scales (Koenderink, 1990).
Vuong et al. (2006) tested these contrasting predictions
by comparing the precision with which a stereoscopic probe
dot could be adjusted in depth along the surface normal to
lie on a curved surface specified by either disparity (of a
sparse stereo-dot array), monocular shading (Lambertian),
or both. In one experiment, the shading was identical in
both eyes, providing no information for the stereo adjust-
ment task; in another experiment, the boundary contours of
the shaded image differed slightly between the left and right
images, providing a weak stereo cue. In both experiments,
the precision (SD) of the depth adjustments averaged
15 arcsec in the disparity-only condition and improved to
11 arcsec in the combined condition, even when the
monocular shading provided no stereoscopic information
at all. When the shaded contours differed slightly between
the two stereo images, the precision of depth adjustments in
the shading-only condition averaged 35 arcsec, and the
statistically predicted performance in the combined condi-
tion (according to the “modified weak fusion” model) was
13 arcsec. Adjustments in the combined-cue condition by
all 4 observers were about 20% more precise than predicted
by the statistical model.
Vuong et al.’s results indicate that surface information
from low-resolution shading and high-resolution disparities
was structurally compatible. Whereas the stereo-dot array
was sparse, the shading was continuous and aided visual
surface interpolation between dots.
Invariance: The strength of the intrinsic constraint model
reflects the use of surface structure that is isomorphic across
image properties (e.g., disparity, shading, motion, etc.) and
invariant with viewing conditions such as viewpoint,
illumination, and resolution.
Summary and conclusion
This article began with the idea that Fechner’s insight about
relations between matter and mind is relevant to the
fundamental problem of vision—concerning how retinal
images yield visually perceived objects. Fechner’s insight
was that relations between matter and mind involve
corresponding structures of variations in the material and
mental worlds. Fechner’s idea is one version of a general
concept of information, involving the representation of a
relational structure in one domain by that in another. Thus,
a basic goal of vision science is to identify the elementary
spatial structure that is shared in common by environmental
objects, optical images, and perceptual discriminations.
Three criteria for evaluating representation of structural
relations are (a)resolution, (b)uncertainty, and (c)invari-
ance. For spatial vision, invariance under changes in
observational parameters such as viewpoint, focal length,
illumination, and context is especially relevant.
Applying these criteria to theoretical and experimental
evidence about spatial vision reveals that the structural
correspondence between environmental objects, optical
images, and perceptual discriminations is the differential
structure associated with local surface shape. Other spatial
Atten Percept Psychophys

properties involving relative depth and surface slant do not
satisfy these criteria and, therefore, do not constitute
reliable information for seeing environmental objects.
The general conclusion is that a modern version of
Fechner’s idea about the relation between matter and mind
is a key to understanding spatial vision.
Author note The authors are grateful to Jan Koenderink and
James Todd for helpful discussions and insightful suggestions on
several aspects of this article; to Ragnar Steingrimsson and Joshua
Solomon for helpful comments on an earlier version; and for the
extensive and insightful recommendations of two anonymous
reviewers. We are also grateful to Douglas Morse for his help
with German-to-English translation of extensive sections of the
book on Foundations and Functions of Information Theory by
Meyer-Eppler, and to Dominic Ali for his expert help in supplying
and editing several photographs.
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