Perceptual Organization 189ABCDFigure 7.11 Steps in determining completion according to Kellman and
Shipley’s relatability theory (see text). Source: From Palmer, 1999.
theories is that they are only as good as the simplicity metric
on which they are based. Failure to predict experimental re-
sults can thus easily be dismissed on the grounds that a better
simplicity measure would bring the predictions into line with
the results. This may be true, of course, but it makes a theory
difficult to falsify.
A third possibility is to explain amodal completion by
appealing directly to ecological evidenceof occlusion. For
example, when a contour of one object is occluded by that of
another, they typically form an intersection known as a
T-junction. The top of the T is interpreted as the closer edge
whose surface occludes those surfaces adjacent to the stem of
the T. The further assumptions required to account for
amodal completion are that the occluded edge (and the sur-
face attached to it) connects with another occluded edge in
the scene and a set of specifications about how they are to be
joined.
One such theory of completion is Kellman and Shipley’s
(1991)relatability theory. It can be understood as a more
complete and well-specified extension of the classic grouping
principle of good continuation (Wertheimer, 1923/1950).
The basic principles of relatability theory are illustrated in
Figure 7.11. The first step is to locate all edge discontinuities,
which are discontinuities in the first derivative of the mathe-
matical function that describes the edge over space. These are
circled in Figure 7.11 (A). The second is to relate pairs of
edges if and only if (a) their extensions intersect at an angle
of 90° or more, and (b) they can be smoothly connected to
each other, as illustrated in Figure 7.11 (B). Third, a new per-
ceptual unit is formed when amodally completed edges form
an enclosed area, as shown in Figure 7.11 (C). Finally, units
are assigned positions in depth based on available depth
information (see chapter in this volume by Proffitt and
Caudek), as depicted in Figure 7.11 (D). In completion, for
example, depth information from occlusion specifies that the
amodally completed edges are behind the object at whose
borders they terminate. This depth assignment is indicated in
Figure 7.11 (D) by arrows that point along the edge in the
direction for which the nearer region is on the right.
Kellman and Shipley’s (1991) relatability theory of
amodal completion is couched in terms of image-based infor-
mation: the existence of edge discontinuities and their two-
dimensional relatability in terms of good continuation. Other,
more complex approaches are possible, however. One is that
completion takes place within a surface-based representation
by relating two-dimensional surfaces embedded in three-
dimensional space (Nakayama, He, & Shimojo, 1995). An-
other is that it occurs in an object-based representation when
three-dimensional volumes are merged (Tse, 1999). Recent
evidence supports the hypothesis that the final perception of
amodal completion is based on merging volumes (Tse, 1999).
Figure 7.12 provides evidence against both image-based and
surface-based views. Part A shows an example in which the
outer contours on the left and right side of the closest object
line up perfectly, thus conforming to the requirements of edge
relatability, yet they fail to support amodal completion. Fig-
ure 7.12 (B) shows the opposite situation, in which there are no
relatable contours (because they are themselves occluded), yet
people readily perceive amodal completion behind the cylin-
der. These examples thus show that relatable contours at the
level of two-dimensional images are neither necessary nor suf-
ficient for perceiving amodal completion.
Figure 7.12 (C) shows a case in which there are relatable
surfaces on the left and right sides of the occluder, and yetA BCD
Figure 7.12 Image-based versus surface-based versus and volume-
based approaches to completion (see text). Source: From Tse, 1999.