Psychol Res (1992) 54:278-285 Psycho gical Research Psycholo-gische Forschung © Springer-Verlag 1992

A critical discussion of Kellman and Shipley's (1991) theory of occlusion phenomena

Frans Boselie and Diederik Wouterlood

Nijmegen Institute for Cognition and Information (NICI), University of Nijmegen, PO Box 9104, 6500 HE Nijmegen, The Netherlands

Received August 16, 1991/Accepted June 27, 1992

Summary. Kellman and Shipley (1991) recently advanced a new theory to explain the perception of partly occluded objects and illusory figures. The theory is a formalization of the Gestalt law of good continuation. In this paper we describe their account of occlusion when the contour of the occluder is completely specified by a display. Next, we outline some critical objections and present a number of counterexamples. Finally, we compare their theory with Wouterlood and Boselie's (in this issue) model of occlu- sion phenomena, which might also be considered as a formalization of the law of good continuation.

Introduction

Kellman and Shipley (1991) recently advanced a new theory to explain the perception of partly occluded objects and illusory figures in a unified framework. The theory is a specification of the Gestalt law of good continuation as applied to edges of surfaces. Central to Kellman and Ship- ley's approach is the claim that the perception of partly occluded objects and the perception of illusory figures both derive from a single boundary-interpolation and unit-for- mation process. In this paper, we do not go into their attempt to integrate occlusion and illusory-contour phe- nomena in one theoretical framework. What is of interest to us here is their theory of occlusion phenomena as applied to the class of occlusion displays in which the contour of the occluder is completely specified by the display, i.e., the classical occlusion pattern studied in the literature. First, we describe Kellman and Shipley' s account of occlusion in these cases. Next, we outline some critical objections and present a number of counterexamples. Finally, we compare their theory with Wouterlood and Boselie's (in this issue) model of occlusion phenomena, which might also be con- sidered as a formalization of the law of good continuation.

According to Kellman and Shipley, the perception of partly occluded objects derives from a unit-formation

Correspondence to: E Boselie

process. When perceptual unit formation takes place, boundaries occur across regions where physical specifica- tion of the boundaries is absent. In some cases, a number of nonadjacent regions of a pattern are all seen to belong together and to form a single figure; in other cases, a single region in a pattern is seen to be incomplete and is amodally completed. In all these cases, the percept contains a unit that does not coincide with the physically specified units of the pattern.

Two concepts are central to Kellman and Shipley's approach to unit formation: discontinuity and relatability. First, discontinuities in the first derivative of edges of surfaces (i. e., corners or very sharp curves) are necessary, though not sufficient, conditions for unit formation. Sec- ond, new contours are perceived only when the edges leading into discontinuities are relatable to others. Spa- tially, an edge that leads into a discontinuity is relatable to another edge when the two can be connected by continuous progress from one edge to the other with no discontinuity in between. Just as the presence of a corner or a very sharp curve defines a discontinuity, its absence defines relatabil- ity. The notion of relatability may be formally expressed in terms of two conditions. Given two surface edges, E1 and E2, and their linear extensions, el and e2, the conditions are (1) el and e2 must intersect; (2) their angle of intersec- tion must be obtuse or 90 ° . When two edges meet the relatability criteria, a first-order continuous curve can be fit between them, tangent to the end points at both edges. Though it is not clear from their presentation whether Kellman and Shipley predict that this curve will be the actual perceived connection of interpolated edges, the idea of a smooth connection forbids the interpolation of a sharp corner between edges. When the edges leading into a dis- continuity are relatable to others, the discontinuity is clas- sified as arising from occlusion. If edges, together with their extensions, completely enclose an area, a new percep- tual unit is formed. The effect of the relatability criteria is that the number of discontinuities in the optic array that must be ascribed to the boundaries of objects in the world is minimized.

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Fig. 1. A, B displays used by Shipley and Kellman (1992) to test their theory; C though, as in A, black edges are collinear in pairs, unlike those in A, the black parts do not easily form a single figure, because of lack of closure

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Fig. 2 A - E. Examples of relatabil- ity and interposition given by Kellman and Shipley (1991); A, B gray edges relatable; C - E gray edges not relatable; A - D inter- position; E no interposition

According to Kellman and Shipley, the boundary-inter- polation process is the same in the perception of partly occluded objects and in the perception of illusory figures. These phenomena differ as far as the depth placement of units formed is concerned. The unit-formation process does not produce a depth ordering of the units involved. Depth ordering of the units formed is based on an analysis of surface-quality relations and on the depth cue of inter- position. For instance, in Figure 1 A the square appears behind the cross-shaped figure, because there is a surface color change between its four visible areas. Besides, the units are ordered in depth by the interposition rule, which states that a contour that continues through an intersection without change of direction, while another contour is cut off, belongs to an object in front.

As we have already noted, the process of boundary interpolation of necessity implies occlusion. On the other hand, according to Kellman and Shipley, not every in- stance of occlusion will be the sequel of a boundary-inter- polation process. In those cases in which boundary inter- polation is not possible because edges are not relatable, an impression of occlusion may arise, caused by the depth cue of interposition. In some cases, interposition will not only produce occlusion, but at the same time trigger a kind of figural-completion process that has to be discerned from the perceptual boundary-interpolation process described thus far. For example, if the visible portions of an object that is partly occluded - because of interposition - are consistent, say, with a triangle, this fact may be adequate for a display to be recognized as consistent with a triangle and reported as such. Here unit formation may be based on the processes of recognition, not on the visual interpolation of contours.

Figures 1 A and 1 B give examples of displays used by Kellman and Shipley to test their theory (Shipley & Kellman, 1992), whereas Figure 1C illustrates the pre- dicted effect of closure on unit formation. Figure 1 A only has aligned black edges, whereas Figure 1 B has some mis- aligned black edges. Kellman and Shipley' s theory predicts that the four black regions appear to be connected by interpolation behind the gray occluder in the Case of Fig- ure 1A (because the extensions of the black edges meet and thus are relatable, and because the gray surface is not interrupted), whereas unit formation will be marginal or impossible in the case of Figure 1 B, because of the sub-

stantial misalignment of the horizontal black edges (mis- aligned edges do not have intersecting extensions). Be- cause of interposition, each of the black regions in Fig- ure 1 B separately may appear to continue behind the gray region, but without forming a unitary figure together with the other three black regions. In Figure 1 C as well as in Figure 1 A, all black edges leading into the discontinuities at the contour of the gray region are collinear to other edges and are thus relatable to the corresponding collinear edge. Nonetheless, the perceptual outcome will be quite different in each case. Whereas in Figure 1 A the collinear edges will be readily connected and form part of one partly occluded square, no such unit formation will take place in the case of Figure 1 C, which is readily seen as three triangles.

Kellman and Shipley's theory is further clarified by their interpretation of the displays in Figure 2 (taken from their paper). In Figure 2 A one sees a black rectangle, partly occluding a cigar-like object. This is because the gray regions are relatable, and because the surface quality of the rectangle is not interrupted. The same holds for Figure 2B, and one therefore sees a partly occluded circle there. The edges of the gray region of Figure 2 C are not relatable. Because of interposition, one nonetheless sees that the gray region continues behind the black one, though, according to the theory, the specific locations of the hidden boundar- ies are not given by the boundary-interpolation process. In Figure 2 D one sees a partially occluded triangle with an acute corner behind the occluder. This perceptual outcome cannot be explained by relatability (edges oriented at sig- nificantly less than 90 ° should not give rise to edge inter- polation at all), nor by interposition (which would not explain the definite impression of an occluded sharp corner). It is explained by taking refuge in familiarity: the gray region is recognized as part of a triangle. Figure 2 E will probably be seen as a case of juxtaposition. This is because both the black and the gray edges are unrelatable, and because the depth cue of interposition is missing. So much for Kellman and Shipley's theory of occlusion phe- nomena.

Shipley and Kellman (1992) report three experiments test- ing their claim that the perception of partly occluded ob- jects and illusory figures are manifestations of a single visual-interpolation process. In each experiment, two mag- nitude-estimation tasks were used. One assessed the per-

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A B C

D E F

G H I

J L M

Ob K

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Fig. 3. A, B these displays are preferably interpreted as instances of occlusion, edges running into each other at an acute angle, producing a kite in A, a trapezoid in B. Note that these solutions are preferred over occlusion interpretations with obtuse angles, for instance, a regular pen- tagon in A, a regular hexagon in B, which are simpler and more regular than the forms with acute corners (Boselie, 1988) C a cross next to a square. D, E though these displays do not differ in the relatability of the black edges leading into the gray region, the shape of the black parts is easily seen in D (crosses), but not in E. F depth ordering in the absence of interposition and surface quality differences; the four three-line junctions are of the arrow type. G, t I though, as in Figure 1 B, the straight black edges leading into the gray region are misaligned, conlrary to Figure 1 B, the perceived tmity of the black parts is strong. I, J though these displays do not differ in the relatability of their edges, only I spontaneously splits up (I taken from Kellman & Shipley, 1991). K Though both regions, as far as relatability is concerned, can be continued behind the other one, observers report juxtaposition here (Boselie & Wouterlood, 1989). L, M though the black edges meet at an acute angle in L, and at an obtuse angle in M, the impression of occlusion is nonetheless stronger in L (Durlinger, 1989). N in order to be readily related perceptually, extensions of El and E2 must meet behind the gray occluder. P, Q the square is phenomenally occluded in Figure 1 A; Q shows a simple figure, possibly occluded in Figure 1 B. Perceptually, Q is more readily partitioned than R and P gives a stronger impression of unity than Q

ceived unity of visible parts of partly occluded figures and the other the perceived clarity of edges in illusory-figure displays. Shipley and Kellman interpret the results a; strongly supporting their identity hypothesis. The specific

conditions under which edges were interpolated in both occlusion and illusory-contour figures fitted well with the predictions based on Kellman and Shipley' s (1991) theory.

Though Kellman and Shipley's theory does a good job in predicting some occlusion phenomena, in our opinion it faces problems in accounting for other ones. We shall not assess their work in detail here, our purpose being just to indicate some respects in which the theory is incomplete or incorrect.

As we noted above, figures with edges oriented at sig- nificantly less than 90 ° should not give rise to edge inter- polation at all; interpolation of sharp corners should not occur. Yet, as Kellman and Shipley themselves note, oc- cluded triangles are reported as triangles. In fact, the phe- nomenon that edges that are oriented at acute angles con- nect phenomenally behind an occluding object, forming an acute angle behind the occluder, has been reported time and again in the literature (Boselie, 1988; Boselie and Wouterlood, 1989; Gerbino and Salmaso, 1987; Kanizsa and Gerbino, 1982, Wouterlood and Boselie, in this issue). Figures 3 A and 3 B give some examples.

How do Kellman and Shipley account for these anomalous findings? They suggest that the fact that the visible portions of a potentially partly occluded object are consistent, say, with a triangle may be adequate for a display to be recognized as consistent with a triangle and reported as such. On this hypothesis, such reports would not be the results of perceptual boundary-interpolation processes, but might involve processes of recognition based on partial information. It is unfortunate that a theory that aims to explain occlusion phenomena in terms of per- ceptual processes has to take refuge in an explanation in terms of familiarity and response biases to account for such a robust occlusion phenomenon as closure at acute angles. In assessing Kellman and Shipley's theory, one first has to take a stand about the status of this explanation. Several notes have to be made here.

First, Kellman and Shipley's suggestion does not explain the interpolation of sharp comers when visible portions of a display are not consistent with a shape that might be recognized on the basis of partial information. In the ex- periments reported by Wouterlood and Boselie (in this issue), the stimuli contained only rather unfamiliar poly- gons, whether or not these polygons were completed amo- dally. Though perceptual completions based on recogni- tion processes thus seem very unlikely in these cases, sharp corners, even between edges oriented at significantly less than 90 ° , are reported frequently.

Second, Gerbino and Salmaso (1987), in a series of experiments, investigated how amodal completion affects pattern recognition and tested possible models of the processes underlying the completion of simple shapes. Ac- cording to these investigators, amodal completion is not categorial in nature and is better conceived of as the prod- uct of a very fast, automatic transformation within the visual code. One of their conclusions is that the completion solution is not a reorganization of the literal solution. These conclusions are based on a series of experiments in which both partly occluded equilateral triangles (one acute angle occluded) and partly occluded diamonds (one obtuse angle

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occluded) were presented. Though, according to Kellman and Shipley's theory, the processes that produce complete triangles and diamonds respectively are quite different, no differences in reaction time between triangles and dia- monds are reported by Gerbino and Salmaso (see also Gerbino, 1989).

Finally, Kellman and Shipley's account of completion in terms of recognition in some special cases can be applied in all cases in which visible parts are consistent with some familiar shape. One only has to replace "triangle" by "circle" or "cigar", respectively, to explain the perception of units with definite shape in cases (Figures 2A and 2B) they explain in terms of perceptual-boundary interpolation. If one accepts a recognition explanation in one case, it has to be accepted in other cases as well! The relatability concept would only be needed to explain unit formation in cases of random shapes and would not be a central concept at all.

Given the experimental data and the considerations pre- sented above, Kellman and Shipley' s account of the inter- polation of acute corners does not seem plausible or attrac- tive to us, the more so as the interpolation of acute corners is correctly predicted by Wouterlood and Boselie's model (see next section).

Another aspect of Kellman and Shipley's work is the status of interposition in their theoretical analysis of occlusion phenomena. Discontinuities and relatability operate solely to form units, whereas depth information - i.e., similari- ties of surface qualities, interposition - determines the positioning of units. Interposition is the name given to the following phenomenon: when a contour continues through an intersection and another contour is cut off (a T junc- tion), the continuous contour generally belongs to an object in front of another object, whose contour is being cut off (Helmholtz, 1866; Ratoosh, 1949). Though many cases can be brought forward in which the perceived depth relations in fact are as specified by this rule of thumb, exceptions to the rule are not rare (see Arnheim, 1971, p. 240; Buffart et al. 1981, p. 242; Chapanis and McCleary, 1953, passim; this paper, Figures 3 C and 3 D). According to Kellman and Shipley's theory, Figure 3 C is an example of complete relatability in which the boundary of the region next to the square is relatable to itself across the gap; because of interposition, this region is in the background. They there- fore predict a preference for an occlusion interpretation with the square in front of the other figure. However, Figure 3 C has been presented to subjects in several studies (Dinnerstein and Wertheimer, 1956; Buffart et al., 1981; Boselie, 1988), and in all cases the preferred interpretation is a nonocclusion interpretation, namely "a cross next to a square".

Or take, for example, the outcomes of the experiments reported by Wouterlood and Boselie (in this issue). They presented patterns with two T junctions, resulting in 50% occlusion interpretations only, despite the fact that in both three-line junctions of the patterns the same contour con- tinues without change of direction through the intersection. Such counterexamples disqualify interposition as a general rule of depth ordering. Besides this, interposition itself is an occlusion phenomenon which itself has to be explained

in the first place (for instance by a likelihood, coincidence, or minimum principle), and which therefore cannot figure in a theory of occlusion phenomena. In addition, depth ordering clearly takes place where interposition is absent (Figure 3 F gives an example). This is, for instance, shown by the results of Wouterlood and Boselie's experiments (in this issue). Patterns with arrow junctions - the type of three-line junction exemplified by Figure 3 F - produce occlusion. Despite the fact that the patterns at issue do not contain contours that continue through an intersection, depth ordering is unequivocal, as is shown by the robust preference for one of the possible occlusion interpretations.

In our view, the arguments summed up thus far make it clear that the concepts of recognition and interposition do not adequately explain occlusion phenomena. We now turn to the concepts that are central to Kellman and Shipley's theory: discontinuity and its complement, relatability. The explanation of unit formation, and thus of a whole class of instances of occlusion, in terms of these two concepts in fact omits other factors - concavity and familiarity are mentioned explicitly by Kellman and Shipley - but, in their view, discontinuities and relatability are not only nec- essary, but also sufficient, conditions for unit formation to occur (1991, p. 202). In our view, however, this claim is open to question, as will be shown by some examples.

Figure 1B illustrates a display used by Shipley and Kellman (1992) to test their theory. Because of the mis- alignment of the horizontal edges, these edges are not relatable to each other, and the formation of a single figure out of the four black regions is thus precluded. This is exactly what they found in their experiment. Though not all black regions can be united to form a single figure, two of them - indicated in Figure 1 B by the numbers 1 and 3, pair (1, 3) - in fact are relatable, together with their extensions they completely enclose an area, and they thus should form a perceptual unit. On the other hand, the combination of regions 1 and 4 (pair (1, 4)), and the combination of regions 2 and 3 (pair (2, 3)) both have edges that are unrelatable, they do not form a closed area, and they thus should not form a perceptual unit. To test this prediction of Kellman and Shipley's theory, we presented Figure 1B, together with a number of other displays, to 20 subjects. Stimuli were shown separately, centered on a white cardboard, at normal viewing distance. The instruction used was similar to the one used by Shipley and Kellman (1992) in their Experiment 3: "Please indicate the pair of black regions that most strongly appears to form a single figure." The results were clear cut: none of the subjects indicated pair [1, 3] as giving an impression of unity, whereas pairs [1, 4], and [2, 3] where chosen equally often. These results do not fit well with Kellman and Shipley's theory, whereas they are correctly predicted by Wouterlood and Boselie's model (see next section).

In addition, variations of the kinds of display used by Shipley and Kellman to test their theory can be easily designed, which probably will lead to perceptual outcomes not predicted by them. As in Shipley and Kellman's Fig- ure 1 B, in Figure 3 G and 3 H too all the straight black edges leading into discontinuities with the gray region are

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parallel to other black edges and substiantially misaligned. Though the conditions for unit formation, in terms of Kellman and Shipley's theory, are thus the same in all three displays, in our view the perceptually achieved unity of the black parts is far greater in Figures 3 G and 3 H than in Figure 1 B. To test this conjecture, Figures 1 B, 3 G, and 3 H were presented together to the same group of 20 sub- jects. The instruction used was similar to the one used by Shipley and Kellman (1992) in their Experiment 1: "Please indicate the pattern that most strongly gives the impression that all black areas are part of one single figure." Again, the results were very clear: none of the subjects indicated Figure 1B, whereas choices were distributed over Figures 3 G and 3 H, with a slight preference for Figure 3 H over Figure 3 G.

In this context, the following distinction may be rele- vant. Percepts may differ in the following respects: (a) the strength of the impression of a subjective unit (separate regions all clearly being seen as part of a single object, as opposed to clearly not being seen as part of a single object); (b) the strength of the impression of the definitiness of the shape of the partly occluded unit (occluded edges are seen in definite positions, as opposed to not being seen in de- finite positions). The black regions in both Figure 1 A and Figure 3 G are unanimously seen by our subjects as form- ing one object, with the object in Figure 1 A having a definite shape, whereas the shape of the black object in Figure 3 G, in those places where it is occluded by the gray region, is seen as less circumscribed. Thus, perceived unity and perceived definitiness of shape can be kept apart, and the question therefore is: Are these two aspects of a percept also distinguished in Kellman and Shipley's theory? Kellman and Shipley present their theory as an attempt to explain how partly occluded objects are perceived as units with definite shape (1991, p. 170). However, in their ex- periments they never asked subjects to rate the definiteness of the shape of the perceived unit, they only asked them to indicate the strength of the impression of unity. Because it seems only natural to assume that Shipley and Kellman designed their experiments in such a way that the experi- ments test the central claim of their theory, contrary to their statement mentioned above, definititeness of shape is not what their theory is about.

Returning now to the difference in perceived unity in Fig- ure 1 B on the one hand, and Figures 3 G and 3 H on the other hand, the difference in phenomenal unity of the black regions in these displays, in our view, at least partially goes back to differences in perceived overall regularities: the overall regularity of all black regions together, as com- pared to the regularity of the separate black regions, is far greater in Figures 3 G and 3 H than in Figure 1 B. Though we agree with Kellman and Shipley that global outcomes in perception may have local causes (Boselie, 1988; Boselie and Wouterlood, 1989), their claim that global simplicity is generally a consequence of unit formation as described by the rules of relatability seems unwarranted. This is demonstrated once more by two pairs of Figures: pair 3 D - 3 E and pair 3 I -3 J.

Although Figures 3 D and 3 E do not differ in the relata- bility of the black edges leading into the discontinuities at

the contour of the gray square, the perceptual outcome is quite different in each case. In Figure 3 D, the visible black regions are easily seen as complete crosses, whereas the shape of the four unoccluded black regions in Figure 3 E can be seen only with great effort. Figure 3 E produces a strong impression of occlusion - as is correctly predicted by Kellman and Shipley - whereas Figure 3 D does not, a result that is not predicted by them.

Despite the fact that the relatability conditions are the same in Figures 31 (presented by Kellman and Shipley as a demonstration of a spontaneously splitting figure) and 3 J: (in both cases the four concave discontinuities can be re- moved because edges are collinear), the perceptual out- comes are quite different: a spontaneously splitting figure in Figure 31 - as predicted by Kellman and Shipley - whereas splitting up does not take place in Figure 3 J - Kellman and Shipley here predict splitting too. This differ- ence in perceptual outcome, in our view, is due to the difference in overall symmetry. In particular, the easily perceived vertical symmetry of Figure 3 J implies a decom- position of the pattern in a left and right part, with col- linearity still playing a role within each of these halves. This decomposition prevents a split up along collinear lines only.

Figure 3 K is taken from Boselie and Wouterlood, 1989. In their study, all subjects saw this display as "two hexa- gons next to each other." Koffka (1935, p. 153) presents a similar pattern as an example of two adjoining hexagons. These results are not compatible with Kellman and Ship- ley's theory. Because the extensions of the edges leading into the discontinuities where the two hexagons meet inter- sect at obtuse angles, their theory predicts a preference for occlusion interpretations.

Figures 3 L and 3 M are taken from a study by Durlinger (1989). In a series of experiments she systematically varied the relationship between the angles at which surfaces meet in three-line junctions and studied the strength of the occlu- sion impression as a function of these variations. In one experiment, pairs of stimuli were presented. Both stimuli were composed of two adjacent regions, one of which possibly could be continued amodally to form a symmetri- cal figure, whereas the other by continuation would form an asymmetrical figure. The stimuli of a pair differed in the angles made by the regions at the three-line junctions (see Figures 3 L and 3 M). Some of the findings of the experi- ment were as follows: all observers report partial occlusion of a black object in Figure 3L; Figure 3M is interpreted either as a juxtaposition of objects, or as partial occlusion of a black object. These results are incompatible with Kellman and Shipley' s theory. For Figure 3 L, their theory predicts just the opposite of what has been found in the experiment: partial occlusion of a gray object behind a black one. This prediction follows simply from the fact that the gray edges, leading into the discontinuities formed by the three-line junctions, meet at an obtuse angle - and thus are relatable - whereas the black edges meet at an acute angle - and thus are unrelatable. Though Kellman and Shipley do not specify what will happen in a case such as in Figure 3 M, in which occlusion can go either side, the gray region occluding the black one or the black region occluding the gray one (because gray and black edges do

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not differ in relatability), the actual outcome of the experi- ment - a clear asymmetry between both regions in the occluder/occluded relation - is not what one expects with their theory in mind.

We conclude this series of demonstrations with Fig- ure 3N. Though the extensions of the collinear edges E1 and E2 in this display meet, and thus, according to Kellman and Shipley' s criteria, are relatable and in addition, togeth- er with other visible edges, completely enclose an area, they are connected with great effort only and the unit formed is very difficult to perceive. This could be due to the fact that in order for these collinear edges to be connected, the black region in Figure 3 N has to be split up into two planes with different positions in depth. So an additional requirement for unit formation to take place may be that extensions must intersect actually behind an oc- cluder.

In sum, in our opinion these arguments make it clear that the current version of Kellman and Shipley's theory does describe a number of occlusion phenomena inade- quately. Admittedly, in some of our demonstrations taken from the literature, lines function as surface boundaries, whereas Kellman and Shipley prefer to study displays composed of filled-in regions. Of course, these findings with line drawings have to be replicated with displays composed of filled-in regions. For the moment, we have no reason to expect that a change in stimulus material in this respect will entail a real change in outcomes. In this con- text, it is worthwhile to draw attention to a weak spot in the methodology of Shipley and Kellman's (1992) experi- ments by which they tested their theory. They presented occlusion displays to assess the effects of the position, orientation, and separation of the physically specified edges on the process of perceiving object unity despite occlusion. Unfortunately, by manipulating the spatial rela- tionship between edges leading into discontinuities, they also varied the unity of possibly occluded figures. It is a well-established fact that single forms may vary considera- bly in perceived unity (Arnheim, 1971; Leeuwenberg and Van der Helm, 1991). The strength of the impression of unity evoked by the simple and regular figure phenome- nally occluded in Figure 1 A (a square) is much stronger than the perceived unity of any figure possibly occluded in Figure 1B (see Figures 3P and 3Q). That is to say: not only does misalignment of edges hypothetically decrease unit formation under occlusion conditions, it almost cer- tainly also affects the perceived unity of forms, irrespective of occlusion. This contamination of effects makes it im- possible to ascribe the 'variance of unity ratings given by Shipley and Kellman's subjects exclusively to differences in the degree to which they succeeded in forming a single figure out of the separate parts of a display. The alternative hypothesis is: subjects always succeeded in forming a single, definite, partly occluded form out of the visible parts of a display; their unity ratings represent the per- ceived unity of the occluded forms.

We conclude by comparing Kellman and Shipley's theory with the model of occlusion phenomena we advanced re- cently (Wouterlood and Boselie, in this issue). Our model too is a formalization of the law of good continuation. It

[0,01 [0,21 [0,1] I I,ll

A B C D

Fig. 4. Wouterlood and Boselie's model: the four possible outcomes of initial grouping at three-fine junctions in terms of number of open seg- ments of regions

assumes that a pattern is described by the smallest number of contour elements possible, and with changes of direction as small as possible. Smooth continuation is pursued

(1) By the splitting up of a pattern in closed regions in front of a continuing background.

(2) By the grouping, if possible, of only those two line segments at three-line junctions that continue direc- tions most smoothly. The nongrouped line segment is called an open segment. Grouping at three-line junc- tions produces one out of four possible pattern descrip- tions, in which the regions, depending on the nature of the three-line junctions on hand, are characterized by 0, 1, or 2 open segments (see Figure 4). If grouping pro- duces incompatible spatial relations between regions - the [0,1] and [1,1] cases - regrouping takes place. In the end, grouping always produces either a [0,0] or a [0,2] description of a pattern. A [0,0] description is always final. A [0,2] description still needs more pro- cessing: open segments have to be closed.

(3) By the continuing, if possible, of open line segments linearly behind a closed region, till they run into anoth- er line segment of the same region, within the boundar- ies of an occluder. If possible, the open region is closed in this way (Closure condition). If closure is not at- tained (No-closure condition) - because open seg- ments do not converge or because they do not converge within the boundaries of the potential occluder - it is achieved in one of two other ways. Either the open- ended region is closed behind the occluder by an addi- tional contour element - a straight line or a curve - or it is closed by the common border in the same plane as the other region.

When a description of a pattern as two closed regions is attained without continuation of open segments behind a bounded region, the final perceptual interpretation is: Jux- taposition, i.e., two regions in one plane. When a descrip- tion as two closed regions results from continuing open segments behind an occluder - linearly or by the addition of another closing contour element - the final perceptual interpretation is: Occlusion, i.e., two regions in different planes.

The model specifies the relative frequency of Occlusion interpretations and Juxtaposition interpretations for the four types of pattern - [0,0], [0,2], [0,1], and [1,1] - in Closure and No-closure conditions.

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Strictly speaking, the current version of our model does not apply to a number of Kellman and Shipley's displays, which are composed of more than two regions and are regular. Even so, the perceptual outcome of these displays does fit well with the gist of the model. In Figures 1 A and 1 B, the combination of each of the separate black regions with the gray region answers the condition [0,2], No-clo- sure condition. Seen separately, each of the combinations will thus produce juxtaposition interpretations and inter- pretations in which the black region is continued behind the gray one. Seen together, continuation of all open seg- ments is possible in Figure 1 A, which will produce "a black square" as the most prominent interpretation in that case. For Figure 1 B one finds that seen together, the verti- cal line segments of the black regions 1 and 4, and of regions 2 and 3, if continued linearly, will run into each other; the same holds for all the line segments in regions 1 and 3. This situation implies that alternative interpretations of Figure 1 B will be viable. Interpretation A: regions 1 and 4, respectively regions 2 and 3, are grouped together, form- ing two separate black objects, each partially occluded by the gray region; the shape of these objects is partially definite (straight edges in definite positions), partially in- definite (indefiniteness starting where two parallel edges continue behind the occluder). Interpretation B: perceptual organization results in three black objects: regions 1 and 3 are grouped together, forming one black rectangle; regions 2 and 4 stand on their own. There will be a preference for Interpretation A over Interpretation B, because the total number of contour elements of the black objects is smaller in Interpretation A than in Interpretation B (8 vs. 10). In Figure 1 C, the combination of each of the separate black regions with the gray region answers the condition [0,2], Closure condition, and thus should result in partial occlu- sion of a triangle.

The perceptually preferred interpretation of all five dis- plays in Figure 2 is as expected by our model. (A) Seen together, the four open-ended gray line segments are elim- inated by linear continuation till they run into each other, forming the cigar-like object. (B) This is a case of [0,2], Closure condition, front-back distinction of L junction on common border compatible with the spatial relations origi- nating from the three-line junctions, producing the circle. (C) This is an instance of [0,2], No-closure condition, resulting in both juxtaposition and occlusion interpreta- tions. (D) An instance of [0,2], Closure condition, produc- ing a triangle. (E) A case of [0,0], resulting in juxtaposi- tion.

Comparing Kellman and Shipley's theory with our model, several differences stand out. First, there is a difference in the intended scope of the two approaches. Kellman and Shipley's theory aims at integrating occlusion and illusory contour phenomena. We do not go into the question whether their attempt to describe both types of phenomena in a unified framework is successful or not. Leaving the identity hypothesis aside their theory attempts to explain how partly occluded objects are perceived. Our model is even more restricted in its aim: it is restricted to the percep- tion of patterns from which regularities have been remov- ed. The difference in scope results in Kellman and Shi-

pley's theory being less specific in some respects than our model. For instance, the theory does not make specific predictions for patterns which are labeled [0,1 ] or [1,1] in our model, which are perceptually ambiguous. On the other hand, because of its wider scope, their theory makes speci- fications, where our model is mute. For instance, Kellman and Shipley are quite explicit in rejecting global simplicity and regularity as causal factors in perceptual outcomes, whereas we kept away of that issue.

Despite these differences, both approaches have a common ground: the application of the Gestalt law of good con- tinuation to the edges of surfaces. Intuitively, according to Kellman and Shipley, relatability requires that two edges be connectable by a smooth, monotonic curve. They there- fore specify the law as: edges leading into discontinuities are continued without change of direction, producing linear extensions, and extensions are continued smoothly (i. e., with change of direction allowed) into each other behind the occluding object. In our model, we start from a different intuition: edges leading into discontinuities are grouped by the criterion of the smallest change of direction (i.e., a change of direction is allowed), and ungrouped edges are, if possible, continued linearly (i. e., a change of direction is not allowed) behind an occluder till they run into another edge. So, whereas Kellman and Shipley stress the smooth continuation of amodal contour elements, our model calls for the smoothest possible connections be- tween visible contourelements, and for the straightforward continuation of amodal parts, irrespective of the outcome (an acute or obtuse comer) this well entail at the point of intersection. This difference in applying the good-con- tinuation notion may have evolved from a difference in the domains of stimuli that both theories seek to envisage in the first place: Kellman and Shipley's work first of all is an attempt to describe the formation of phenomenal units out of several non-adjacent visible parts. So, naturally, the question "When do edges of various regions connect?" is central to their approach. We, on the other hand, concen- trate on the question "When do edges of a single region continue behind an adjacent region?" Both specifications of the notion of good continuation may seem plausible on a-priori grounds, and the question thus is: is there a differ- ence in fit with available data? We think there is, and, in our view, at the moment the difference is in favour of our approach.

(1) Edges oriented at acute angles in the Closure condition phenomenally connect behind an occluding object, and moreover, form an acute angle behind the occluder. This has been reported repeatedly in the literature, lastly by Wouterlood and Boselie (in this issue) for irregular patterns, and is demonstrated by Figures 1 C, 2D, 3A, 3B, and 3L. This finding is predicted by our model, and disproves Kellman and Shipley's theory.

(2) Edges oriented at obtuse angles or at angles of 90 ° are not always amodally completed. This is demonstrated by Figures 1B, 3C, 3D, 3K, and 3L. This finding is predicted by our model in case of Figures 1 B, 3 K, and 3 L, whereas it is not predicted by Kellman and Ship- ley' s theory in any of these cases.

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(3) Occlusion interpretations do occur when interposition is absent, and depth ordering cannot be based on an analysis of surface qualities either. This has been pro- ven by the experiments reported by Wouterlood and Boselie (in this issue), and is demonstrated by Fig- ure 3F. This finding is predicted by our model, and disproves Kellrnan and Shipley's theory.

As we said above, most of the patterns published in the occlusion literature are very regular, and most of the coun- terexamples of Kellman and Shipley' s theory, presented in Figure 3, therefore from sheer necessity are regular too. Because our model, strictly speaking, does not apply to regular patterns, it cannot be disproved by such patterns either, which makes a comparison with Kellman and Ship- ley's theory an unbalanced one. On the other hand, those cases in which the preferred interpretation is one in which edges oriented at acute angles form an acute angle behind an occluder, whereas the interpretation is not the overall regular one, can be taken as support for our model. Figures 3A and 3B are instances of this situation. We conclude with a note on Figures 3 L, and 3 M. Experiments (Durlinger, 1989) ctearly show that the impression of oc- clusion produced by these patterns is strongest for Fig- ure 3 L. This result cannot be explained by a global mini- mum tendency, because the black region can be completed to form a vertically symmetrical object in both cases, and the occlusion tendency thus has to be equally strong for both patterns. Kellman and Shipley predict the occlusion of a gray object in Figure 3 L, a prediction which is clearly wrong. Starting from our model, the experimental outcome makes sense: According to our model, the common border of Figure 3 L is grouped with the contour elements of the gray region, because the change of direction between the common border and the gray contour elements at both junctions is the smallest one; the ungrouped black edges are continued behind the gray region; consequently the pattern is unequivocaily interpreted as "occlusion of black." In Figure 3 M, contrary to Figure 3 L, the common border is grouped equally well with both gray and black edges, whereby the impression of "occlusion of black" is decreased. Though our model thus accounts for most of the experimental outcomes of Figures 3 L and 3M, it does not explain the observed asymmetry of gray/black occlusion and black/gray occlusion in the case of Figure 3 M. In our opinion, this asymmetry is due to the asymmetry in the overall regularity of both regions. The vertical mirror sym- metry of the visible gray region makes this region less

liable to occlusion, whereas the lack of vertical symmetry enhances the tendency to complete the black region to form a symmetrical object.

As the confrontation of both models with available data makes clear, much remains to be learned about occlusion phenomena. It will be a challenge for further research to design experiments that will be crucial to competing models.

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