Perception & Psychophysics1999, 61 (5), 943-951

Completing visual contours: The relationshipbetween relatability and minimizing inflections

MANISH SINGH and DONALD D. HOFFMANUniversity ofCalifornia, Irvine, California

Visual completion is a ubiquitous phenomenon: Human vision often constructs contours and sur­faces in regions that have no sharp gradients in any image property. When does human vision inter­polate a contour between a given pair of luminance-defined edges? Two different answers have beenproposed: relatability and minimizinginflections. Westate and prove a proposition that links these twoproposals by showing that, under appropriate conditions, relatability is mathematically equivalent tothe existence of a smooth curve with no inflection points that interpolates between the two edges. Theproposition thus provides a set of necessary and sufficient conditions for two edges to be relatable. Onthe basis ofthese conditions, we suggest a way to extend the definition of relatability (1) to include therole of genericity, and (2) to extend the current all-or-none character of relatability to a graded mea­sure that can track the gradedness in psychophysical data.

Visual completion is a process ofhuman vision that canconstruct contours and surfaces in regions that have nosharp gradients in any image property. Researchers dis­tinguish two kinds ofvisual completion (Michotte, Thines,& Crabbe, 1964/1991). Inmodal completion, the percep­tion ofa completed surface is accompanied by a perceivedbrightness gradient along its completed contour. For ex­ample, in Kanizsa's (1979) illusory triangle (Figure 1b),the region inside the triangle is seen as being whiter thanthe surrounding white. In amodal completion, by con­trast, there is no perceived brightness gradient, eventhough one is aware of the shape of the completed sur­face. For example, in Figure 1a, one is aware ofa unifiedelliptical shape passing behind the rectangle-eventhough one sees no brightness edges in the region wherethe ellipse is occluded by the rectangle.

Human vision does not, however, always completecontours to form unified objects. InFigures 1c and 1d, forexample, the displays of Figures la and Ib have been re­spectively modified so that one no longer has the percep­tion of unified, completed shapes. What spatial factorsdetermine when human vision will, or will not, completea visual contour? .

This question has been addressed in the literature intwo different ways. One proposal, owing to Kellman andShipley (1991), is that the process of interpolation be­tween two luminance-defined edges proceeds when thetwo edges are relatable to one another. The precise defi-

This work was supported in part by National Science FoundationGrant SBR-90 14278. For discussions and suggestions, we thank BruceBennett, Myron Braunstein, Mike D'Zmura, and Chetan Prakash. Wealso thank Jan Drosler, Shinsuke Shimojo, and two anonymous review­ers for helpful comments on an earlier version of the paper. Addresscorrespondence to M. Singh, Department of Brain and Cognitive Sci­ences. NE20-45I , Massachusetts Institute of Technology, Cambridge,MA 02139 (e-mail: manish@psyche.mit.edu).

nition of relatability is as follows (see Kellman & Ship­ley, 1991, pp. 174-175):

DEFINITION I. Let Eland E2 be two surface edges, with05 1 and S2 as their respective linear extensions (see Fig­ure 2). Then the two edges are said to be relatable if(1.1) the two linear extensions SI and s2 intersect; and(1.2) their outer angle of intersection (angle c/J in Fig­ure 2) is acute. 1

For example, in Figures la and lb, the various edgepairs are relatable, whereas in Figures lc and l d, they arenot. In a number of experimental studies, Shipley andKellman (e.g., Shipley & Kellman, 1992a, 1992b) haveshown the psychological relevance of the notion of re­latability. For example, they found that the strength of aperceived interpolated contour decreases steadily as agiven edge pair deviates from perfect relatability (as inthe displays in Figures lc and l d),

A different proposal, owing to Takeichi, Nakazawa,Murakami, and Shimojo (1995), involves the role ofinflec­tion points (i.e., points where a curve reverses its sign ofcurvature). According to this proposal, (1) human visioncompletes visual contours in a way that minimizes thenumber of inflection points on the interpolated contour,and (2) the more inflection points an interpolated contourneeds to have (while maintaining continuity ofdirection),the lower its perceptual strength-and the less likely it isthat visual completion will take place.? Takeichi et al.have presented evidence that subjects indeed complete vi­sual contours in a way that is consistent with these re­quirements. For the displays in Figures Ic and l d, for ex­ample, any smooth curve interpolating between thevarious edge pairs must have at least one inflectionpoint-and this might explain the absence, or weakness,ofperceived interpolated contours in these figures. Take­ichi et al. also note that modal completion (e.g., in illusoryfigures, such as Kanizsa's triangle) takes place only whenit is possible to have a smooth interpolating curve that

943 Copyright 1999 Psychonomic Society, Inc.

944 SINGH AND HOFFMAN

Figure 1. Two examples of visual completion: (a) amodal com­pletion of an ellipse behind the rectangle, and (b) modal comple­tion ora triangle in front of the disks. In (c) and (d) the percep­tion of completed figures is greatly reduced.

(1995). We end with some constructive suggestions onpossible ways to extend the notion of relatability.

RELATABILITY AND INFLECTIONS:A MATHEMATICAL RESULT

In Kellman and Shipley's definition of relatability, itis clear that an edge is essentially thought of as an or­dered pair (a, t), where a is a point in 9\z (the point at whichinterpolation must begin), and a unit direction vector t(giving the direction ofthe edge at point a), along whichthe interpolated contour must continue. So, for example,neither the length of the edge, nor its curvature, affectsthe definition of reiatability (see Definition 1).3

For any such ordered pair (a, t), with a, tE9\Z andII t II = 1, let R( a, t) denote the open ray originating at a,and in the direction of t. That is,

R(a, t) = {xE9\zlx = a + rt; r> O}.

Then the definition ofrelatability can be reformulatedas follows.

DEFINITION la.Zera., az,t l , t zE9\Z,with II till = II tzll = 1.The two edges (a., t l ) and (a z, t z) are relatable (see Fig­ure 3) if

(b)

(d)

(a)

(e)

R

has no points of inflection-hence modal completionmay have a higher threshold than amodal completion. Inaddition to the requirement ofminimizing inflection points,Takeichi et al. also require that the end points of the in­terpolated contour (i.e., the points where the interpolatedcontour meets the luminance-defined edges) not be pointsof inflection on the completed curve, and that the inter­polated contour continue the curvature of the edges.

The theory ofrelatability and the theory ofinflections,therefore, both provide a possible explanation for thesame phenomenon-namely, the presence or absence, aswell as the strength, of perceived interpolated contours.Furthermore, the two proposals seem related. For exam­ple, in Figures 1c and 1d, making the various edge pairsnonrelatable also introduces inflection points on anysmooth curve interpolating between them. (Note, however,that the edge pairs in Figure 1 have the special propertythat the two edges in each pair are parallel.) In this paper,we address the question of what, in general, the preciserelationship between these two proposals is.

In the next section, we state and prove a propositionwhich shows that, under appropriate conditions, relata­bility is in fact mathematically equivalent to the condi­tion that there exists a smooth curve with no inflectionpoints that interpolates between the two edges. This propo­sition therefore provides a set ofnecessary and sufficientconditions (and hence, an alternative definition) for re­latability, in terms of properties that an interpolatingcurve must satisfy-and this includes the property ofhav­ing no inflections. Following this, we consider some otherpoints of difference between the theoretical frameworkofKellman and Shipley (1991) and that ofTakeichi et al.

and

(1a.2) Icf> 1< ~,

where cf> is the directed angle from t I to - t z.With this simple reformulation, we can now state the

proposition. Recall that a CI curve is one that is first-ordercontinuous everywhere; and a piecewise CZ curve is one

II

II

II

II

II

II

, I: I I

-----::---------;'j~----~ 51I

II

II

I~

Figure 2. Kellman and Shipley's (1991) definition ofrelatabil­ity: The linear extensions of the two edges must meet, and theirouter angle of intersection must be acute.

cb·.... (; 82: cj> ..•••••p I

....... t2-- ......

bI

II

II

---"I~ - - - - -l~P----~81 t1 " Xo

II

I

IFigure 3. A Cl and piecewise C2 curve that interpolates be­

tween two relatable edges (Kellman & Shipley, 1991, Appendix B).This curve satisfies the conditions of our proposition.

that is second-order continuous everywhere except, pos­sibly, for finitely many points.

PROPOSITION. Two edges (ai' t l) and (a2, t 2) are relat­able ifand only ifthere exists a regular C I, and piecewiseC2,plane curve a: [0, L] ---t 9\2, with curvature Ka , thatsatisfies the following three properties: (l) a(O) = a];a(L) = a2; a'(O) = t l; a'(L) = -t2; (2) Ka(S) doesnot re­versesignfor 0 < s < L; and (3) f~ IKa(S) Ids < ,./2.

Condition I simply says that the interpolating curvemust start at the point aI' with tangent t l, and terminateat the point a 2, with tangent -t2. Condition 2 says that thecurve should have no points of inflection. Condition 3 saysthat the interpolating curve must not turn through a totalangle ofmore than ,./2. We will comment further on Con­dition 3 immediately following the proofofthe proposition.

PROOF. (I) Let us first assume that the two edges (a., t I)and (a 2, t 2) are relatable. We need to show the existenceof a C I, and piecewise C2, plane curve a that inter­polates between these two edges, and satisfies Properties1-3 of the proposition.

If t. and t 2are parallel, relatability (1) forces R(a l , t l )

and R(a2, t 2) to be colinear, for otherwise R(a I' t 1') nR(a 2, t 2) = 0; and (2) forces I/J = 0, for otherwise Con­dition 2 of Definition 1a is violated. In this case, we cansimply define a to be the straight-line segment joining alto a 2, as follows,

a(s) = a] + st], SE [0, L],

where L = II a, - a211. Then a trivially satisfies Proper­ties 1-3 of the proposition.

If t I and t 2 are not parallel, we use the construction ofSkeath (Kellman & Shipley, 1991, Appendix B),4 andshow that this curve satisfies the three properties.

Assume without loss ofgenerality that a l = 0, the ori­gin of the coordinate system, and t l = i = (l, 0), the unitvector along the positive x-direction. Then, -t2 = (cosl/J,sine). Invoking Property 1 of relatability, let R(a], t l) nR(a2' t 2) = {xo}. (Two nonparallel rays either are dis­joint or intersect in a point.) Assume also that II a l -

RELATABILITY AND INFLECTIONS 945

Xo II ~ II a 2 - Xo II and I/J> O. (The proofs in the other casesare similar.)

The curve a interpolating from a l to a2 is now con­structed by piecing together an arc of a circle and astraight-line segment (see Figure 3), as follows.

If II a, - Xo II < II a2 - XoII, there exists a point b onR(a2' t 2), lying between a 2and xo, such that II a] - Xo II =

lib - xoll· Iflla1 - xoll = II a 2 - xoll,takeb = a2· Let t 3 =j = (0, I), the unit vector along the positive y-direction,and

Then, the two rays R(a l, t 3) and R(b, t 4) must intersect.Call their point of intersection e. Then, by our construc­tion, we have

II a l - ell = lib - ell (= p),

since the two triangles ~exoal and ~exob are congruent.Also, let II b - a211 = I.

Now, define a: [0, L] ---t 9\2 by:

a(s)=fc+p[c0s( 3; +*).sin( 3; +*)Jo~s~PI/J,lb-(s-pt!»t2 , pt!>$,s$,pl/J+I,

where L = PI/J + I.Then, we have

a'(s) = {[-sin(3; +; ).cos(3; +; )}o<s<PI/J,

-t2 , PI/J < s < L.

Therefore,

a(L)=b-(L-pl/J)t2 =b-It2 =a2'

a'(O) = ( -sin 3; , cos 3; ) = (I, 0) = t.,

a'(L) = -t2 •

Clearly, aisunit speed (II a'(s) II = I forO<s<L),and it is C2 on each of its two components (arc of circle,and straight-line segment). Also, letting a: denote theleft-hand limit and a+ denote the right-hand limit, then

a_(pl/J)=e+ p[cos (3; +I/J)' sin (3; +I/J)] = b,

a+(pl/J) =b,

a~(pl/J)= [-Sin e; +I/J). cos e; +I/J)]

= (cosl/J,sin e) = -t2,

a: (pl/J) = -t2 ·

946 SINGH AND HOFFMAN

Hence, a' is continuous at s = PI/J, and therefore a is C',and piecewise C2, on [0, L] (s = pi/J is the only point ofC2 discontinuity for °< s < L).

Since a is a unit speed curve, the plane curvature of ais defined by

lCa(S) = O'(s),

where O(s) is the slope angle of a at S.5

We have,

O(s)= {SiP,°s s <S, pi/J .I/J, pi/J <S, s <S, L

Therefore,

( ) = {l/ p,°< S< pi/JlCa S .0, pI/J<s< L

Hence lCa(S) does not reverse sign for °< S< L.Also,

rL rpt/> 1 1Jo IlCa(S) Ids =Jo p ds= p (pI/J) = I/J,

and we know, by relatability, that

11>< n'I' 2'

This proves the first half of the proposition.(II) Let us now assume that we are given a C', and

piecewise C2, curve a: [0, L] ~ 9\2 which interpolatesbetween two edges (a" t,) and (a2' t 2), and satisfies Prop­erties 1-3 of the proposition. We can assume that a isunit speed. (Otherwise, it is always possible to reparam­etrize a to make it unit speed.) We show that (a" t,) and(a2' t 2) must be relatable.

Assume, as before, that a, = 0, and t, = i = (1,0). Sincea is C', it is possible to define a continuous function 0on [0, L], such that O(s) gives the slope angle of a at s[with 0(0) = 0]. Since a is piecewise C2, lCa(S) = O'(s)is defined for all but finitely many points in [0, L]. ByProperty 2, since lCa(S) does not reverse sign, let us as­sume, without loss of generality, that lCa(S) ~ 0, VSE

[0, L] [wherever lCa(S) is defined]. Hence, Ois monoton­ically increasing on [0, L].

Let O(L) = I/J. By Property 1 ofthe proposition, we musthave -t2 = a'(L) = (cos I/J, sin I/J), so that

I/J = cos-'(t, . -t2) ·

Since 8(0) = 0, and Ois monotonically increasing on [0,L],it follows that I/J ~ 0. If I/J = 0, it must be the case, by mono­tonicity, that O(s) = 0, VSE [0, L], so that a is a straight­line segment. In this case, (a" t,) and (a 2, t 2 ) are triviallyrelatable. Let us assume, therefore, that I/J> 0.

By Property 3 of the proposition, we have

f: O'(s)ds < ~,

since lCa(S) = O'(s) ~°for °<s's <S,L [except, of course,for the finitely many points where O'(s) is not defined-

but these do not affect the integral]. By Cauchy's funda­mental theorem of calculus, we have

f: O'(s)ds =O(L)- 0(0) = I/J.

Therefore,

and this proves the second condition in the definition ofrelatability.

Now, we show that the two open rays R(a" t,) andR(a2' t 2) must intersect. Let a2/11a211 = (cos{3, sin{3).Hence {3 is the slope of the chord joining a, and a2 (re­call that a, = 0). It is clear (see Figure 4) that R(a" t.)and R(a2' t 2) will intersect if and only if

{3 < I/J < x.

We have already shown that

tt11><-'I' 2'

Hence we need only show that {3 < I/J.Since a is C', we know by the mean value theorem

that there exists an So in the open interval (0, L) such thata'(so) is parallel to the chord joining a, and a2 . In otherwords,

3soE(O, L) : O(so) = {3.

Since 0 is monotonically increasing on [0, L], we musthave

{3 = O(so) <S, O(L) = I/J.

To start a reductio ad absurdum, assume that {3 = I/J.Then, we must have (again, by the monotonicity of 0),

O(s) = {3, VSE [so' L].

,,

I~~- - - - - - - - - - - -

.--.~ ----_:.~p-----81 t1 "

II

I,Figure 4. Two open rays interesect if and only if fJ < ", < 1T.

RELATABILITY AND INFLECTIONS 947

Define the set,

T = {tE[O, L) I8(s) = {3, \fSE[t, L]}.

Since SOE T, we know that Ti:- 0. Also, it is clear that Tisa closed set, and it is bounded below since 0 i: T [recall that8(0) = 0]. Therefore, Tmust have a minimal element. Let

t* = min T(> 0).

Since t* E T, a(t*) lies on the chord joining a 1 and a2 ,

and therefore {3 is also the slope of the chord joining aland a(t*). Again applying the mean value theorem, weconclude that

(a) (b)

Figure S. Examples that clarify the role of J~ IlCa(S)Ids in theproposition.

3s~ E (0, t*) : 8(so) = {3.

Now we have So < t* with

8(s~) = 8(t*) = {3.

Since 8 is monotonic, this forces

8(s) = {3, \fsE [s~, t*],

and therefore,

8(s) = {3, \fsE [s~, L].

But this means that So E T, which is a contradiction, sincet* = min T, and So < t*. This contradiction proves that{3< t/J, and hence, the two open raysR(a l, t l) andR(a2, t 2)

must intersect.QED

(c)

81

(d)

Some Remarks1. Condition 3 of the proposition-namely,

-is strictly stronger than Condition 2 in the definition ofrelatability (Definition la)-namely,

1rIt/JI<T

Whereas It/J Imeasures the magnitude ofthe angle betweenthe two tangents a'(O) = tt and a'(L) = -t2, f~1 /(,s) 1

ds measures the total angle through which the curve aturns as it traverses its path from s = 0 to s = 1. Figure 5shows three examples that clarify this difference. For thecurve in Figure 5a,

but

For the curve in Figure 5b,

but

And for the curve in Figure 5c,

t/J=~«~}but

2. The proposition is no longer true if Condition 3,

is replaced with

It/JI< ~.

A counterexample is provided by Figure 5c; this satisfies

It/JI< ~,

and the interpolating curve has no inflections, but thetwo edges are nevertheless not relatable.

3. The proposition is no longer true ifCondition 2 (i.e.,having no inflections) is removed. A counterexample isprovided by Figure 5d, in which the interpolating curvesatisfies

948 SINGH AND HOFFMAN

but the two edges are nevertheless not relatable.4. Kellman and Shipley have also extended the defin­

ition of relatability to three dimensions, by adding therequirement that the two edges be coplanar. The proposi­tion is easily extended to the 3-D case: One need onlyadd a fourth condition requiring that the interpolatingcurve a be planar (i.e., have torsion 0).

EXTENDING RELATABILITY

One should be cautious in interpreting the propositionproved above. Although it provides a precise statementofthe link between relatability and the role of inflectionpoints, it should not be construed as claiming that theframeworks of Kellman and Shipley (1991) and Takeichiet al. (1995) are equivalent. One difference between thetwo frameworks, ofcourse, is that Kellman and Shipley'sdefinition of relatability requires that

11/>1< ~,

whereas Takeichi et al.'s framework does not. Another dif­ference, however, is that the framework ofTakeichi et al.explicitly takes into account the role of genericity.

Genericity is a powerful constraint used by human vi­sion. Like other visual constraints, it serves to reduce the

(a)

(c)

countless number of interpretations that are always con­sistent with any given images. In its simplest form, thisprinciple says to reject unstable interpretations of visualstimuli-those interpretations which, ifperturbed slightly,would lead to a qualitative change in the image. It has beenapplied to provide theories ofa number of visual capaci­ties, such as the 3-D interpretation ofline drawings (Bin­ford, 1981), the perception of subjective contours (Al­bert & Hoffman, 1995), the perception of object parts(Biederman, 1987; Hoffman, 1998; Hoffman & Rich­ards, 1984; Hoffman & Singh, 1997; Singh & Hoffman,1997; Singh, Seyranian, & Hoffinan, 1999), the perceptionofshape from shading (Freeman, 1994), the phenomenonofcolor constancy (Brainard & Freeman, 1997), and theperception of transparency (Singh & Hoffman, 1998).

As pointed out earlier, the definition ofrelatability (Def­inition 1, or Ia) does not take into account the curvatureof an edge-only its direction, and the location of thepoint where interpolation must begin. This allows for thepossibility that the point where the interpolated curvemeets the luminance-defined edge may be (I) a point ofinflection (see Figure 6a), or (2) a point ofdiscontinuity ofcurvature (i.e., a second-order discontinuity) of the com­pleted contour (see Figure 6c).6 Both of these cases arenongeneric (Takeichi et aI., 1995). For example, assumethat the end points of the interpolated curve are points ofinflection on the completed contour (as in Figure 6a). Inthis case, a slight change in viewing position would per­turb the occluding surface and the occluded surface slightly

(b)

(d)

Figure 6. It is nongeneric for the end points of an interpolated curve to be pointsofinflection, or C2 discontinuity, on the completed contour.

with respect to each other-so that the points where occlu­sion begins would no longer be points of inflection (seeFigure 6b). Another way to think of this is as follows: Ifwe have two surfaces, one occluding the other, and thecontour ofthe occluded surface has points ofinflection onit, then only for a few special viewing positions will suchan inflection point be a point where occlusion begins. Sim­ilarly, it is nongeneric for the end points of the interpolatedcurve to be points of C2 discontinuity on the completedcontour (see Figure 6d). However, it is less clear to whatextent human vision is sensitive to C2 discontinuities.

Future work on relatability needs to address these is­sues. One possibility might be to take the conditions ofthe proposition proved above as an alternate definition ofrelatability, and then to extend relatability by adding fur­ther conditions that an interpolating curve must satisfy.Some possible examples ofsuch conditions are as follows:The end points of the interpolating curve should not bepoints of inflection, and the interpolating curve should beC2 continuous everywhere, including at the endpoints.In order to do this, a luminance-defined edge should bedefined not only by its direction and end point, but alsoby its curvature.

Another issue is that the current definition of relata­bility is all or none: A pair ofedges either does, or does not,satisfy the two conditions that define relatability (Def­inition I). Psychophysical data, however, show graded­ness in subjects' responses (Kellman & Shipley, 1991;Shipley & Kellman, 1992a). For example, the perceived

Figure 7. A graded measure ofrelatability based on If; (dK/ds)2dsand If; IKa(S) Ids can track the gradual decrease in subjects' re­sponses with increase in angle between two relatable edges (a-e),and with increase in offset between two parallel edges (d-f).

(a)

(b)

(c)

----~

(d)

_/

(e)

(f)

RELATABILITY AND INFLECTIONS 949

strength of boundary interpolation decreases graduallywith increasing angle (see Figures 7a-7c) between a pairof relatable edges (Kellman & Shipley, 1991). Moreover,for edge pairs that are parallel (see Figures 7d-7f), per­ceived interpolation strength decreases gradually with in­creasing degree of offset between the edges (Shipley &Kellman, 1992a), even though only the colinear pair isrelatable. Also, for textures composed ofGabor patches,the detectability ofvirtual lines connecting the patches de­creases with increasing angle between the patches (Field,Hayes, & Hess, 1993); similar effects have been foundwith textures composed ofdots (Pizlo, Salach-Golyska, &Rosenfeld, 1997; Smits & Vos, 1987; Uttal, 1975). Hence,it would be desirable to have a measure of relatabilitythat is graded. Takeichi et al.'s (1995) framework doesallow for some gradedness: The more inflections an in­terpolated curve needs to have, while maintaining conti­nuity ofdirection, the weaker the interpolation is. How­ever, for the two cases mentioned above (see Figure 7),even this measure does not explain the gradedness in per­ceived strength ofinterpolation: For the relatable edge pairsshown in Figures 7a-7c, the number of inflection pointsneeded on the interpolated contour is zero--irrespectiveof the angle between the edges. Similarly, for the paralleledge pairs shown in Figures 7d-7f, each interpolatedcontour needs exactly one point of inflection-irrespec­tive of the degree of (nonzero) offset between the edges.

In the computer vision literature, a number ofmeasuresof smoothness for curves (and surfaces) have been pro­posed, and some ofthese have also been applied to percep­tual psychology (e.g., Blake & Zisserman, 1987; Grim­son, 1981; Sha'ashua & Ullman, 1988; Ullman, 1976;Weiss, 1990). A commonly used measure is variation ofcurvature. For a curve defined on [0, L], the variation ofcurvature is given by

The lower this variation, the smoother the curve. Hence,interpolated curves are typically chosen in order to min­imize this (or some such similar) measure.

One way to develop a graded notion of relatability isas follows: Given two edges, first choose a maximallysmooth curve that interpolates between them-say, byminimizing variation of curvature. (This will also mini­mize the number of inflection points on the curve.) Then,for this curve, use the values

f:1 ICa(S) 1ds

(the measure used in Condition 3 ofthe proposition) and

fL(dICa )2Jo ds ds

(variation ofcurvature) to obtain a graded measure ofre­latability. Note that these two measures can vary inde-

950 SINGH AND HOFFMAN

pendently ofeach other, and that both may be determinantsof the degree of relatability. For example, the curves inFigures 7a-7c all have

( )

2L dICa

fo ~ ds=O,

but they differ in

f:1 ICa(S) Ids

(the values being 0, n13, and n12, respectively). Con­versely, the curves in Figures 7c and 7e both have

rL nJo !ICa(s)lds=2

-that is, they both turn through a total angle of n12, butthey differ in

f:( d:aJds

(which gives value 0 for Figure 7c, but a positive value forFigure 7e).

A graded measure of relatability based on both

f:1 ICa(S) Ids

and

f:( d:aJds

is consistent with the currently available data on bound­ary interpolation that shows gradedness in subjects' re­sponses. It can explain the decrease in perceived strengthofboundary interpolation with both (1) increase in anglebetween the two edges (Figures 7a-7c), and (2) increasein offset between two parallel edges (Figures 7d-7f). Itcan also explain why having more inflection points onan interpolating curve decreases its perceived strength:Adding inflection points on a curve increases both

and

iL( d ICa )2d

ds.o s

Furthermore, in the case of parallel edges,

provides a scale-invariant measure of the degree of offsetbetween the two edges. Such a measure is desirable, be­cause, within a reasonable range ofscalings, one does notexpect the perceived strength ofinterpolation (and hence,the degree of relatability) to change with scaling (e.g., if

the entire figure is expanded to twice-or half-its orig­inal size). The precise way in which these two measurescombine in order to give an overall measure ofrelatabil­ity remains, however, a matter ofempirical investigation.

CONCLUSION

Relatability and minimizing inflections are importantfactors in determining whether or not human vision willinterpolate a contour between two luminance-definededges. They can also serve to formalize the notion of"good continuation" (see, e.g., Kellman & Shipley, 1991).We have stated and proved a proposition that shows that,under appropriate conditions, relatability is mathemati­cally equivalent to the existence of a smooth curve withno inflection points that interpolates between two edges.The conditions in the proposition thus provide an alter­nate definition of relatability, in terms of properties thatan interpolating curve must satisfy, This characterizationis useful because it allows for an easy way to extend thedefinition of relatability (1) to include the role ofgener­icity, and (2) to extend the current all-or-none characterof relatability to a graded measure that can track thegradedness in psychophysical data.

REFERENCES

ALBERT, M. K., & HOFFMAN, D. D. (1995). Genericity in spatial vision.In R. D. Luce, M. D'Zrnura, D. D. Hoffman, G. 1. Iverson, & A. K.Romney (Eds.), Geometric representations ofperceptualphenomena(pp. 95-112). Mahwah, NJ: Erlbaum.

BIEDERMAN, I. (1987). Recognition-by-components: A theory ofhumanimage understanding. Psychological Review, 94, 115-147.

BINFORD, T. O. (1981). Inferring surfaces from images. Artificial lntel­ligence, t 7, 205-244.

BLAKE, A., & ZISSERMAN, A. (1987). Visual reconstruction. Cambridge,MA: MIT Press.

BRAINARD, D. H., & FREEMAN, W.T. (1997). Bayesian color constancy.Journal ofthe Optical Society ofAmerica A, 14, 1393-1411.

FIELD, D. J., HAYES, A., & HESS, R. F. (1993). Contour integration bythe human visual system: Evidence for a local "association field." Vi­sion Research, 33, 173-193.

FREEMAN, W. T. (1994). The generic viewpoint assumption in a frame­work for visual perception. Nature, 368, 542-545.

GRIMSON, W. E. L. (1981). From images to surfaces: A computationalstudy ofthe human early visual system. Cambridge, MA: MIT Press.

HOFFMAN, D. D. (1998). Visual intelligence: How we create what wesee. New York: Norton.

HOFFMAN, D. D., & RICHARDS, W. A. (1984). Parts of recognition. Cog­nition, 18, 65-96.

HOFFMAN, D. D., & SINGH,M. (1997). Salience of visual parts. Cogni­tion, 63, 29-78.

KANIZSA, G. (1979). Organization in vision. New York: Praeger.KELLMAN, P. J., & SHIPLEY, T.F.(1991). A theory ofvisual interpolation

in object perception. Cognitive Psychology, 23, 141-221.MICHOTTE, A., THINES,G., & CRABBE, G. (1991). Amodal completion

of perceptual structures. In G. Thines, A. Costall, & G. Butterworth(Eds.), Michotte s experimental phenomenology of perception(pp. 140-167). Hillsdale, NJ: Erlbaum. (Original work published1964)

MILLMAN, R. S., & PARKER, G. D. (1977). Elements ofdifferentialgeometry. Englewood Cliffs, NJ: Prentice-Hall.

PIZLO,Z., SALACH-GOLYSKA, M., & ROSENFELD, A. (1997). Curve de­tection in a noisy image. VisionResearch, 37, 1217-1241.

SHA'ASHUA, A., & ULLMAN, S. (1988). Structural saliency: The detec­tion of globally salient structures using a locally connected network.

In Proceedings ofthe Second International Conference on ComputerVision (pp. 32I-327). Washington, DC: IEEE Computer SocietyPress.

SHIPLEY, T. E, & KELLMAN, P.J. (I 992a).Perceptionof partly occludedobjectsand illusoryfigures:Evidenceforan identityhypothesis. Jour­nalofExperimental Psychology: Human Perception & Performance,18, 106-120.

SHIPLEY, T.E, & KELLMAN, P.J. (l992b). Strength of visual interpola­tion depends on the ratio of physicallyspecified to total edge length.Perception & Psychophysics, 52, 97-106.

SINGH, M., & HOFFMAN, D. D. (1997). Constructing and representingvisual objects. Trends in Cognitive Sciences, 1(3), 98-102.

SINGH, M., & HOFFMAN, D. D. (1998). Part boundaries alter the per­ception of transparency.Psychological Science, 9, 370-378.

SINGH, M." HOFFMAN, D. D., & ALBERT, M. (in press). Contour com­pletion and relative depth: Petter'srule and support ratio. Psycholog­ical Science.

SINGH, M., SEYRANIAN, G., & HOFFMAN, D. D. (1999). Parsingsilhou­ettes: The short-cut rule. Perception & Psychophysics, 61, 636-660.

SMITS, 1.T.S., & Vos,P.G. (1987).The perceptionof continuouscurvesin dot stimuli. Perception, 16, 121-131.

TAKEICHI, H., NAKAZAWA, H., MURAKAMI, 1., & SHIMOJO, S. (1995).The theory of the curvature-constraint line for amodal completion.Perception, 24, 373-389.

ULLMAN, S. (1976).Filling-inthegaps:Theshapeof subjectivecontoursand a model for their generation.Biological Cybernetics, 25, 1-6.

UTTAL, W. (1975). An autocorrelation theory ofform detection. Hills­dale, NJ: Erlbaum.

WEISS, 1.(1990). Shape reconstructionon a varying mesh. IEEE Trans­actions on Pattern Analysis & Machine Intelligence, 12, 345-362.

NOTES

I. Whilemotivatingtheir definitionof reiatability,Kellmanand Ship­ley (1991) point to the need for a monotonicity constraint: "Intuitively,it requires that the connection must progress continuously from oneedge to the other. The connection cannot extend outward and then re­turn, or double back on itself, etc." (p. 174).Their example of violationof the monotonicityconstraint showsa curve with constant sign of cur­vaturethat turns through a full circleand a half, and which crossesitself(Figure 22, p. 174).They go on to say that the monotonicity constraint

RELATABILITY AND INFLECTIONS 951

is incorporated formally in the two conditions that define relatability(Definition I). Therefore,we shall simplyuse the twoconditionsin De­finition I as defining relatability.

Kellman and Shipley(1991) also offer a different formulationof re­latability (giventhat the edges are not parallel}-namely, 0"; Rcos ~.,; r(see Figure 2), whereR and r are chosen so that R ~ r (p. 176).They as­sert that this is equivalent to Definition 1. Again, therefore, we shallsimply use Definition I for relatability.

2. The idea of minimizing the number of inflection points on an in­terpolating contour is perfectly consistent with standard approaches tocurve interpolation in computer vision that minimize variation in cur­vature. For example, curves that minimize

also minimize number of inflection points.3. However, the length of the edges may affect a different factor,

called support ratio (Shipley & Kellman, 1992b; see also Singh, Hoff­man, & Albert, in press).

4. In Appendix 8 of Kellman and Shipley (1991), Skeath uses thisconstruction to show that "when two edges meet the relatability crite­rion, a first-order continuouscurve can be fit between them, tangent tothe end point at both edges" (p. 176).This is undoubtedlytrue (and theconstruction gives a concrete example). However, it is not a criterionthat can serve to characterizerelatability,since a first-order continuouscurve can also be fit between two edges that are not relatable. The pur­pose of our proposition is to givejust such a characterization-s-one thatis both necessary and sufficient.

5. For any Cl plane curve a defined on some interval [0, L], it is al­ways possible to define O(s) such that 0 is continuous on [0, L] (see,e.g., Millman & Parker, 1977, p. 55). In general, this may involveadding integral multiples of 21C in places, in order to ensure continuity;but this is not the case here.

6. These twocases are independent.Guaranteeingthat the end pointsof the interpolated contour are not points of C2 discontinuitydoes notguarantee that they will not be points of inflection-and vice versa.

(Manuscript received February 17, 1998;revision accepted for publication July 30, 1998.)