Handbook of Psychology, Volume 4: Experimental Psychology

112 Foundations of Visual Perception

In the remainder of this chapter we explore an instance of
covariational analysis applied by Geisler et al. (2001) to
grouping by good continuation (Field, Hayes, & Hess, 1993;
Wertheimer, 1923). We see how Geisler et al. used this analy-
sis to ask whether the statistics of contour relationships in
natural images correspond to the characteristics of the per-
ceptual processes of contour grouping in human observers.

Co-occurrence Statistics of Natural Contours

Geisler et al. (2001) used the images shown in Figure 4.18 as
a representative sample of visual scenes. In these images they
measured the statistics of relations between contour segments.
In every image they found contour segments, callededge
elements,using an algorithm that simulated the properties of
neurons in the primary visual cortex that are sensitive to edge
orientations. This produced for every image a set of loca-
tions and orientations for each edge element. Figure 4.19A
shows an example of an image with the selected edge ele-
ments (discussed later). Geisler et al. submitted these data to a
statistical analysis of relative orientations and distances be-
tween every possible pair of edges within every image. We
now consider what relations between the edge elements the

authors measured and how they constructed the distributions

of these relations.

The geometric relationship between a pair of edge elements

is determined by three parameters explained in Figure 4.20.

The relative position of element centers is specified by two pa-

rameters: distance between element centers,d, and the direc-

tion of the virtual line connecting elements centers,. The

third parameter,, measures the relative orientation of the ele-

ments, calledorientation difference.For every edge element in

an image, Geisler et al. (2001) considered the pairs of this

element with every other edge elements in the image and,

within every pair, measured the three parameters:d,, and.

The authors repeated this procedure for every edge element in

the image and obtained the probability of every magnitude of

the three parameters of edge relationships. They called the

resulting quantity the edge co-occurrence (EC) statistic,

which is a three-dimensional probability density function,

p(d,,), as we explain later. Geisler et al. used two methods

to obtain edge co-occurrence statistics: One was independent

of whether the elements belonged to the same contour or not,

whereas the other took this information into account. The

authors called the resulting statisticsabsoluteandBayesian,

respectively. We now consider the two statistics.

Absolute Edge Co-occurrence

This EC statistic is called absolute because it does not depend

on the layout of objects in the image. In other words, those

edge elements that belonged to different contours in the

image contributed to the absolute EC statistic to the same ex-

tent as did the edge elements that belonged to the same con-

tour. As Geisler et al. (2001) put it, this statistic was measured

“without reference to the physical world.”

Figures 4.19B and 4.19C show two properties of absolute

EC statistic averaged across the images. Because the covaria-

tional analysis used by Geisler et al. (2001) concerns a relation

between three variables, the results are easier to understand

when we think of varying only one variable at a time, while

keeping the two other variables constant.

Consider first Figure 4.19B, which shows the most fre-

quent orientation differences for a set of 6 distances and 36

directions of edge-element pairs. To understand the plot,

imagine a short horizontal line segment, called a reference el-

ement,in the center of a polar coordinate system (d,). Then

imagine another line segment—atest element—at a radial

distancedtand direction tfrom the reference element. Now

rotate the test element around its center until it is aligned with

the most likely orientation difference at this location. Then

color the segment, using the color scale shown in the figure,

to indicate the magnitude of the relative probability of this

most likely orientation difference. (The probability is called

Figure 4.18 The set of sample images used by Geisler et al. (2001).
Source:From “Effect of stimulus contrast on performance and eye move-
ments in visual search,” by R. Näsänen, H. Ojanpää, and I. Kojo, 2001,
Vision Research, 41,Figure 2 (partial). Copyright 2001 by Elsevier Science
Ltd. Reprinted with permission.

[Image not available in this electronic edition.]