Kernel principal component analysis network for image classification 1

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1 Kernel prncpal component analyss network for mage classfcaton Wu Dan 4 Wu Jasong 34 Zeng Ru 4 Jang Longyu 4 Lotf Senhadj 34 Shu Huazhong 4 ( Key Laboratory of Computer Network and Informaton Integraton of Mnstry of Educaton Southeast Unversty Nanjng 0096 Chna) ( Insttut Natonal de la Santé et de la Recherche Médcale U 099 Rennes France) ( 3 Laboratore Tratement du Sgnal et de l Image Unversté de Rennes Rennes France) ( 4 Centre de Recherche en Informaton Bomédcale Sno-franças Nanjng 0096 Chna) Abstrac t: In order to classfy the nonlnear feature wth lnear classfer and mprove the classfcaton accuracy a deep learnng network named kernel prncpal component analyss network (KPCA Net) s proposed. Frst mappng the data nto hgher space wth kernel prncpal component analyss to make the data lnearly separable. Then buldng a two-layer KPCANet to obtan the prncpal components of mage. Fnally classfyng the prncpal components wth lnearly classfer. Expermental results show that the proposed KPCANet s effectve n face recognton object recognton and hand-wrtng dgts recognton t also outperforms prncpal component analyss network (PCANet) generally as well. Besdes KPCANet s nvarant to llumnaton and stable to occluson and slght deformaton Ke y wor ds: deep learnng; kernel prncpal component analyss net (KPCA Net); prncpal component analyss net (PCANet); face recognton; object recognton; hand-wrtten dgt recognton Bographes: Wu Dan(990 ) female graduate; Shu Huazhong male Ph.D professor shu.lst@seu.edu.cn. Foundaton tems: The Natonal Natural Scence Foundaton of Chna (No ) the Research Fund for the Doctoral Program of Hgher Educaton (No ) the Program for Specal Talent n Sx Felds of Jangsu Provnce (No. DZXX-03) by produce-learn-research projects of Jangsu Provnce (BY047-) by the 333 project (No. BRA0588) and by the Hgh-end Foregn Experts Recrutment Program (GDT ) and by the Open Fund of Jangsu Engneerng Center of Network Montorng (KJR404).

2 A major dffculty of mage classfcaton comes from the consderable ntra-class varablty arsng fro m dfferent llu mnatons rgd deformatons non-rgd deformatons and occlusons whch are useless for classfcaton and should be elmnated. Deep learnng structures lke deep convolutonal networks have the ablty to learn nvarant features []. Bruna et al [] bult a scatterng network (ScatNet) that s nvarant to both rgd and also non-rgd deformatons. Chan et al. [3] constructed a prncpal component analyss network (PCANet) whch cascaded prncpal component analyss (PCA) bnary hashng and block-wse hstogram. PCANet acheves the state of the art accuracy n many datasets of classfcaton tasks lke extended Yale B dataset AR dataset and FERET dataset. Kernel PCA (KPCA) [4-5] s a nonlnear generalzaton of PCA n the sense that t performs PCA n the feature spaces of arbtrary large dmenson. KPCA can generally provde better recognton rate than the ordnary PCA by the followng two reasons: ) KPCA uses arbtrary number of nonlnear components however ord nary PCA uses only lmted number of lnear prncpal components; ) KPCA has more flexb lty than ordnary PCA snce KPCA can choose dfferent kernel functons (for example Gaussan kernel Polynomal kernel etc.) for d fferent recognton tasks however ordnary PCA uses only the lnear kernel. In ths paper we propose a new deep learnng network named kernel prncpal component network (KPCANet) whch cascades two KPCA stages and one poolng stage. When the kernel functon s lnear the proposed KPCANet degrades to the PCANet [3]. Expermental results show that the proposed KPCANet s nvarant to llumnaton and stable to slght non-rgd deformaton and generally outperforms PCANet n both face recognton and object recognton tasks. KPCANet Fg. shows the whole structure of the proposed KPCA Net whch conssts of two KPCA stages and one poolng stage. Suppose that the patch sze s k k at all stages and all the nput mages are of sze m n. Input I Output of the frst stage: I l I l n feature space Hashng convoluton convoluton I n feature space The frst stage Flters n the frst stage Flters n the second stage The second stage Output of the second stage: I ls Poolng stage Features of I Fg. The detaled block dagram of the proposed KPCANet. The frst stage of KPCANet k k Inputtng N mages I ( = N) that belong to c classes. We take a patch p j R kk centered n the j -th ( j = mn ) pxel of mage I and vectorze the patch as x R. j Collectng all the vectorzed patches x j j = mn of I we obtan a

3 kk mn matrx X = x x x mn R. Construct the same matrx for all nput mages and put them Nmn X = X X X R. For convenence the p-th column of X s denoted kk together we get [ ] N kk kk as x p p = Nmn. We then map X fro m the nput space to the feature space by T X X () kk : kk To fnd the prncpal component of T( x p ) we need to dagonalze the covarance matrx C: Nmn T p ( p) ( p) C = T x T x () Nmn = To smplfy the dagonalzaton of C we could dagonalze K nstead where K = ( T( x p ) T( x q )) T ( x p ) denotes the centralsed ( ) T x p and symbol " " denotes the dot product. Snce the dmenson of could be arbtrarly large even nfn te [4-5] t would be d ffcult to compute dot product ( T( x p) T( x q) ) drect ly therefore we substtute dot product wth kernel functon k and obtan ( k ( p q )) K = x x. After that K s centralzed wth K = K K K + K and Nmn Nmn Nmn Nmn K s dagonalzed to get the prncpal egenvectors W l = L wh ch s the KPCA flters n the frst stage as well where ( ) Nmn j =. Nmn l Zero-paddng the boundary of I and convolve t wth W we get the l -th flter output of the l frst stage * n Nl ; L m Il = I Wl R = = where denotes D convoluton and L denotes the amount of flters n the frst stage... The second stage of KPCANet By repeatng the same process as n the frst stage on I l = Nl ; = L we obtan L kernel PCA flters W s s = L of the second stage. Convolve I wth l W we get the output s of the second stage Ils = Il * Ws = Nl ; = L; s = L ;.3 The poolng stage of KPCANet Every L nput mage s bnarzed and converted to an mage wth: L s l = H s= P I where H s the Heavsde step (lke) functon [3]. ( ls) L s H ( ls ) Il * Ws Nl ; L ; s= = I = = (3) Each of the L mages P l ( l = L) s then parttoned nto B blocks. We compute the hstogram of the decmal values n each block and concatenate all the B hstograms nto one vector denoted as ( l ) Bhst P. At last the KPCANet features of I are gven by

4 L T ( ) [ Bhst ( ) Bhst ( ) Bhst ( )]? f LB = P P P R (4) L Snce deep archtectures are composed of multple levels of nonlnear operatons such as n complcated propostonal formulae re-usng many sub-formu lae [6] the frst two stages of KPCANet are set to be the same n ths paper we could re-use the whole structure of the frst stage as well. Kernel functon Tab. Varous kernel functons used n ths paper Lnear k( ) Gaussan k ( ) Expresson Polynomal k ( ) = ( + ) Exponental ( ) Laplacan k ( ) k Value of parameters x y = xy + c c = 0 x y = exp σ = σ x y xy d d = 3 x y = exp σ = σ x y = exp σ = σ Sgmod k( x y) = tanh ( α xy + c) Ratonal quadratc k ( x y) = exp x y + c α = c = c = Inverse multquadrc k ( x y) = x + c c = Crcular k ( ) x x acos x y x y = π σ π σ x y < σ σ = 0. 0 otherwse 3 3 x x + 0 otherwse Sphercal k ( x y) = σ σ x y < σ σ = 0. Expermental Results We now evaluate the performance of the proposed KPCANet on varous databases ncludng MNIST USPS Yale face dataset COIL-00 objects dataset and AR dataset. Besdes we compared KPCANets that cascade varous (fro m one to three) stage(s) of KPCA layer n ths paper as well. All the features

5 learned by KPCANet are classfed wth SVM classfer. In ths secton we use varous kernel functons to evaluate the performance of the proposed KPCANet n recognton tasks ncludng hand-wrtten dgts recognton face recognton and object recognton. Kernel functons that are used n ths paper are presented n Tab.. MNIST [7] and USPS [8] are used to evaluate the performance of KPCANet on hand-wrtten mages. MNIST contans tran mages and 0000 test mages all mages are of sze 8 8. USPS contans 998 mages of sze 6 6 n total 5000 of them are chosen randomly to tran KPCANet and the rest are for testng. The Yale Face Database [9] s used to evaluate the performance of proposed KPCANet on face mages t contans 65 grayscale mages n GIF format of 5 ndvduals each ndvdual contans mages wth dfferent facal expresson or confguraton: center-lght wear glasses happy left-lght wear no glasses normal rght-lght sad sleepy surprsed and wnk. All mages of ths database are cropped to sze and 90 of them are chosen randomly to tran the proposed KPCANet the rest are for testng. COIL-00 (Columba Object Image Lbrary) [0] s a database of color mages of 00 objects. Images of the objects are taken at pose ntervals of 5 degrees ths corresponds to 7 poses per object. All mages are transformed nto gray mages and cropped to sze 3 3. Half mages of each object are chosen randomly to tran KPCANet and the others are for testng. The performances of dfferent kernel functons on datasets ncludng MNIST USPS Yale face dataset and COIL-00 dataset are presented n Tab.. Both the patch sze and the block sze are set to 8 8 and the flter numbers are set to 8 at all stages the overlappng rato of block s 0.5. It can be seen from Tab. that the performance of PCANet performs better than KPCANets n hand-wrtten dgt recognton generally whle the latter outperforms the former n face recognton and object recognton. Tab. Comparson of error rates of KPCANet wth varous kernel functons on dfferent datasets % Kernel functon MNIST USPS Yale face COIL-00 objects dataset dataset Lnear Gaussan Polynomal Exponental Laplacan Ratonal quadratc Sgmod Inverse multquadrc Crcular Sphercal Face recognton on AR face dataset The propertes of KPCANet that nvarant to llumnaton and stable to slght deformatons and occlusons are tested by performng KPCANet on AR dataset [] n ths secton. AR dataset contans about 4000 color mages from 6 ndvduals. The subset of the data that contans 00 ndvduals consstng of 50 males and 50 females of sze 65 0 s chosen. The color mages are converted to gray scale ones. Each ndvdual conssts of mages wth frontal llumnaton and neutral expresson whch s used as the tranng sample the other mages ncludng 4 mages varaton from llumnaton to dsguse are used for testng.

6 The patch sze and the block sze are set to be 7 7 and 8 8 respectvely. The overlappng rato of block s 0.5. We compare the proposed KPCANet wth LBP [] and P-LBP [3] n Tab.3. KPCANet wth lnear kernel functon and Laplacan kernel functon s used n ths experment. From Tab.3 one can see that when the mages only undergo the change of llumnaton the testng accuracy rate acheves 00% wth both lnear kernel KPCANet and Laplacan kernel KPCANet ths demonstrates that KPCANet s nvarant to llumnaton. Besdes KPCANet outperforms LBP [] and P-LBP [3] on dfferent expressons and dsguses wth varous llumnaton condtons whch show that KPCANet s robust to small deformaton and occluson. Tab. 3 Comparson of accuracy rates of the methods on AR face database % Test sets Illumnaton Expresson Dsguse wth llumnaton LBP [] P-LBP [3] KPCANet (lnear kernel) KPCANet (Laplacan kernel) KPCANet w th varous stages n AR face dataset In ths secton we perform KPCANet whch cascade dfferent number of KPCA flter bank layer and a poolng layer wth AR face dataset we used n Secton. and all mages are cropped to sze 3 3. Lnear kernel sg mod kernel and crcular kernel are chosen here n order to smplfy the result. The patch sze and the block sze are set to be 7 7 and 8 6 respectvely. The overlappng rato of block s 0.5. The results are shown n Tab.4. Tab.4 Comparson of accuracy rates of KPCANet wth dfferent stages number on AR face dataset % Flter bank layer number Lnear kernel Sg mo d kernel Crcular kernel One Two three Fro m Tab.4 we can see that the accuracy rate ncrease as the number of KPCA flter bank layer ncrease n KPCANet however tranng tme grows exponentally at the same tme. 3 Concluson In ths paper we propose the KPCANet whch s an extenson of PCANet for mage classfcaton. The proposed KPCANet cascades kernel PCA bnary hashng and block-wse hstogram. Experments prove that KPCANet wth dfferent kernel functons s stable n general and also s nvarant to llumnaton and stable to slght deformaton and occluson. Moreover KPCANet s sutable for the recognton of hand-wrtten mages face mages and object mages. References [] Yann L C Koray K Cl F. Convolutonal networks and applcatons n vson[c]// Crcuts and Systems (ISCAS) Proceedngs of 00 IEEE Internatonal Symposum on. Pars France 00: [] Bruna J Mallat S. Invarant scatterng convoluton networks[j]. IEEE Transactons on Pattern Analyss and Machne Intellgence 03 35(8): [3] Chan T H Ja K Gao S et al. PCANet: a smple deep learnng baselne for mage classfcaton?[j]. arxv preprnt arxv:

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