Learning Topological Image Transforms Using Cellular Simultaneous Recurrent Networks

Transcription

1 Proceedings of Inernaional Join Conference on Neural Neworks Dallas Texas USA Augus Learning Topological Image Transforms Using Cellular Simulaneous Recurren Neworks J. Keih Anderson Deparmen of Elecrical & Compuer Engineering Universiy of Memphis Memphis TN USA Khan M. Ifekharuddin Deparmen of Elecrical and Compuer Engineering Old Dominion Universiy Norfolk VA USA Absrac In his work we invesigae cellular simulaneous recurren neworks (CSRNs) o learn opological image mappings paricularly hose of he affine ransformaions. While affine image ransformaion in convenional image processing is a relaively simple ask learning hese ransformaions is an imporan par of having neural neworks (NNs) funcion as generalized image processors. We inroduce he CSRN and discuss is adapaion for image processing asks. We repor resuls for ranslaion roaion and scaling of boh binary and grey-scale images. Our resuls sugges ha he CSRN is capable of learning and performing hese basic opological ransformaions. Keywords- Cellular Simulaneous Recurren Nework; CSRN; affine ransformaion I. INTRODUCTION Arificial neural neworks (NNs) have been inspired by he abiliies of humans and animals o learn and adap. Feedforward neworks are boh fas and powerful and are paricularly useful for saisical paern recogniion. These neworks are inspired by working principles of brain sensory processing areas such as he visual corex. However hese ypes of neworks have been shown inadequae for complex applicaions such as long-erm opimizaion reinforced learning and image processing involving opological mapping. A fundamenal challenge o he heory underlying NNs posed by Rosenbla in his early work on perceprons is he recogniion of opological relaions [1]. Minsky and Paper [] have shown ha perceprons are incapable of solving his class of problems. Pang and Werbos [9] demonsrae ha mulilayered perceprons (MLPs) in spie of being more powerful han perceprons are also unable o solve hese ypes of problems due o heir exponenial complexiy. Topological image mappings such as affine ransformaions and image regisraion are par of his class of problems. Of ineres in his work are he affine ransformaions. Muli-layered feed-forward neworks are universal funcion approximaors capable of learning he linear funcions associaed wih affine ransformaions. However he exponenial complexiy and D naure of images require large complex feed-forward soluions. MLPs and feed-forward NNs in general have no been shown o perform affine ransformaions on images. Bebis e al [6] show ha feedforward neworks can learn o esimae parameers for affine ransformaion. In heir work hey exrac criical poins from an image objec. These poins are ransformed via sandard affine ransformaion mehods hrough a range of affine parameers. Principal componen analysis (PCA) is applied o hese ransformed poins o produce a reduced se of feaures. These feaures are used o rain he feed-forward nework o recognize he affine parameers used in ransforming he objec. Their resuls indicaes ha NNs are capable of learning he -D funcions associaed wih affine ransformaions however he nework does no acually perform he affine ransformaion. A ype of recurren neural nework known as a Cellular Neural Nework (CNN) has been shown capable of performing fracional and single pixel ranslaion [4][5]. By exension he CNN can be used o perform roaion by firs decomposing he roaion ino muliple single pixel ranslaions and one fracional pixel ranslaion hen applying he CNN o he image once for each single pixel or fracional ranslaion. This approach does no lend iself o pracical applicaion. In addiion he CNN relies on predeermined weighs referred o as emplaes [6][7][8] and herefore does no learn o perform hese ransformaions. Neuroscieniss have uncovered evidence ha he brain is highly recurren. There is also evidence ha he neurons in he brain are conneced in cellular srucures. Cellular simulaneous recurren neworks (CSRNs) mimic boh he recurrency and cellular srucure of he brain. CSRNs have been shown capable of solving he ype of opological relaion problems previously discussed. The CSRN was firs exploied o solve he D maze raversal problem [9] which is a long-erm opimizaion problem. From he CSRNs incepion boh is creaors and poenial users have speculaed ha he CSRN may have imporan implicaions in image processing. However o dae very lile work has been done o adap CSRNs o simple image processing asks such as opological mapping. The applicaion of he CSRN o image processing was proposed in a subse of he connecedness problem [10]. We demonsraed simple regisraion of in-plane roaed binary and low complexiy gray scale image images using CSRN [11][1]. We furher exploied sae ransiion propery of CSRNs o solve large scale pose invarian facial recogniion [13]. In hese works CSRNs are used o perform facial recogniion on image sequences. Noe ha none of hese works addressed he crucial quesion of he CSRN s abiliy o learn opological image mappings /13/$ IEEE 90

2 In his work we invesigae he CSRN s abiliy o learn opological image mappings paricularly hose of he affine ransformaions. Applicaion of he CSRN o image processing ask and in paricular opological mapping of images is moivaed from boh he modeling and applicaion perspecives. From a modeling perspecive he recurrency and cellular srucure of he CSRN make i a more biologically plausible soluion. From an applicaion perspecive he CSRN is capable of performing affine ransformaions direcly on images wihou expensive pre-processing and once rained can perform hese ransformaions very quickly. This paper is organized as follows. In secion II we review he fundamenals of affine ransformaions. In secion III we inroduce he CSRN and in secion IV presen he proposed opological image processing algorihm using CSRN for affine ransformaion. In secion V we presen he resuls for ranslaion roaion and scaling of boh binary and grey-scale images. Convergence and compuaion imes are also discussed. In secion VI we draw conclusions and discuss fuure works. II. AFFINE TRANSFORMATION OF IMAGES Affine ransformaions are an imporan class of linear -D geomeric ransformaions which map pixel inensiy values from heir locaion in an inpu image o heir new locaion in an oupu image [14]. These ransformaions are characerized by he fac ha hey preserve sraigh lines. Affine ransformaions in images consis of ranslaion roaion scaling and verical/horizonal shear. We focus our aenion on ranslaion roaion and scaling. The general form of an affine ransformaion is ha of a - D linear equaion x a = y a X A X + B = 1 a x b +. a y1 b where X 1 is he curren pixel s locaion and X is i s new locaion. A and B are he slope and inercep marices respecively. In he case of pure ransformaions we can furher reduce he form o where T is referred o as he ransformaion marix. A. Transformaion Equaions The equaion for ranslaion is given by 11 1 X x = y x 1 = y 0 = T X x y1 (1) () (3) 1 x + x y1 y0 where x 0 and y 0 are he amoun of ranslaion in he x and y direcions respecively[15]. By augmening B o A in (1) we can achieve he form in (). x1 x 1 0 x0 (4) = y y 0 1 y0 1 The equaion for roaion is given by x y cosθ = sinθ where θ is he amoun of roaion abou he cener of he image[15]. The equaion for scaling is given by sinθ 1 cos x θ y1 (5) x 0 sx x1 = (6) 0 y sy y1 where s x and s y are he scale parameers in he x and y direcions respecively[15]. B. Image Re-sampling Image re-sampling is an imporan par of image ransformaion. Re-sampling includes wo relaed opics: mapping mehods and inerpolaion. Several surveys have been conduced on image re-sampling mehods. A horough comparison of hese echniques was done in [16]. 1) Mapping mehods The affine funcions above can be implemened by forward or inverse compuaion. In he forward mehod each pixel in he image is ransformed o is new locaion via he forward equaions (given above). This mehod produces significan error in he new image due o overlaps and/or holes caused by discreizaion and rounding [17]. In he inverse mehod he coordinaes of he new image are used along wih he inverse equaions. In his way one and only one value is calculaed for each pixel posiion in he new image hus eliminaing overlaps and holes. In his work we uilize inverse mapping. ) Inerpolaion Mapping funcions wheher forward or inverse resul in fracional pixel locaions. To complee he mapping sep we mus perform some ype of inerpolaion. Neares neighbor bilinear bi-cubic quadraic splines and cubic b-splines are a few of he more popular mehods. Even hough higher order mehods produce increased accuracy and beer visual resuls bilinear inerpolaion offers possibly he bes rade-off beween accuracy and compuaional complexiy [7]. Since he emphasis in his work is on he efficacy of he CSRN o perform affine ransformaions we consisenly use he neares neighbor mehod for is ease of implemenaion. 91

3 III. CELLULAR SIMULTANEOUS RECURRENT NETWORKS A. Inroducion o CSRNs Pang e al. inroduced CSRN uilizing a self-recurren nework (SRN) in a cellular srucure effecively combining SRNs wih CNNs. They showed ha his NN could be successfully applied o opimizaion problems by approximaing a soluion o he Bellman equaion [9]. To demonsrae his abiliy hey applied NNs o he maze raversal problem. Their works also suggesed ha MLPs are incapable of solving such opimizaion problem [18]. B. Biological Basis From a compuaional sandpoin he human brain is a highly complex nonlinear and parallel processing device. I is composed of billions of neurons. I organizes neurons o perform specific funcions. The number and organizaion of neurons o perform a specific funcion as well as he synapic weighs beween neurons are learned hrough experience [19]. Neuroscieniss speculae ha reinforced learning akes place in he hippocampus of he brain. Populaions of neighboring neurons form cell ensembles which form he basic building block for he enire sysem [0]. These ensembles are inerconneced in a cellular srucure similar o ha found in CSRNs. Neurobiologis have long undersood ha local recurrency plays a criical role in he higher par of he human brain and recen evidence indicaes ha he brain is highly recurren [0][1]. CSRNs resemble he brain in boh cellular srucure and recurrency. The reinforced learning in he CSRN is in fac quie similar o ha of he human brain. C. CSRN Archiecure Figure 1 shows he cellular srucure of he CSRN. Noe ha he cellular srucure of he nework maches ha of he underlying inpu paern. Each cell shown as a grey box conains an SRN core. The core nework is shown in Fig. and is referred o as a generalized muli-layered percepron (GMLP) core. The core consiss of 18 nodes five of which are acive neurons. Nodes n0-n1 are inpu nodes. Node 0 is he bias inpu. The nework shown here has 3 exernal inpus. The 4 neighbor inpus come from he oupus of node 13 of he neighboring cells. This node is referred o as he connecor node. The inerconnecions provided by hese inpus form he cellular srucure of he nework. The final 5 inpus are self-recurren inpus consising of he oupus of he five acive neurons fed back as inpus. Noe ha in he GMLP archiecure depiced here he inpus o each acive node consis of he oupus from all previous nodes. For he purpose of clariy no all inerconnecs beween nodes have been shown. The oupu of node 17 is muliplied by a scaling weigh W s o produce he cell s ulimae oupu. CSRNs uilize weigh sharing ha is he weighs for each cell are equivalen. This significanly reduces he amoun of memory required o sore he CSRN s weighs. Figure. Core nework of he CSRN. This paricular core is a GMLP core. D. Learning in CSRNs CSRNs learn by adaping heir weighs via supervised learning. If he CSRN oupu is given by Y i and i corresponding arge image is T i hen he error beween he wo is given by ei = Ti Yi. The oal error in he nework oupu is given by (7) Figure 1. Cellular srucure of he CSRN. c 1 E i = e i (8) i = 1 where c is he number of pixels in he image. For a given raining se he error becomes N c 1 E = E i (9) n= 1 i= 1 where N is he number of raining images. Equaion (9) is he sum-squared error and represens he cos funcion of he nework. The objecive of he learning process is o adjus he nework s weighs o minimize his cos funcion. Wih his 9

4 simple cos funcion he CSRN is capable of approximaing he -D funcions associaed wih affine ransformaions. E. Training he CSRN Parameer esimaion wih Kalman filers has been applied o he raining of NN in he pas []. Ilin e al. apply parameer esimaion o he raining of CSRNs using an exended Kalman filer (EKF) [3]. In his mehod he required Jacobian is compued via backpropagaion hrough ime (BPTT) and he weighs are adaped via EKF. The reducion in raining ime is a breakhrough in he raining of CSRNs which make hem racable in complex asks such as image processing. We uilize EKF mehod for parameer esimaion of nonlinear sysems [4]. The process and observaion equaions are given as and (10) (11) W is he curren sysem sae a ime and is represened by he mean and covariance of he sysem weighs (µ K ). γ is he process noise. In his case he rue weighs do no change and he sae ransiion is effeced only by he process noise. Y +1 is he observaion a ime +1 F is he forward funcion of he NN and η is he measuremen noise. The EKF mehod compues a new mean and covariance for he sae given he exising sae and a new measuremen updae using he equaions given as (1) (13) (14) (15) where µ is he mean sae marix K is he sae covariance R and Q are he covariance of he zero-mean process and measuremen noise respecively. The innovaion a is he error beween he prediced measuremen (based on he curren sae) and he new measuremen. C is he sysem Jacobian and G is he Kalman gain. IV. Y T Γ = C K C + R W G = +1 W + γ = F( W u ) + η. + 1 T 1 = KC Γ µ + = µ + G a 1 K = K G C K + Q + 1 ALGORITHM FOR TOPOLOGICAL IMAGE PROCESISNG USING CSRN Figure 3 shows a flowchar of he CSRN image processing algorihm. Alhough for he purposes of his work we are ineresed in affine ransformaions his algorihm is generic and can be applied o any image processing ask. If sub-image processing is no uilized he sub-image loop is removed from he main program. We discuss sub-image processing sep in a subsecion below. and Furher insigh ino his algorihm is gained by examining he raining module. The raining loop performs he following seps on each raining image: CSRN forward compuaion Calculae error using Eq. (7) Compue he learning mehod (parameer esimaion via EKF) Adjus nework weighs. Similarly he esing loop performs he following seps on each esing image: CSRN forward compuaion Calculae error using Eq. (7) Selec bes generalizing ne based on lowes sumsquared error (SSE). n Main Ini. parameers Generae Training/ Tesing Images Sub-Image Loop Call Training Module Sore Sub-Image Oupu Done? y Perform Image Trans. Display Resuls End Training Module Creae CSRN objec (radom ini. weighs) Epoch Loop Training Loop Tesing Loop Done? Resuls Loop reurn Figure 3. Flowchar for he CSRN image processing algorihm. Since he CSRN was iniially developed o solve he maze raversal problem we mus adap he nework o perform image processing asks in his case affine ransformaions. The following seps are involved for such adapaion of CSRN. A. Inpus/Oupus An examinaion of Equaions (3) - (6) for affine ransformaions given above shows ha we need he following inpu variables for CSRN: Pixel locaion: The x and y locaions of he pixel wihin he image. Transformaion parameer: The basic parameer for all affine ransformaions is he amoun of ransformaion. This may be he number of pixels o ranslae an image he angle of roaion or a scaling facor. The oupu of he nework will be he new locaion of he designaed pixel. y n 93

5 B. Funcion Approximaion To rain he CSRN o perform a given affine ransformaion we generae a se of raining images by ransforming a es image by various degrees. For example in he case of roaion we migh use º 15º and 0º. For each of he raining images we consruc wo ransformaion marices which encode he ransformaion for he x and y direcions respecively. These marices can be encoded wih a posiion or movemen funcion. A posiion funcion conains he new posiion of he pixel while a movemen funcion conains he amoun he pixel needs o be moved from is curren locaion. In he course of our work we find ha he CSRN is beer able o approximae he movemen funcion and have uilized i in his work. These ranslaion marices become he arges by which we rain he CSRN. C. Sub-Image Processing The complexiy of he CSRN grows wih image size. In a pracical implemenaion a radeoff exiss beween he number of raining images and he size of he images. Ilin e al limi heir work o 7x7 images [][3][5]. In [11] we exend he use of CSRNs o 15x15 images. We find his image size o be he pracical limi for CSRNs using he EKF raining mehod wih our hardware/sofware configuraion (secion V-C). This is due o he size of he marices required for compuaion of he Jacobian via BPTT. To overcome his limiaion we uilize sub-image processing. We divide each raining image ino smaller 5x5 sub-images. Nex we rain a separae CSRN for each of he sub-images. Once raining is complee o perform a given ransformaion on an image he image is divided ino subimages each of which is processed by is corresponding CSRN. The oupus of each CSRN are hen combined o form he final ransformed image. The addiion of sub-image processing provides a dramaic decrease in raining ime. As repored in [1] addiion of sub-image processing decreases he processing ime for a 15x15 image by 6.5%. This ime savings allows he user o increase he number of raining images for improved nework performance or o increase image size. V. RESULTS A. Performance Merics Before discussing resuls for learning opological affine ransformaions using CSRNs we define several merics which will allow us o evaluae he efficacy of he CSRN for his sudy. 1) Funcion Approximaion Merics The following merics are used o evaluae how well a nework learns a given funcion J which is hen used o produce a final ransformed oupu image. J mse funcion mean squared error. MSE beween he arge funcion and he acual nework oupu. J acc funcion accuracy. Compares he oupu of each cell o he known funcion value for ha cell. If he values are equal we consider ha cell a mach. The oal number of maches normalized by he oal number of cells yields he funcion accuracy in percen. ) Image Comparison Merics The following merics provide a quaniaive mehod for comparison of a desired arge image wih he oupu image produced by he CSRN. IM acc image accuracy. Compares he final oupu image wih he known arge image on a pixel by pixel basis. If he corresponding pixels have equal values we consider ha pixel a mach. The oal number of maches normalized by he oal number of cells yields he image accuracy in percen. Primarily used for binary images. IM cr - correlaion raio. A common meric for comparing he similariy of wo images [15]. Is value is compued using Eq. (16) below. IM cr = ( A( i) A )( B ( i) B ) ( A ( i) A ) ( B ( i) B ) i i (16) A(i) and B(i) represen he individual pixels of image A and B respecively. A and B are he mean pixel values for image A and B respecively. The closer his raio is o 1 he more similar he images. Primarily used for grey-scale images. 3) Oher Merics In his secion we discuss some addiional merics uilized in his work. T r - raining ime. Measure of he ime (in secs) required for raining a given nework. Compued using MATLAB s ic-oc funcions. Normalized for image size and number of raining images. T c - convergence ime. The ime (in epochs) required for he SSE of he nework o converge during he raining process. B. Transformaion of Binary Images This secion examines he use of CSRNs o perform affine ransformaions on binary images. Due o he aforemenioned radeoff beween he number of raining images and image size we have chosen o work wih simple binary images wih a size of 15x15 pixels for proof of concep. This size image allows up o 11 raining images which allows for adequae resuls. 1) Translaion Simulaions wih a simple cross image indicae ha he CSRN performs ranslaion wih 100% accuracy. Figure 4 shows he resuls of raining he CSRN o perform ranslaion. In his es a GMLP core is rained o perform ranslaion on a 5x5 cross embedded in a 15x15 image. The nework is rained using 11 images shifed o he righ from 0 o 10 pixels. Training images uilize zero padding wih a widh of 5 pixels. 94

6 SSE - over all es images a) 400 SSE Figure epochs Figure 4. Resuls of ranslaing a 5x5 cross embedded in a 15x15 image via a CSRN wih a GMLP core. The cross is ranslaed a disance of 10 pixels in he x direcion. The nework achieved a final raining Jmse = 0.0 Jacc = 100% and an IMacc = 100%. 5 0 SSE - over all es images b) Movemen encoding is used wih inverse mapping. The inpu image shown is he cross image shifed o he righ by 10 pixels. The oupu image shows he cross correcly ranslaed o he far lef side of he image. The nework achieves a final Jmse = 0.0 a funcion accuracy of 100% and an image accuracy of 100%. In his es he CSRN is simulaed 100 imes in order o compue he necessary saisics. Table I summarizes he resuls of our ranslaion es. SSE epochs Figure 5. a) Plo of he sum-squared error during raining of he CSRN wih a GMLP core for he case of ranslaion shown in Fig. 3. b) Error plo rescaled for clariy. TABLE I. BINARY TRANSLATION RESULTS Meric CSRN w/ GMLPCore Ave +/- sd dev Bes Jacc 100 +/ Jmse 0.0 +/ IMacc 100 +/ IMcr 1.0 +/ T r (norm).84 +/ Tc Figure 5 shows he convergence of he CSRN s SSE during he raining process. The figure clearly indicaes ha he nework is learning and converges in approximaely 150 epochs. This plo is ypical for all hree ransformaions and is shown only for his case. ) Roaion Translaion is local in naure however roaion is global resuling in a more complex ransformaion. Our es resuls indicae ha he CSRN is capable of performing roaion. Figure 6 shows he resuls of raining he GMLP core o perform roaion. In his es he nework is rained using 11 images wih he cross roaed from 0 o 0 in seps of. The inpu image shown is he cross image roaed 16. In his example he cross was correcly roaed back o a 0 angle. The nework achieves a final raining MSE of 0.0 a funcion accuracy of 97.3% and an image accuracy of 100%. Figure 6. Resuls of roaing an 11x11 cross embedded in a 15x15 image via a CSRN wih a GMLP core. The cross was roaed o an angle of 16. The nework achieved a final raining J mse = 0.0 J acc = 97.3% and an IM acc = 100%. Table II summarizes he resuls of our roaion simulaion. Saisics are compued on a series of 100 simulaions. Meric TABLE II. BINARY ROTATION RESULTS CSRN w/ GMLPCore Ave +/- sd dev Bes Jacc / Jmse 0.0 +/ IMacc / IMcr / T r (norm).88 +/ Tc

7 3) Scaling Similar o ranslaion scaling is a local ransformaion. However i presens is own challenges. In he cases of ranslaion and roaion he final ransformed image remains he same size as hose of he raining images. However wih scaling he sizes of he final image and each individual raining image are differen. Our applicaion was no coded o handle individual image sizes. We resolved his issue by zero padding each raining image o make i he same size as he arge image. In addiion when working wih such small images (15x15) o much informaion is los when down-scaling hese images o produce he necessary raining images so much so ha even sandard scaling mehods(such as MATLAB s resize) produce blank images when he raining images are up-scaled o produce he corresponding ranslaion marices required o rain he nework. In order o obain a sufficien number of raining images we limi ourselves o 4 scales and 1.66 and apply hese scales o 3 differen es images: a cross a box and a simulaed eye pach. These images are shown in Fig. 7. Figure 7. Images used for raining he CSRN for up-scaling. Tess wih he simple cross image indicae ha he CSRN is capable of performing scaling. Fig. 8 shows he resuls of raining he GMLP core o perform up-scaling. In his es he nework is rained using 1 images. The inpu image shown is he cross scaled by a facor of.87. In his example he cross is perfecly scaled. The nework achieves a final raining MSE of 0.0 a funcion accuracy of 81.8% and an image accuracy of 100%. Meric TABLE III. BINARY SCALING RESULTS CSRN w/ GMLPCore Ave +/- sd dev Bes Jacc / Jmse 0.0 +/ IMacc / IMcr / T r (norm) / Tc C. Transformaion of Grey-Scale Images In he previous secion we show ha CSRNs are capable of performing affine ransformaions in small binary images. Wha abou more realisic images? In his secion we apply CSRNs o he affine ransformaion of larger more ineresing images specifically grey-scale facial images. To achieve his we use a 5x5 face image. The image is zero-padded wih a 5 pixel widh resuling in a 35x35 image. Sub-image processing (secion 4.) is employed wih a sub-image size of 5x5. 1) Translaion The ranslaion resuls shown here uilize a GMLP core EKF raining movemen encoding and inverse mapping. We uilize eleven raining images wih he images ranslaed from 0 o 10 pixels. A single image ranslaed 10 pixels o he righ is used for esing. Figure 10 shows he resuls of our ranslaion es. Table IV shows resuls for ranslaion of our face image. The nework achieves an image correlaion raio of 100% wih a funcion accuracy of only 100%. Figure 9. Resuls of ranslaing a 5x5 face embedded in a 35x35 image via a CSRN wih a GMLP core. The face is ranslaed by 10 pixels. The nework achieved a final raining J mse = 0.0 J acc = 100% and an IM cr = 100%. Figure 8. Resuls of scaling an 11x11 cross embedded in a 15x15 image via a CSRN wih a GMLP core. The cross is scaled by a facor of The nework achieved a final raining J mse = 0.0 J acc = 81.8% and an IM acc = 100%. Table III summarizes he resuls of our scaling es wih he GMLP core. The nework is simulaed 100 imes. From hese resuls we see ha he CSRN is capable of performing scaling. Noe ha he resuling funcion accuracy for scaling is lower han ha of ranslaion and roaion. This is due o he lack of resoluion in raining scales limied by he image size. TABLE IV. TRANSLATION RESULTS FOR GRAY-SCALE IMAGE Translaion Meric CSRN w/ GMLP Core & EKF Training J acc 100% J mse 0.00 IM acc 100% IM cr 100% 96

8 Noe also ha while we sill repor he image accuracy we uilize he image correlaion raion IM cr raher han he image accuracy when comparing grey-scale images. ranslaion and roaion. Noe he low image accuracy once again indicaing loss of informaion in he inpu image. ) Roaion To perform roaion on he face image we uilze eleven raining images wih he images roaed from 0 o 0 in seps of and a single image roaed by an angle of 16 for esing. Figure 11 shows he resuls of our roaion ransformaion. The face image has clearly been roaed back o a zero degree angle. Noe ha he oupu image appears slighly blurred as compared o he original arge image. This is due o loss of informaion in he roaed inpu image. Table V conains he resuls of performing roaion on he face image. The CSRN successfully roaed he image achieving a funcion accuracy of 90.3% and a high image correlaion raion IM cr = 99.4%. These resuls indicae ha he CSRN has successfully learned he roaion ransform. Resuls of scaling he 5x5 face image via a CSRN wih a GMLP core. The image was up-scaled by a facor of The nework achieved a J mse = 1.13 J acc = 64.4% and an IMcr = 93.%. TABLE VI. SCALING RESULTS Scaling Meric CSRN w/ GMLP Core & EKF Training J acc 66.4% J mse 1.13 IM acc 45.0% IM cr 93.% Figure 10. Resuls of roaing he 5x5 face image embedded in a 35x35 image. Transformaion performed via a CSRN wih a GMLP core. The es image was roaed by 16. The nework achieved a final J mse = 0.70 J acc = 90.3% and an IMcr = 99.4%. TABLE V. ROTATION RESULTS FOR GRAY-SCALE IMAGE Roaion Meric CSRN w/ GMLP Core & EKF Training J acc 90.3% J mse 0.70 IM acc 9.0% IM cr 99.4% 3) Scaling For our scaling es we perform up-scaling. Once again we use a GMLP core rained via EKF wih movemen mapping and neares-neighbor inerpolaion. 7 raining images are used wih scales from.5 o 1.0 in seps of A single image is used for esing. The es image is scaled by a facor of Figure 1 shows he resuls for up-scaling he face image. The image has been scaled up wih some loss of informaion. Table VI liss merics for his es. While he image appears o be properly scaled achieving an correlaion raio of 93.% he nework seems o have more difficuly in learning he scaling ransform as indicae by a mediocre funcion accuracy of jus under 65%. This is in par due o he difficuly in obaining sufficien raining image for such small images. In his case we rain wih only seven images vs. he eleven used for D. Experimenal Plaform All simulaions are performed on a Dell Precision PWS690 worksaion wih an Inel Xenon X core CPU GHz and 3.0 GB of ram. The sysems operaing sysem is Windows XP SP3. Simulaions are run in MATLAB Version 7 (R14) SP. Forward and backpropogaion compuaions for he CSRN are cusom C++ funcions called from MATLAB. VI. DISCUSSIONS AND CONCLUSION In his work we invesigae he capabiliy of CSRN o learn opological image mappings. We adap he CSRN for image processing and rain his NN o perform affine ransformaions. We firs demonsrae he CSRN s abiliy o successfully learn and perform ranslaion roaion and scaling using small binary images. Average funcion accuracies of 100% 83.3% and 73% and highs of 100% 97.3 and 81.8% for each of he respecive ransformaions indicae ha he CSRN is capable of learning he affine ransformaion funcions. As expeced he CSRN yields beer resuls for he simpler case of ranslaion. Average image accuracies a or near 100% indicae he qualiy of he ransformaions performed by he CSRN. The addiion of sub-image processing allows us o uilize he CSRN on larger more ineresing grey-scale images. The CSRN achieves a funcion accuracy of 90.3% for roaion while only 14.3% and 64.4% were achieved for ranslaion and scaling cases respecively. Examinaion of he resuling ranslaion image clearly indicaes ha he CSRN has learned o perform his ransformaion; however he funcion accuracy meric is degraded by a single pixel misalignmen of he image. The mediocre funcion accuracy for he scaling ransform is 97

9 primarily due o he difficuly in obaining sufficien raining images for image of his size. In spie of hese difficulies he CSRN is sill able o achieve image correlaions raios of 95.8% 99.4% and 93.% suggesing ha he nework can indeed learn and perform he desired ransformaion mappings. In he fuure we plan o examine how he CSRN will learn opological mappings on larger more pracical images. We also plan o invesigae generalized image processing and learning asks such as pixel level ransformaions filering and opological image regisraion using CSRN. REFERENCES [1] F. Rosenbla The percepron: A probabilisic model for informaion sorage and organizaion in he brain Psychological Review vol.65 no.6 pp [] M.L. Minsky and S.A. Paper Perceprons Cambridge MA: MIT Press [3] M.Minsky S. Paper Perceprons-Expanded Ediion: An Inroducion o Compuaional Geomery MIT Pres [4] Q.Gao P.Messmer G.Moschyz Binary Image Roaion Using Cellular Neural Neworks IEEE Inernaional Symposium on Circuis and Sysems Vol.3 pp [5] G. Cosanini D. Casali R. Perfei Cellular neural nework emplae for roaion of grey-scale images Elecron. Leers Vol. 39 Issue 5 pp [6] L. Chua L.Yang Cellular Neural Neworks: Theory IEEE Transacions on Circuis and Sysems Vol.35 No [7] L. Chua L.Yang Cellular Neural Neworks: Applicaions IEEE Transacions on Circuis and Sysems Vol.35 No [8] T.Yang Handbook of CNN Image Processing: All You Need o Know abou Cellular Neural Neworks Yang s Scienific Research Insiue LLC. YangSky.com Monographs in Informaion Sciences 00. [9] X. Pang P. Werbos Neural Nework Design for J Funcion Approximaion in Dynamic Programming arxiv:adap-org/ v1 June [10] R. Ilin R. Kozma P.Werbos Beyond Feedforward Models Trained by Backpropagaion: A Pracical Training Tool for a More Efficien Universal Approximaor IEEE Transacions on Neural Neworks Vol.19 No [11] K.Anderson K. Ifekharuddin E.Whie P.Kim Binary Image Regisraion Using Cellular Simulaneous Neworks. Conference proceedings IEEE CIMSVP 009. [1] K.Anderson K.Ifekharuddin P.Kim Gray Scale Image Regisraion using Cellular Simulaneous Recurren Neworks pp: 7440 C1-C1 Proc. Of 54h SPIE Annual Meeing Vol. 744 San Diago CA. 009 [13] Y. Ren K. M. Ifekharuddin and W. E. Whie Large-scale poseinvarian face recogniion using cellular simulaneous recurren nework Applied Opics vol. 49 no. 10 pp. B9-B [14] R. Fisher S. Perkins A. Walker and E. Wolfar. Affine Transformaion. Hypermedia Image Processing Reference. hp://homepages.inf.ed.ac.uk/rbf/hipr/affine.hm.003 [15] R. Gonzales R. Woods Digial Image Processing. Addison-Wesley 199. [16] J.A. Parker R.V. Kenyon D.E. Troxel Comparison of inerpolaing mehods for image resampling IEEE Transacions on Medical Imaging pp [17] B. Ziova J. Flusser Image regisraion mehods: a survey Image and Vision Compuing 1 pp Elsevier [18] P.Werbos Supervised learning: can i escape is local minimum WCNN93 Proceedings Erlbaum Reprined in V. Roychowdhury e al(eds) Theoreical Advances in Neural Compuaion and Learning Kluwer [19] F. Rosenbla Principles of Neural Dynamics New York Sparan 196. [0] C. von der malsburg W. Schneider. Cyberneic Vol.54 pp [1] H. Chang W. Freeman Parameer opimizaion in models of he olfacory neural sysem Neural Neworks Vol.9 No.1 pp [] S. Haykin Kalman Filering and Neural Neworks Wiley 001. [3] R. Ilin R. Kozma P. Werbos Cellular SRN Trained by Exended Kalman Filer Shows Promise for ADP 006 Inernaional Join Conference on Neural Neworks. Sheraon Vancouver Wall Cenre Hoel Vancouver BC Canada. July 006. [4] Thrun S. BurgardW. and Fox D. Probabilisic Roboics The MIT Press [5] R. Ilin Learning and Parameerizaion of Recurren Neural Nework Arrays for Brain Models and Pracical Applicaions PhD disseraion Dep. of Compuer Science Universiy of Memphis Augus 008. [6] G.Bebis M. Georgiopoulos N. Lobo M. Shah Learning affine ransformaions Paern Recogniion Volume 3 Issue 10 Ocober 1999 pp [7] J.A. Parker R.V. Kenyon D.E. Troxel Comparison of inerpolaing mehods for image resampling IEEE Transacions on Medical Imaging pp