WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia IJCNN A Group-decision Making Model of Orientation Detection

Hui Wei and Yuan Ren Zheyan Wang Lab of Cognitive Model and Algorithm School of Mathematical Sciences School of Computer Science Fudan University Fudan University Shanghai 200433, China Shanghai 200433, China Email: [email protected] Email: {weihui,renyuan}@fudan.edu.cn

Abstract—The feedforward model proposed by Hubel and B. Motivation and contribution Wiesel partially explained orientation selectivity in simple cells. This classical hypothesis attributed orientation preference to Although partially supported by some physiological evi- idealized alignment of geniculate cell receptive fields. Many dence [4]–[6], the Hubel-Wiesel model is more like a geo- scholars have been either revising this model or putting forward metric sketch based on template-matching in essence and has new theories to account for more related phenomenon such as contrast invariant tuning. None of the previous neural models been questioned in many aspects [7], [8]. One of the major is complete in implementation details or involves strict compu- difficulties in this model is the explanation of contrast invariant tational strategies. This paper mathematically studied a detailed orientation tuning [9]–[11], which has been widely discussed but vital question which has long been neglected: the possibility of at all times. Many scholars have therefore been modifying massive variable-sized, unaligned geniculate cell receptive fields this model [12], [13], while others have been putting forward producing the orientation selectivity of a simple cell. The response curve of each afferent neuron is fully utilized to obtain a local distinct theories with intra-cortical connections including re- constraint and a group-decision making approach is then applied current models, feedback models and hybrid models [14]– to solve the constraint satisfaction problem. Our new model does [18]. Literatures [15], [19] comparatively studied the most not achieve just consistent experimental results with physiological popular theories on orientation selectivity and summarized data, but consistent interpretations of several illusions with the main differences between them. Nevertheless, all the observers’ perceptions. The current work, which supplemented the previous models with necessary computational details, is previous research has neglected several vital problems and based on ensemble coding in essence. This underlying mechanism implementation details, and is therefore incomplete to some helps to understand how visual information is processed in from extent. the to the cortex. Essentially, the foundation of Hubel-Wiesel model is the appropriate alignment of the RFs of all the LGN cells from I. INTRODUCTION which a simple cell receives input, and the centers of these A. Orientation selectivity in simple cells RFs should therefore be similar in size. Nevertheless, the According to the neurobiological studies on the visual RF size varies surprisingly among neighboring neurons in mechanism, ten layers of cells in the retina and six layers the LGN [20] but the center-surround ratio remains almost in the lateral geniculate nucleus (LGN) form a multi-level constant [21]. Even if all LGN cell RFs were identically big network with feedback control and can achieve orientation in their coverage of the photoreceptor layer, it would be too detection instantaneously, and the results are eventually stored idealized to hypothesize such a kind of alignment. It is widely in the primary (V1) [1]. The retinal ganglion accepted that LGN cell RFs are identical to GC RFs [22], cells (GCs) and the LGN cells play important roles in ori- we therefore just review the anatomical structure of a GC entation detection. Their concentric receptive fields (RFs) RF. As is shown in Figure 2, the RF of a GC is composed are responsible for primarily integrating physical stimuli and of the photoreceptors connected by the dendrite fields of the all the subsequent visual tasks are accomplished with the horizontal cells and the bipolar cells. The distributions of the representations produced by these neurons. dendrite fields are quite diverse. These RFs are most likely to Nobel Prize winners Hubel and Wiesel discovered orien- distribute irregularly. tation selectivity in V1 simple cells (SCs) and proposed a As far as the responses of LGN cells to stimuli are con- feedforward model accounting for this phenomenon [2], [3]. cerned, the Difference of Gaussian (DoG) function, which Under the famous hypothesis, the RF of an SC is produced had been found to be a desirable description of GCs with by the collinearly aligned RFs of the afferent LGN cells in concentric RFs [23], [24] more than three decades ago, was certain orientation, and the SC is therefore the most sensitive extended to quantitatively study LGN cell RFs [25], [26]. But to stimuli in that orientation. Figure 1(a) briefly illustrates the the individual effect of one LGN cell in orientation selectivity connectivity pattern of the classical Hubel-Wiesel model. has seldom been investigated. Moreover, most experiments

U.S. Government work not protected by U.S. copyright Fig. 1. Hubel and Wiesel’s feedforward model and the proposed model in this paper.

Fig. 2. The sketch of a GCs RF and the neuronal connections. in the earlier researches were done with less natural stimuli processing, since lines are just defined by collinear edges including light bars, spots and gratings. In the real world, in an image, our model also applies to line detection with either an isolated bar, a spot or a grating can be rarely seen. little alternation required. This model can also explain the Factually, it is contrast edges that are the most natural type of generation of several famous visual illusions. Another major stimuli. All these problems outline the topics of this paper: contribution of this paper is a proposed neural network with 1) How do LGN cells respond individually to a contrast specific computing strategies which is realizable for the neural stimulus with a linear edge? system. 2) How can SCs utilize the response curves of LGN cells? 3) How will the orientation of the edge be determined in C. The organization of this paper case of variable-sized, unaligned LGN cell RFs with In the introduction above, we concisely described the popu- contrast invariance? lar models of orientation selectivity, their incompleteness and In answer to the above questions, this paper brought forward the anticipation of our work. The remainder of this paper is a computational model. The structure and the core mechanism organized as follows. Section II proposes a multiple guesses- of our model is shown in Figure 1(b) by contrast with the based fitting model and discusses the underlying constraint classical hypothesis. The current work mainly aims to relieve satisfaction problem. Section III gives a numeric solution and the rigid constraints and enrich the features of existing models. analyzes its properties. Section IV designs a computational Most discussions are purely mathematical and do not conflict network for neural implementation. Section V simulates SC with universally recognized facts in neural science. For image RFs using the new model. Section VI illustrates the appli- cations in explaining visual illusions. Finally Section VII of the neuronal response curve which embodies significant concludes the paper. information.

II. A COMPUTATIONAL MODEL BASED ON CONSTRAINT SATISFACTION The classical feedforward model and other related work are incomplete and have several difficulties in the implemen- tation details. This paper focuses on the questions listed in the introduction under a new computational framework with strict definitions and proofs. To keep consistent with neuronal connections physiologically, the basic structure of the network consists of an SC, a layer of afferent LGN cells and their unaligned RFs of variable sizes. The stimulus takes the form of a contrast edge completely covering these RFs. (a) The RF of a neuron and a (b) For one cell, all the possible solutions contrast stimulus. correspond to the tangent lines to a con- The previous models emphasize the regular arrangement centric circle of its RF. of LGN cell RFs of similar sizes. Such a rigid constraint will be totally unnecessary as long as it can be proven that (1) Key information about the contrast edge can be inferred from individual cells’ responses. (2) The SC can synthetically analyze all this information to decide the orientation of the stimulus. The whole process is based on group decision- making. This section mainly deals with the first question. Following traditional methods, we also use DoG to quanti- tatively describe LGN cell RFs. The DoG function had been originally recognized as a successful descriptor of GC and was thereafter modified for fitting a wide range of neurons [27]. According to the report in [28], the normalized response 𝑅 of an LGN cell to a contrast edge can be written by a simplified DoG with respect to the distance 𝑟 from the center point of the RF to the edge, and is independent of the contrast. For the example shown in Figure 3a, 𝑑 denotes the width of coverage, 𝑟𝑠 denotes the RF radius, and therefore 𝑟 = ∣𝑑 − 𝑟𝑠∣. 𝑅 𝑑 For simplicity, this paper transformed against 2𝑟𝑠 .One response curve is shown as an example in Figure 3c. Usually, (c) Usually there are two intersections generated by a response 𝑅0 on the for a given value 𝑅0 of response (represented by the solid line), response curve. there exist two values 𝑟1 and 𝑟2 of the corresponding distance Fig. 3. The Procedures of generating a speculation as a local constraint. (represented by the dashed lines) and both can be considered as a guess for the distance. In fact, numeric experiments revealed that the computational accuracy was hardly affected by the two III. THE SOLUTION AND THE IMPLEMENTATION candidates and the smaller value denoted just by 𝑟0 is therefore A. The numeric solution selected in the program. A possible edge should lie on a line According to the previous section, the constraint satisfaction tangent to the concentric circle of radius 𝑟0. For multiple LGN problem can be written as the following mathematical ques- cells, the edge should lie on the common tangent line to all tion: to find the common tangent line(s) 𝑦 = 𝑎𝑥 + 𝑏 of the such circles, which can be explained by Figure 3b. circles of radii 𝑟𝑖 centered at (𝑥𝑖,𝑦𝑖) ,𝑖=1, 2, ⋅⋅⋅ ,𝑛 respec- As a result, for each LGN cell alone, the location of its tively. Usually the line does not exist exactly due to the errors RF and the guessed distance from the edge form a local of guesses, we instead seek for the optimal one minimizing 𝑛 ( )2 ∑ ∣𝑦 −𝑎𝑥 −𝑏∣ constraint independently while all of them together turn into 𝑖√ 𝑖 − 𝑟 𝑎2+1 𝑖 . The problem is then transformed into a global constraint satisfaction problem for the SC. Since 𝑖=1 the distance is defined by RF size and independent of the a generalized least square. Thus contrast, contrast invariant orientation selectivity in the SC ∑𝑛 ( )2 ∣𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏∣ with variable-sized LGN cell RFs can be realized as long min √ − 𝑟𝑖 = 𝑎2 +1 as the constraint satisfaction can be solved. Although this is 𝑖=1 ∑𝑛 ( )2 really not a novel approach, to our knowledge, little effort has (𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏) min 𝑠𝑔𝑛 (𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏) √ − 𝑟𝑖 (1) been made to associate constraint satisfaction with neuronal 𝑎2 +1 connections correspondingly or to solve it within neuronal 𝑖=1 capabilities. More importantly, our approach made best use where 𝑠𝑔𝑛 (𝑥)=1for 𝑥 ≥ 0 and otherwise 𝑠𝑔𝑛 (𝑥)=−1. −→ −→ Let 𝑟 =(𝑟1,𝑟2, ⋅⋅⋅ ,𝑟𝑛), 𝑦 =(𝑦1,𝑦2, ⋅⋅⋅ ,𝑦𝑛), 𝑎∗ and suppose that 𝐷𝑘 will no longer change after the 𝑘th −→ 𝑇 𝑡 = √ 1 𝑧 =(𝑎, 𝑏) 𝐷 = 𝑎2+1 , , iteration. Then we have 𝑑𝑖𝑎𝑔 ((𝑠𝑔𝑛 (𝑦1 − 𝑎𝑥1 − 𝑏)), ⋅⋅⋅ ,𝑠𝑔𝑛(𝑦𝑛 − 𝑎𝑥𝑛 − 𝑏)) and  (√ √ )  𝑡 −→ 2 2  𝑥 𝑥 ⋅⋅⋅ 𝑥 𝑢 𝐷𝑘 𝑟 𝑎 +1− 𝑎 +1  1 2 𝑛 ∣𝑎𝑘+1 − 𝑎𝑘∣ 1 𝑘 ∗ 𝐴 = Then we get a pair of equivalent =  (√ √ ) 11⋅⋅⋅ 1 ∣𝑎𝑘 − 𝑎∗∣  𝑡 −→ 2 2  𝑢1𝐷𝑘 𝑟 𝑎 +1− 𝑎∗ +1  minimizations 𝑘−1 ( ) ∣𝑎𝑘 − 𝑎∗∣ ∑𝑛 (𝑦 − 𝑎𝑥 − 𝑏) 2 ≈ (8) 𝑖 𝑖 ∣𝑎𝑘−1 − 𝑎∗∣ min 𝑠𝑔𝑛 (𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏) √ − 𝑟𝑖 𝑎2 +1 𝑖=1 −→ −→ −→ 2 The convergence is therefore of the order 1.LetΔ= ⇔ min ∥𝑡𝐷 𝑦 − 𝑟 − 𝑡𝐷𝐴 𝑧 ∥2 (2) 𝑡 −→ 𝑢1𝐷𝑘 𝑟 , and then the convergent rate is dependent on Δ. and it can be easily proven that the optimal solution to (2) Similarly we can prove the stability under the convergent will also satisfy the following equation condition. Specifically,  ( ) 𝑇 −→ 𝑇 −→ −→  √ √  (𝑡𝐷𝐴) (𝑡𝐷𝐴) 𝑧 =(𝑡𝐷𝐴) ⋅ (𝑡𝐷 𝑦 − 𝑟 ) (3)  𝑡 −→ 2 2  ∣𝑎𝑘+1 − 𝑎∗∣ = 𝑢1𝐷𝑘 𝑟 𝑎𝑘 +1− 𝑎∗ +1  Since 𝑡 is a function of 𝑎, an iterative method is needed to −→ < ∣𝑎𝑘 − 𝑎∗∣ (9) find the solution to 𝑧 : iteratively substitute the newest value of 𝑎 into 𝑡 and re-solve (3) for a more accurate value of 𝑎 until the change becomes negligible. Specifically, by introducing Since the iterative process (6) involves computing the QR 𝐴 pseudo-inverse matrix, we have factorization of the( matrix) 2×𝑛, the complexity of this iterative process is 𝑂 𝑛2 . −→ + −→ −→ 𝑧 𝑛+1 =(𝑡𝑛𝐷𝑛𝐴) ⋅ (𝑡𝑛𝐷𝑛 𝑦 − 𝑟 ) = 𝐴+−→𝑦 − 1/𝑡 𝐴+𝐷 −→𝑟 𝑛 𝑛 (4) IV. IMPLEMENTATION WITH NEURAL NETWORK +−→ where the initial vector takes 𝑧0 = 𝐴 𝑦 . It has been made clear that a single LGN cell can contribute B. Convergence Analysis relative position of the contrast edge the same distance to its RF center. Since a simple cell receives signals from multiple It can be easily verified that ⎛ ⎞ LGN cells, there always exists a unique solution best satisfying ∑𝑛 ∑𝑛 all the constraints. This group decision-making mode is similar ⎜ 𝑛𝑥1 − 𝑥𝑖 ⋅⋅⋅ 𝑛𝑥𝑛 − 𝑥𝑖 ⎟ ⎜ 𝑖=1 𝑖=1 ⎟ to voting: a superior neuron collects information from the ∑𝑛 ∑𝑛 ∑𝑛 ∑𝑛 ⎝ 2 2 ⎠ inferior cells and then makes a balancing decision. Although 𝑥𝑖 − 𝑥1 𝑥𝑖 ⋅⋅⋅ 𝑥𝑖 − 𝑥𝑛 𝑥𝑖 𝑖=1 𝑖=1 𝑖=1 𝑖=1 the computing process provides a theoretically desirable so- 𝐴+ = ( ) ∑𝑛 ∑𝑛 2 lution to the problem, it seems fit for a computer rather 2 𝑛 𝑥𝑖 − 𝑥𝑖 than a neural system which can hardly accomplish complex 𝑖=1 𝑖=1 equation solving. Instead, the neural system requires a feasibly (5) + 𝑇 𝑇 equivalent method. Denote the first and the second rows of 𝐴 by 𝑢1 and 𝑢2 , +−→ From the bottom up, our model is extended by a five- and the first and the second elements of 𝐴 𝑦 by 𝑐1 and 𝑐2.By substituting them into (4), we explicitly obtain the following layer neural network and this extension is affordable to neural computing process: system and can be easily implemented on computers. The { √ neuronal connections in the network are strictly localized. As 𝑎 = 𝑐 − 𝑢𝑇 𝐷 ⃗𝑟 𝑎2 +1 is shown in Figure 4, on the bottom are the photoreceptors 𝑘+1 1 1 𝑘 √ 𝑘 𝑇 2 (6) which constitute the GC RFs; the second layer consists of the 𝑏𝑘+1 = 𝑐2 − 𝑢2 𝐷𝑘⃗𝑟 𝑎 +1 𝑘 GCs/LGN cells clustered into different groups; the third layer After the 𝑘 times iteration, if 𝐷𝑘 does not change remark- consists of clustered voting cells; the fourth layer consists of ably, the approximate optimal solution is then found and we fewer voting cells; on the top is a simple cell. When a stimulus have activates the RFs, GCs/LGN cells output their responses. Each ∣𝑢𝑡 𝐷 −→𝑟 (𝑎 + 𝑎 )(𝑎 − 𝑎 )∣ of the third-layer voting cells obtains a large solution set which ∣𝑎 − 𝑎 ∣ = 1 𝑘 𝑘 𝑘√+1 𝑘 𝑘+1 𝑘+1 𝑘 √  satisfies the local constraints imposed by several neighboring 𝑎2 +1+ 𝑎2 +1  𝑘 𝑘+1  neurons; then the senior voting cells further seek for the locally    𝑡 −→ optimized solutions in a similar way. The SC makes a final < 𝑢1𝐷𝑘 𝑟 ∣𝑎𝑘 − 𝑎𝑘−1∣ (7) determination by finding a unique solution that satisfies all Clearly, a necessary condition for the convergence of (6) is senior voting cells equally to the greatest extent. Take the cells 𝑡 ∣𝑢1𝐷𝑘⃗𝑟∣≤1. numbered ’A’,’1’,’2’ and ’3’ as an example (Figure 4), voting An important factor evaluating the solution is the convergent cell#A guesses the orientation of the local stimulus activating rate. Denote the accurate solution of the current problem by the RFs of cell#1, cell#2 and cell#3. Fig. 4. The computational network of hierarchical group-decision making.

V. S IMULATION OF SIMPLE CELL RECEPTIVE FIELDS Numerical simulations were performed to examine whether the proposed model can mimic the RF of an SC and achieve orientation selectivity, in other words, whether this approach can better detect stimuli in the preferred orientation. Given a Fig. 5. The distributions of LGN cell RFs and the produced SC RFs. The set of LGN cell, contrast stimuli are randomly generated on numbers of LGN cells are 20, 30 and 40 respectively from the top down. the rectangular grids and we statistically calculate the average Circles: LGN cell RFs. The lighter a grid looks, the more sensitive the SC is response (sensitivity) of the SC at each grid to all edges to the contrast edges passing through there, and vice versa. passing through that grid. The sizes and the distributions of the RFs are both variable. The response magnitude, which represents the sensitivity to stimuli at different positions, is transformed into a graylevel image. Figure 5 shows three examples obtained with different numbers of LGN cells. The SC modeled in each of them will be the most sensitive to horizontally stimuli in the middle of its RFs while the least sensitive to vertical stimuli. Notably, these plots are quite similar to the physiologically result (Figure 6) reported in [29]. Factually, our results seem more natural since the changes in the sensitivities take place gradually. These experiments met our anticipations and strongly verified the Fig. 6. Relative locations of the LGN cell RFs with respect to the SC RF. conception of this paper. Gray: idealized SC RF Gabor. Red circles: LGN RFs of the same polarity to the cortical zone; blue circles: LGN RFs of opposite polarity [29]. VI. EXPLANATIONS OF VISUAL ILLUSION In digital images, lines always appear in the form of contrast edges. The proposed model of orientation selectivity can in a slightly inward manner, thus shortening the horizontal therefore be developed as a line detector. Specifically, each line. Those below occurred in a slightly outward manner, thus network with a number of LGN cell and the associated SC lengthening the horizontal line. The results turn out to be composes a window in which a line will be detected, while in accordance with observers’ perceptions. The Geometrical a large image may require many detection windows. Since analysis is presented on both images. many visual illusions are caused by combined lines of different The Hering illusion. As is shown in Figure 8a, the image orientation, we use several illusive images as complex stimuli of the Hering illusion contains two long parallel lines densely and check whether the detection results are consistent with intersecting many segments. It seems that the two parallel observers’ perceptions. lines are bending away toward each other at the center: all The Muller-Lyer¨ illusion. This first experiment tests how our the acute crossing angles are exaggerated by the eyes. We model can be used to explain the occurrence of the Muller-¨ simplify the image by reducing several segments and enlarge Lyer illusion. As is shown in Figure 7a, two line segments be- two segmentations containing all the intersections. Again, tween a pair of arrows have an identical lengths but the above we examine our algorithm over corners where parts of the line seems shorter than the one below. Obviously, the illusion long lines are to be detected with short distracting segments. must appear somewhere around the corners. The strokes of the Figure 8b shows the result: the detected angles are greater than two shapes are increased by a certain percentage. Then we run the actual angles, producing two curved virtual lines. the algorithm over several corner regions. Figure 7b illustrates The Zollner¨ illusion. As is shown in Figure 9a, the image the results: The results obtained with the above image occurred of the Zollner¨ illusion contains several long parallel bars (a) The Muller-Lyer¨ illusion.

(b) The results obtained with the enlarged image of the Muller-Lyer¨ illusion and the geometric analysis. Selected corners are designated with green windows (more false edges) and blue windows (more true edges). The obtained lines are drawn in red. Two examples of them are denoted by ’a’ and the corresponding actual lines are drawn in red denoted by ’b’. Fig. 7. The Muller-Lyer¨ illusion and the explanations.

(a) The Hering illusion. (b) The results obtained with the simplified image of the Hering illusion. Fig. 8. The Hering illusion and the explanations intersecting many short bars. It seems that the long bars are nonparallel. As with the Hering illusion, the acute crossing angles seem greater than they really are. We randomly select a segmentation of the image: two parallel bars with distracting bars; then enlarged it and also examined our algorithm over corners where parts of the long lines were detected with short (a) The Zollner¨ illusion distracting segments. Figure 9b shows the results: the detected lines diverge slightly away from their real positions, making the left bars seem as if they were rotated clockwise while the right bars counterclockwise, so every two parallel bars in the original image appear to be nonparallel. The Orbison illusion. As is shown in Figure 10a, the rectangle and the circle appear distorted. Specifically, the left side of the rectangle looks shorter than the right and the two lines look like two less-than signs. The left part of the circle seems to be pushed toward the center while the right part seems to be pulled away from the center. The example in Figure 10b verifies that a single circle can be properly detected (b) The results obtained with the simplified image of the Zollner¨ illusion. piecewise by our approach when the detection window is Fig. 9. The Zollner¨ illusion and the explanations small. Similar to the previous experiments, we perform the (a) The Orbison illusion

(b) The piecewise de- (c) The results obtained with the enlarged image of the Orbison illusion. tection of a circle Fig. 10. The Orbison illusion and the explanations. detection around corners. The results, as shown Figure 10c, is consistent with our observation. The Cafe´ wall illusion. As is shown in Figure 11a, the parallel gray lines between staggered rows of alternating black and white ”bricks” appear to be sloped. Specifically, for the first two lines, their left endpoints look lower than their right ones, and, for the following two lines, the trend is reversed. The perceived orientation of one line is influenced by nearby segments. Again, our algorithm was performed around (a) The Cafe´ wall illusion. intersections in the enlarged image. The results shown in Figure 11b are in accordance with our observations. In order to strengthen reliability, we tried different windows around several corners. The lines obtained in nearby windows were quite similar.

VII. CONCLUSION Orientation selectivity has always been a hot topic in the research on cortical SC. The feedback model proposed by Hubel and Wiesel, the modified versions and many distinct theories all partially succeeded in explaining part of the related facts observed in physiological experiments, including contrast invariance. Meanwhile, they cannot account for or even contradict some phenomenon. In this paper, they are all considered as sketches with little implementation details and computational strategies. Concerning the incompleteness and difficulties of the previ- ous models, except the first one of the three questions summa- rized in the beginning of this paper, which has already been investigated in our previous work, the others were analyzed and answered with mathematical verifications throughout the discussion: 1) The response curve of a single GC/LGN cell can be (b) The results obtained with the enlarged image of the Cafe´ wall employed to infer the relative position of the contrast illusion. edge with respect to the RF. For each LGN cell, the Fig. 11. The Cafe´ wall illusion and the explanations estimated distance and the location of the RF form a local constraint. All those constraints are independent of [9] G. Sclar and R. D. 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