Contours information and the perception of various visual illusions
CContours information and the perception of variousvisual illusions
Shu Tian Eu and Ee Hou Yong Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang TechnologicalUniversity, Singapore Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA * [email protected] ABSTRACT
The simplicity principle states that the human visual system prefers the simplest interpretation. However, conventional codingmodels could not resolve the incompatibility between predictions from the global minimum principle and the local minimumprinciple. By quantitatively evaluating the total information content of all possible visual interpretations, we show that theperceived pattern is always the one with the simplest local completion as well as the least total surprisal globally, thus solvingthis apparent conundrum. Our proposed framework consist of (1) the information content of visual contours, (2) direction ofvisual contour, and (3) the von Mises distribution governing human visual expectation. We used it to explain the perception ofprominent visual illusions such as Kanizsa triangle, Ehrenstein cross, and Rubin’s vase. This provides new insight into thecelebrated simplicity principle and could serve as a fundamental explanation of the perception of illusory boundaries and thebi-stability of perceptual grouping.
Introduction
A fundamental issue in human perception research is how human subjects show a clear preference for a specific interpretationof visual stimulus from many different ways in which a stimulus could be possibly interpreted. In particular, this phenomenoncould be clearly observed when we examine various cases of visual illusions (eg. the Kanizsa illusion which consists of anillusory triangle with three occluded circles instead of three Pac-man inducers) . This gives rise to several interesting questions:does human perception choose an interpretation which is more probable or which is simpler? Are these two interpretationsactually equivalent and how do we define the notion of the most probable interpretation and the simplest interpretation?Over the last century two different paradigms were proposed to explain the human perception of visual stimuli: thelikelihood principle and the simplicity principle . The likelihood principle states that visual system has the tendency toperceive the most probable (the one with maximum probability) interpretation while the simplicity principle states that visualsystem has a preference towards the simplest interpretation of a visual stimulus. These two principles seem to be incompatible,and were always regarded as competitors historically . This debate continued until 1996 when Chater suggested aninteresting new point of view that these two principles could actually be reconciled using Kolmogorov complexity theory.He showed that the visual interpretation which objectively is the most likely (likelihood principle) to be correct is in fact theone with the minimum length of description (simplicity principle). Under this perspective, the most appealing aspect of bothprinciples, namely, the veridicality of perception in terms of external world and the efficiency of the visual system in termsof internal resources are preserved . In recent years the Bayesian model has become a popular choice in unifying these twoprinciples . For example, Feldman has shown that the most likely visual interpretation should be the one which is lowestin the partial order of the hierarchical interpretation space (simplicity principle) which maximizes the Bayesian posteriorprobability (likelihood principle).In this paper, we aim to address another important question within the framework of simplicity paradigm, which is thecontradiction between the global minimum principle and the local minimum principle. The simplicity principle is derived fromthe well known law of Pragnänz. The law of Pragnänz is the most general Gestalt rule that states that people will perceive andinterpret ambiguous or complex images as the simplest form(s) possible . This implies that the visual system, like any otherphysical system, will respond to the stimuli with a tendency to evolve into the equilibrium state involving minimum energyloading. Hochberg and McAllister claimed that in the case of vision, this energy load is, in fact, the information load, andhence proposed that the less the amount of information needed to define a given interpretation of pattern as compared to otheralternatives, the more likely that interpretation is perceived . In their original paper, information is defined as the number ofdifferent items (eg. number of different line segments, number of different angles) to be described in order to preserve the“figural goodness", which is a rather vague definition without much mathematical rigour. The global minimum principle was a r X i v : . [ q - b i o . N C ] N ov mplemented in several perceptual coding languages , in which it was rephrased and referred to as the minimum principleor the principle of descriptive economy . In this context the principle states that the perceptually preferred interpretation is theone that requires the fewest predicates in coding models.However, the global minimum principle was not supported by some theorists . Kanizsa, for example, asserted that it isthe role of good continuation that governs perceptual organization . He believed that the minimum principle should only beapplied to local regions of the figure. Our perception and interpretation of figure will not be affected by all global regularitiesand symmetries within the figure predicted by the global minimum principle. Fig 1 shows two clear examples of how patternswould be completed according to global minimum principle and local minimum principle respectively. It is clear the predictedoutcomes are incompatible when used with conventional coding models e.g. structural information theory’s formal codingmodel , Kolmogorov complexity measure, etc. Figure 1.
Global completion and local completion of patterns. (a) An example taken from Boselie, 1988 . Theinterpretation predicted by global completion is preferred. (b) Example proposed by Kanizsa to show that the local completionpattern is more prevalent, thus disproving the global minimum principle .In this paper, we aim to resolve this conundrum using the framework of information content of visual contours. Bycomparing the total information content, namely the surprisal of all possible interpretations, it is observed that the perceivedpattern is always the one with the least surprisal. It should be noted that this evaluation of information content is carried outpurely geometrically on the final description of the illusory pattern. Unlike conventional coding model, our method showsthat the local completed pattern in fact possesses the least amount of information globally, thus reconciling the two seeminglyincongruent principles. Human perception selects the pattern with the least total information content based on informationalong contours. The vagueness of the simplicity paradigm is hence removed. This result also agrees with previous empiricaland experimental studies.This information analysis is also adopted in analyzing three different types of visual illusions (Fig 2), including Kanizsaillusion, Ehrenstein illusion, and Rubin’s vase . Kanizsa illusion (Fig 2a), which has not yet been explained within theframework of global minimum principle, is a class of illusions involving illusory contours, modal and amodal completion .Modal completion refers to the process of perceptual completion without occlusion whereas amodal completion refers to thecompletion of occluded objects . Amodal completion is highly variable since the visual system is free to choose the formof continuity based on local interpretations of the cues provided by the visible part . In the presence of three Pac-maninducers (the circle with a missing piece) on a homogeneous background, we will perceive the figure as a central modalcompleted triangle that appears brighter and three occluded circles. Although there is no presence of physical boundaries,illusory boundaries will arise along the edges of the triangle. We will show that this illusion arises as perceiving the figure asa triangle and three circles is more economical compared to perceiving it as three circles with missing pieces, as the formercontains less information.Unlike Kanizsa illusions in which illusory contours are induced approximately collinear to the direction of the edges,Ehrenstein illusion is another type of illusion where the illusory contours are induced perpendicularly to the end of the lines(Fig 2b). It involves the central modal completion of a bright plate and a cross behind the plate. The emergence of the illusorycontour will be discussed in detail in a later section. Finally, we discussed the Rubin’s Vase, which is a bi-stable two dimensionalfigure. The Rubin’s Vase in Fig 2c could either be perceived as a central vase or two faces. This bi-stability of figure-groundrelation will also be studied within the framework of contours’ information. igure 2. Three examples of optical illusions. (a) Kanizsa triangle, (b) Ehrenstein cross and (c) Rubin’s Vase.
Methods
The concept of surprisal was first introduced by Shannon . In a recent work, Feldman and Singh used Shannon’s conceptof information content and surprisal to quantify contours and object boundaries, a framework that we will adopt. Givena continuous measure M and a probability distribution p ( M ) that represents our expectation of the value of M before anymeasurement is taken, what is the information gained by measuring M ? Shannon claimed that the more the measured valuedeviates from the expected value, the more “surprising" it is, and hence the more information it contains. This surprisal, alsoknown as information content, is defined as s ( M ) = − log [ p ( M )] , (1)is always positive for a discrete measure and can take on arbitrary real values when the measure M is continuous . To computethe surprisal along the contours, the targeted curve is sampled at interval of length ∆ s , so that a curve of length L will have N = (cid:98) L ∆ s (cid:99) number of straight edges, where (cid:98) x (cid:99) is the floor function, e.g. (cid:98) . (cid:99) =
3. The curve is represented by equally spaced N + x , . . . , x N + connected by N straight edges, with unit tangents ˆt , . . . , ˆt N , such that ˆt i = ( x i + − x i ) / || x i + − x i || and || x i + − x i || = ∆ s as shown in Fig 3a. For closed curves, x = x N + . In general, the path length will not be integer multipleof ∆ s , i.e. N ∆ s ≤ L < ( N + ) ∆ s , in which case, we let the last straight edge be of length || x N + − x N || = L − ( N − ) ∆ s .The human eye can resolve a periodic signal (e.g. alternating black and white bars) at a spatial frequency of up to about θ =
60 cycles per degree. Assuming that one is viewing the image at a distance of D = RP = θ D = × π ≈ × − m. For all our calculations, we will set the sampling distance to be the humanresolution limit i.e. ∆ s ≈ RP . Every curve in a visual image, including both closed and illusionary ones, can be broken into acollection of discrete straight edges. From point to point along this sampled curve, the turning angle ∆ φ , which is the anglebetween two adjacent edges, is measured and serves as the parameter of interest:cos ( ∆ φ i ) = ˆt i · ˆt i + . (2)Either ˆt N + = ˆt (for closed curves) or ˆt N + may be specified as a boundary condition (for open curves). This choice ismotivated by the fact that ∆ φ is invariant under translational and rotational symmetry, which is the nature of visual perception.Now, having chosen the turning angle ∆ φ as the continuous measure, we need to choose a probability distribution p ( ∆ φ ) thatrepresents our expectation of the value of ∆ φ before any measurement is taken. In the spirit of simplicity paradigm, it isnatural to think that our visual system will expect any curve to continue along its last tangent, since straight line is the simplestextension to any existing pattern. Let us assume that the change in tangent direction on a smooth curve follows the von Misesdistribution centered at ∆ φ = (Fig 3b) defined on the interval [ − π , π ] : p ( ∆ φ ) = A (cid:48) ( b ) exp [ b cos ( ∆ φ )] , (3)where b is the spread parameter that is a measure of the concentration and acts as the inverse of the variance. A (cid:48) ( b ) is anormalizing constant independent of ∆ φ (s.t. (cid:82) π − π p ( ∆ φ ) d ∆ φ = b as shown: A (cid:48) ( b ) = π I ( b ) , I ( b ) = ∞ ∑ m = m ! Γ ( m + ) (cid:18) b (cid:19) m , (4) here I ( b ) is the modified Bessel function of the first kind of order zero. This distribution is chosen as it agrees well withseveral empirical and experimental studies, including studies conducted by Feldman regarding human subjects’ expectation ofhow a curve is most likely to continue . Moreover, based on existing research on orientation selectivity of cortical neurons, thetuning curve of the neuronal spike response is best fitted by the von Mises function, which possesses Gaussian-like propertiesfor angular measurement . Figure 3.
Discrete curve and von Mises distribution. (a) A curve that is discretized into N nodes x i that are connected byedges of length ∆ s and tangent ˆt i . The turning angle between two successive tangent is ∆ φ i . (b) The von Mises distribution p ( x ) with different values of b centered at µ = b is increased. Note how p ( ) > b > . x i and x i + the information gained, which is the surprisal, is simply: s ( ∆ φ i ) = − log [ p ( ∆ φ i )] = − log ( A (cid:48) ( b )) − b cos ( ∆ φ i ) = − log ( A (cid:48) ( b )) − b ˆt i · ˆt i + . (5)The first term − log A (cid:48) ( b ) has no dependence on the turning angle ∆ φ . The negative cosine dependence on ∆ φ in the secondterm shows that the larger the angle deviating from zero (regardless of the direction), the larger the surprisal would be. Thetotal surprisal of a curve, S, is given by S = N ∑ i = s ( ∆ φ i ) = − N ∑ i = log [ p ( ∆ φ i )] = − b N ∑ i = ˆt i · ˆt i + − N log ( A (cid:48) ( b )) (6)Interestingly, this has the same mathematical form as the discretized energy of the wormlike chain model used to describeDNA . For a pattern consisting of M disjoint contours (both real and illusionary), each composed of N j segments s j ( ∆ φ i ) ,where j = , · · · M and i = , · · · , N j , the total surprisal S is S = M ∑ j = N j ∑ i = s j ( ∆ φ i ) = − M ∑ j = N j ∑ i = log [ p j ( ∆ φ i )] . (7) Results and Discussions
Kanizsa Illusions
To show that the perception of illusory contours and modal completion are due to the minimum principle of information, wewill start with the famous Kanizsa triangles, which has two possible perceptual interpretations. The total surprisal of the twodifferent scenarios: (i) to view the three Pac-man inducers as it is (without modal and amodal completion), and (ii) to view it asa modal completed N -polygon and amodal completed full circles. This is shown in Fig 4a where N = Kanizsa triangle
Kanizsa illusion shows a white colored equilateral triangles with side of length R in the foreground, and three black coloredcircles of radius r in the background. The distance between the centers of two circles is R . For example, the number of edges igure 4. Two ways of interpreting Kanizsa triangles. (a) We can either perceive the illusion as (i) three Pac-man inducersor (ii) one modal completed triangle and three amodal completed full circles. (b) Total surprisal as a function of the von Misesparameter b for case (i) (blue curve) and case (ii) (black curve).on a circle of radius r is N = (cid:98) π r ∆ s (cid:99) . For typical values r = R = ∆ s = × − cm, we find that N is of order 10 .The total surprisal for case (i), which consists of three Pac-man inducers is: S (i) (cid:52) = − N (cid:18) + π (cid:19) log ( A (cid:48) ( b )) − Nb cos (cid:18) π N (cid:19) − Nb π . (8)The total surprisal for case (ii), which consists of three full circles and an illusory triangle is: S (ii) (cid:52) ≈ − N log ( A (cid:48) ( b )) (cid:20) + R π r (cid:21) − Nb cos (cid:18) π N (cid:19) − NbR π r . (9)The total surprisal for this two cases (with the aforementioned parameter values) as a function of the von Mises parameter b isshown in Fig 4b. By inspection, we see that if b ≥ . S (i) (cid:52) > S (ii) (cid:52) .Alternatively, we can find the difference in surprisal between the two visual interpretation is ∆ S (cid:52) = S (i) (cid:52) − S (ii) (cid:52) analytically: ∆ S (cid:52) ≈ N (cid:20) R − r + π r π r (cid:21) (cid:2) log ( A (cid:48) ( b )) + b (cid:3) . (10)In order for ∆ S (cid:52) >
0, we find thatlog ( A (cid:48) ( b )) + b ≥ . (11)This can be further simplified into b ≥ .
551 as before. This spread parameter b controls the width of von Mises distribution:the larger the value of b , the narrower the distribution (see Fig 3b). At the critical value b = . x = p ( ) =
1. For b > . p ( ) > s starts to become negative. As long as the von Mises distribution is sharply peaked, outcome (ii) will be preferred. Kanizsa square and polygons
Following the same treatment, the above results could be generalized to Kanizsa square and even polygons. For the Kanizsasquare, there are also two ways of interpreting the stimulus: (i) 4 Pac-man inducers and (ii) a modal completed square with 4amodal completed circles. The total surprisal difference between this two case is found to be: ∆ S (cid:3) = S (i) (cid:3) − S (ii) (cid:3) ≈ N (cid:18) R − r + π r π r (cid:19) (cid:2) log ( A (cid:48) ( b )) + b (cid:3) . (12)This eventually leads to the same inequality for the spread parameter of the von Mises distribution: b ≥ . N -sided Kanizsa polygon will have the following total surprisal difference between two differentinterpretation: ∆ S N -polygon = N N (cid:18) R − r + α r π r (cid:19) (cid:2) log ( A (cid:48) ( b )) + b (cid:3) , (13)where α = π / N . This leads to the same inequality b ≥ . hrenstein figures The same analysis can be extended to another type of illusion, Ehrenstein figures, which involves modal completion induced byendpoints. While looking at the Ehrenstein figure, human subjects tend to interpret it as a central bright circle appearing on topof two crossing lines. Again, we will try to explain this preference of interpretation using the surprisal method introduced.Since line width is found experimentally to have positive effect on the clarity of the contour induced , it is natural to considerthe inducers as 2D rectangles with certain thickness instead of 1D lines. Fig 5a shows 4 rectangular inducers with length l andwidth d ( l (cid:29) d ) and a central gap of radius r . There are two ways to interpret the figure: (i) 4 inducing rectangles and (ii) acentral modal completed circle with an occluded cross (Fig 5b). The total surprisal difference between case (i) and (ii) is: ∆ S E = ( N + r − d − )[ log ( A (cid:48) ( b )) + b ] + b ≈ N [ log ( A (cid:48) ( b )) + b ] . (14)We see that we arrive at the same inequality as the Kanizsa illusions. As long as b ≥ . Figure 5.
Ehrenstein cross and two ways of interpreting Ehrenstein cross. (a) The parameters used for Ehrenstein crosscalculation. (b) Two ways to interpret the figure: (i) Four inducing rectangles, (ii) one central modal completed circle with anoccluded cross.
Global minimum and local minimum
From previous discussion, it is clear that the perception of illusory contours could be explained by comparing the total surprisalof different possible figural interpretations. Now, the difference and advantage of this framework compared to previous codingmodels shall be discussed. As shown in Fig 1, global minimum principle and local minimum principle predict differentinterpretations. For the example shown in Fig 1(a), the interpretation predicted by global minimum principle is preferredexperimentally. On the other hand, for the example shown in Fig 1(b), the result predicted by local minimum principle isshown to be more prevalent experimentally . This introduces an ambiguity into the selection process of the most plausibleinterpretation as these two principles predict different outcomes. However, by calculating information content under ourframework, it could be proven that the preferred interpretations in both cases in fact possess the lowest surprisal comparedto their counterparts. The case in Fig 1(b) is especially worth noting as we show that the result predicted by local minimumprinciple in fact possesses lower surprisal than the other, and hence it is the minimum information configuration globally. Thissheds a light into combining these two principles, as they could be equivalent under the new way of evaluating the informationcontent of the interpretation of figure.
Rubin’s Vase
Other than explaining illusions involving illusory contours and resolving the difference between global and local minimum, thismethod can also be used to explain the figure-ground illusion, which is the bi-stability of perceptual grouping. In our previousdiscussion, the mean of von Mises distribution is zero, which implies that our perception is insensitive to the propagationdirection of tangent, i.e. turning clockwise ∆ φ < ∆ φ > ig 6a. The resulting surprisal is p ( ∆ φ ) = A (cid:48) ( b ) exp [ b cos ( ∆ φ ± δ )] , (15)where δ > . Consecutivestraight edges, i.e. ˆt i = ˆt i + , rather than being the most expected case as before, are now slightly surprising. For the minuscase (preference for CCW), this means that consecutive tangents that turn in the CCW sense carry greater information thanotherwise equivalent CW tangents and vice versa. This broken asymmetry means that the way we “draw" the visual contours isnow important. Consider a person biased towards CCW rotations with skewness δ = ∆ φ = (cid:98) ∆ sr (cid:99) . By tracing the circle in a CWmanner ( ∆ φ = − δ < ) , the total information content is S CW = − N log (cid:0) A (cid:48) ( b ) exp ( b cos ( −| ∆ φ | − δ )) (cid:1) = − N log (cid:0) A (cid:48) ( b ) exp ( b cos ( δ )) (cid:1) . (16)On the other, when visually tracing in a CCW manner ( ∆ φ = δ > ) , the total information content is S CCW = − N log (cid:0) A (cid:48) ( b ) exp ( b cos ( | ∆ φ | − δ )) (cid:1) = S CW = − N log (cid:0) A (cid:48) ( b ) exp ( b ) (cid:1) . (17)The difference in surprisal is ∆ S = S CW − S CCW ≈ Nb δ > . (18)Thus we see that the direction of the contour lines will determine the total surprisal for the case of a skewed von Mises Figure 6.
Total surprisal for circle and Rubin’s vase. (a) The total surprisal for circle drawn in clockwise andcounterclockwise sense will be different if the underlying von Mises distribution is skewed. (b) Adding directions to the twocontour lines in the Rubin’s vase image. There are a total of four different combinations, leading to different total surprisal.distribution. For a general picture with multiple contour lines, the choice of contour directions with the least total informationcontent would be preferred.Referring back to the Rubin’s vase illusion, there are four ways to add directions to the two contour lines, enumerated by(1) to (4) as shown in of Fig 6b. A person with a preference for CCW rotations will prefer to follow the contours in the senseshown in (1) since it has the lowest total surprisal. In fact, we find that S CCW ( ) < S CCW ( ) = S CCW ( ) < S CCW ( ) . (19)The continuity of the contour directions together with the preference for CCW rotations meant that we get two disjoint parts.The directions of the contours is such that the face becomes the “figure" and the vase becomes the “ground" and the resultinginterpretation is that of two faces. On the other hand, a person who has a predisposition for CW rotations, we get S CW ( ) < S CW ( ) = S CW ( ) < S CW ( ) . (20)Hence, the contours directions of (3) would be the preferred outcome. In this case, the continuity of the contour directions andthe predilection for CW rotations results in one connected big part. In this case, vase is now the “figure" while the two faces arethe “ground" and the resulting interpretation is that of a vase. Here we see that the formation of “figure" and “ground" thatresults from the choice of contour directions will tilt the optical illusion towards different interpretations.In summary, in this paper we give a brief review on the status of the long celebrated simplicity principle, which is derivedfrom the law of Pragnänz and later referred to as the global minimum principle. We have shown that by employing Feldmanand Singh’s method in calculating the surprisal along contours and choosing a suitable spread parameter for the von Misesdistribution governing human visual expectation, visual illusions involving the perception of illusory contours such as Kanizsa llusions and Ehrenstein figures could be well explained. Unlike conventional coding model in which the contradiction betweenthe global minimum principle and local minimum principle is inevitable, this method naturally resolves the contradictionto some extent by showing that the prediction by the local minimum principle, which is experimentally proven to be moreprevalent, in fact possesses the globally minimum surprisal. The bi-stability of perceptual grouping, for example in the case ofthe Rubin’s vase is also studied. Individual with biased visual expectation, in which his or her perception is governed by skewedvon Mises distribution, has a tendency towards perceiving one interpretation over another based on the choice of contourdirections. References Kanizsa, G. et al.
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S.T.E. and E.H.Y. acknowledge support from Nanyang Technological University, Singapore, under its Start Up Grant Scheme(04INS000175C230).
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