Optical Illusion. Sara Bolouki, Roger Grosse, Honglak Lee, Andrew Ng

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1 Optical Illuion Sara Bolouki, Roger Groe, Honglak Lee, Andrew Ng. Introduction The goal of thi proect i to explain ome of the illuory phenomena uing pare coding and whitening model. Intead of the pare coding method propoed by Olhauen in [], tranient invariant pare coding i ued. Tranient in variant pare coding can learn larger bae compared to ordinary pare coding, which can help a lot in tudying illuion phenomena. Uing mall bae implie uing mall image, which in turn might diminih illuory effect that we are looking for in the image. In thi report we firt briefly explain image-proceing ytem in human brain, then the elected model for modeling retina and V part of the brain are dicued. Part 4 of the report explain ome of the illuion effect that are going ued for experiment. Part 5 contain the experiment performed and reult. Finally the lat part decribe the future work.. Human image proceing ytem A we know, mammalian viual ytem begin from the eye. Eye can be conidered a the len of a camera that tranmit the light it receive from the world obect to the retina. Retina conit of a large et of photoreceptor cell that polarize the bipolar cell. Bipolar cell in turn pa information to gangolian cell which are the outermot layer of the retina. Thi information i then paed to lateral geniculate nucleu (LGN) part of the brain. The neuron of the LGN relay the viual image to the primary viual cortex (V) which i located at the back of the brain. Neuron repone in V ditinguih between orientation, patial frequencie and color in the image. The third tep of information proceing take place in V area of the brain. V neuron repond to contour timulu and are not enitive to phae change in the image. 3. Spare coding model for V For modeling V part of the cortex, tranient invariant pare coding [] model i ued. The reaon of thi election i that tranient invariant pare coding can learn larger bae, which will help u in tudying illuory phenomena. In ordinary image pare coding, each patch of image form a column in a matrix, X. The goal i to find a matrix A, in which each column i a bai, and matrix S where each column correpond to the activation of bae relative to each patch. Ordinary par coding trie to minimize the recontructing error by achieving the highet poible parity. The problem can be formulated a bellow []: X AS + β S i ubect to a, for all i

Tranient in variant pare coding can learn larger bae compared to ordinary pare coding, which can help a lot in tudying illuion phenomena.

2 In tranlation invariant par coding, image, bae and activation are repreented a two-dimenional array intead of vector. In thi algorithm, convolution and point-wie product i ued intead of matrix multiplication. Applying thee change the new problem can be defined by the following formula, which trie to olve for A by minimizing the um []: m n ( i) ( ) ( i, ) X A S + β i= = i, S ( i, ), ( ) ubect to A, for all i, The ret of the algorithm i imilar to ordinary pare coding; the firt tep i learning the bae and the econd tep i recontruction of the image. Thi proce i continued until bae are converged. 4. Optical illuion According to literature, An optical illuion i any illuion that deceive the human viual ytem into perceiving omething that i not preent or incorrectly perceiving what i preent [3]. Studying illuion phenomena would help u to undertand the viual ytem in the brain. While there are lot of different type of viual illuion, thi proect focue on brightne illuion, orientation illuion and watercolor illuion. Watercolor illuion Brightne Illuion Illuory contour Zollner Effect Different type of optical illuion may correpond to characteritic of different tep in the viual ytem hierarchy. For intance, brightne illuion might be explained by lateral inhibition in retina and V receptive field. There i alo lateral interaction between neuron activitie in V area, which may explain orientation illuion uch a Zollner effect, Orbion and Ehrentein illuion. The third type of illuion

Applying thee change the new problem can be defined by the following formula, which trie to olve for A by minimizing the um []: m n ( i) ( ) ( i, ) X A S + β i= = i, S ( i, ), ( ) ubect to A, for all

3 that i poibly explained by thee hierarchical model i illuory contour. According to neurocience literature, illuory counter caue activation of neuron in V area of the brain, therefore we expect to explain them by V model. Since we only ue V model in thi proect, we are not conidering illuory contour. 5. Experiment Two different experiment have been performed in thi proect. The firt experiment trie to utify Zollner illuion and econd experiment concentrate on brightne and watercolor illuion. 5.. Zollner Illuion A hown by the figure, in Zollner effect, parallel line eem to become cloer to each other. The reaon of thi effect might be miudgment of the value of the angle between horizontal and diagonal line, which could be explained by lateral inhibition between V neuron. The tep of our experiment are: - Learn bae from natural image - Oberve bae activation for illuory image 3- Study bae activation conidering what might caue illuion in V area In order to oberve illuion in the image, we need a et of bae uch that the difference between their orientation i very mall. Bae learned from natural image do not have thi characteritic. Therefore, 360 hand made bae that have 0.5 degree difference in the orientation are ued intead. Unfortunately, the reult of the experiment could not explain the illuory phenomena, ince all the activation correponding to a horizontal line had a degree of orientation between 0 and. The expected reult would be activation value with 3 to 4 degree of orientation, at leat. 5.. Brightne and Watercolor Illuion The goal of the econd experiment i to explain brightne and watercolor illuion. Our hypothei i that whitening i performed by the early viual tage, but brain itelf doe a form of invere-whitening operation in lateral viual ytem in order to compenate the whitening of the early tage. The tep of thi experiment are: - Learn bae from natural image - Recontruct a group of natural image with tranient invariant pare coding uing natural bae 3- Learn invere whitening filter for thi group of image 4- Recontruct illuory image with tranient invariant par coding, uing natural bae 5- Apply learned invere whitening filter to the recontructed illuory image 5... Learn invere whitening filter In thi tage, the goal i to minimize the error between the initial image and the recontructed image from the pare coding, by applying a filter to the recontructed image. The invere whitening filter i learned in frequency domain. The obective i to minimize the following equation: min X Z g 3

Since we only ue V model in thi proect, we are not conidering illuory contour. 5. Experiment Two different experiment have been performed in thi proect.

4 Where X i equal to FFT of th image in the training data et, Z i equal to FFT of the th recontructed image and g i the invere whitening filter which we are going to learn. Auming that there i no dependency between frequency repone of each filter, we can rewrite the above formula a the following: ( X Z g ) min,, Where correpond to each pixel of the image. Linear regreion method i ued to olve the above problem. Parameter that hould be learned are different component of filter g. The cloed form olution i calculated by taking the derivative according to each component of g, and etting the derivate equal to zero: l( ) = min l( ) = g = X Z ( X,, ( X,, Z Z,, g g ) ) Z, = 0 g i the firt component of the g filter. Other component of g are calculated with the ame formula. 60 image with 00x00 reolution were ued in order to learn the invere filter. The learned invere whitening filter ha the ame dimenion a the training image. Figure how amplitude of different frequencie of the learned filter. A hown, the filter ha larger value at lower frequencie, therefore amplifie lower frequencie and can be conidered a the invere of whitening filter Reult Figure - Invere whitening filter To do the experiment we needed to recontruct the illuory image by the parenet code and then convert them to frequency domain. In frequency domain, we can apply the learned g filter to the recontructed image. The next tep i to apply invere Fourier tranform and bring the image back to the patial domain. Figure 3 illutrate one example of performing thee tep on a ample image in the training data et. 4

Linear regreion method i ued to olve the above problem. Parameter that hould be learned are different component of filter g.

5 Figure- (left) Input gray cale image, (center) Recontructed Image, (right) Recontructed image after applying invere whitening A hown in Figure (left), invere whitening trie to make the image moother o it pread out white and dark border. We were expecting to be able to demontrate brightne and watercolor illuion by thi moothening. The ame tet ha been performed on brightne illuion and a group of watercolor illuion. Figure 3 how brightne illuion image recontructed by the pare coding before and after applying parecoding. A hown in the Figure, before applying invere whitening we can ee a lighter and a darker bar, but what we expected from invere whitening wa to make the left part of the image light with a lighter bar at the center border and the right ide darker with a darker bar at the ame place. A hown in Figure 3(right), the expected reult i not obtained. Figure 3- (left) Before applying invere-whitening, (right) After applying invere whitening The ame tet wa performed on a group of watercolor illuion image, expecting to ee preading of the inide color or detecting the illuory contour. Unfortunately the recontructed image did not have either of thee characteritic. To make the experiment more accurate the ame tet wa performed on 00x00 image, which had the ame reult a 00x00 image. 6. Future work There are two uggetion for future reearch. The firt i to tudy other model of whitening which can alo help learning and parametrizing invere whitening filter. The econd uggetion i to re-learn the bae according to the un-whitened image, meaning that the coefficient are derived baed on old bae and whitened image, and then bae are re-learned for the un-whitened image and obtained coefficient. 7. Reference [] Emergence of imple-cell receptive field propertie by learning a pare code for natural image, Bruno A.Olhauen, David J.Fild, Letter to Nature, June 996 [] Tranlation Invariant Spare coding, Roger Gore, Augut 006 [3] 5

The ame tet ha been performed on brightne illuion and a group of watercolor illuion. Figure 3 how brightne illuion image recontructed by the pare coding before and after applying parecoding.

Two Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL

Two Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL Excerpt from the Proceeding of the COMSO Conference 0 India Two Dimenional FEM Simulation of Ultraonic Wave Propagation in Iotropic Solid Media uing COMSO Bikah Ghoe *, Krihnan Balaubramaniam *, C V Krihnamurthy