Perceptual Coherence : Hearing and Seeing

394 Perceptual Coherence
negative-curvature points carry greater information than equivalent positivecurvature
points. Negative curvature is associated with boundaries between
objects, while positive curvature is associated with one object only.
When two objects meet or overlap, the overall contour (i.e., the outline
of the combined shape) contains concave regions; there are surface discontinuities.
Singh and Hoffman (2001) proposed that the natural dividing
points occur at the maximum curvature of the concave edge, what they call
points of negative minima. Examples are shown in figure 9.7.
This rule can help explain one curious aspect of figure-ground organization.
If we take a simple black region with a complex boundary between
the two regions and pull the regions apart, the identical boundary looks
quite different when attached to the two parts (figure 9.8). In fact, it is often
difficult to see that the two parts can be fitted back together. If we apply the
negative minima rule, the boundary will split into different parts in the two
objects, and that makes it difficult to imagine them as identical. Moreover,
the negative minimum rule can help explain why it is easier to detect visual
symmetry than visual repetition. Symmetrical objects have the identical
placement of minimum points, so that the parts determined by the minima
on the left-side and right-side contour lines will be identical. In contrast,
objects formed by repeating a shape will not have identical minimum points
on the two sides, so that it will be difficult to detect the repetition.
Although the minima rule defines the possible part boundaries, the rule
by itself does not predict which parts will, in fact, be formed. To do that,
Figure 9.7. Singh and Hoffman (2001) proposed
that the natural splitting or dividing
points of complex objects occur at points of
maximum concavity. In each of the three
examples, arrows point to the points of
maximum negativity. Adapted from “Part-
Based Representations of Visual Shape and
Implications for Visual Cognition,” by M.
Singh and D. D. Hoffman, 2001, in T. F.
Shipley and P. J. Kellman (Eds.), From Fragments
to Objects: Segmentation and Grouping
in Vision (pp. 401–459). Amsterdam: Elsevier-
Science B.V.