Subspace LDA Methods for Solving the Small Sample Size Problem in Face Recognition

Transcription

1 Proceedings of he Inernaional Conference on Compuer and Informaion Science and Technology Oawa, Onario, Canada, May 11 12, 2015 Paper No. 126 Subspace LDA Mehods for Solving he Small Sample Size Problem in Face Recogniion Ching-Ting Huang, Chaur-Chin Chen Deparmen of Compuer Science/Naional Tsing Hua Universiy 101 KwanFu Rd., Sec. 2, Hsinchu, Taiwan Absrac In face recogniion, LDA ofen encouners he so-called small sample size (SSS) problem, also known as curse of dimensionaliy. This problem occurs when he dimensionaliy of he daa is quie large in comparison o he number of available raining images. One of he approaches for handling his siuaion is he subspace LDA. I is a wo-sage framework: i firs uses PCA-based mehod for dimensionaliy reducion, and hen he LDA-based mehod is applied for classificaion. This paper invesigaes four popular subspace LDA mehods: Fisherface, Complee PCA plus LDA, IDAface, and plus LDA and compare heir effeciveness when handling he SSS problem in face recogniion. Experimenal resuls esed on hree publically available face daabases: JAFFE, ORL, and FEI, show ha LDA wihou reducing image size by PCA projecion is he wors and plus LDA performs beer han he oher mehods for a huge size of daabase. Keywords: Face Recogniion, Linear Discriminan Analysis (LDA), Principal Componen Analysis (PCA) 1. Inroducion Face recogniion has recenly received significan aenion and been an imporan issue in paern recogniion and image analysis over he las few decades as shown by Samaria e al. (1994), Zhao e al. (2003), and Li e al. (2011). The ulimae goal in face recogniion is o develop a compuer-based auomaed sysem which works reliably under unconsrained condiions, runs quickly and requires minimum raining daa. However, i is sill an on-going research area. Face recogniion mehods are of wo ypes, he geomery-based approach and he appearance-based approach. The geomery-based approach aims o locae disincive feaures such as eyes, nose, mouh, and chin. Properies of he feaures and relaions such as disances and angles beween he feaures are used as descripors for face recogniion. The appearance-based approach operaes direcly on he inensiies of pixels wihin he face images and processes an image as a wo-dimensional holisic paern. I exracs feaures in a subspace derived from he raining images. Compared o he geomery-based approach, he appearance-based approach performs more robusly under siuaions like noise, blurring, andhe variaions in illuminaion, facial expressions and occlusion ec. in images. During he las few decades, many appearance-based approaches have been developed based on Principal Componen Analysis (PCA), called Eigenface, such as Sirovich e al. (1987), Kirby e al. (1990), Turk e al. (1991), and Linear Discriminan Analysis (LDA) by Fisher e al. (1936), called Fisherface such as Belhumeur e al. (1997), Yu e al. (2001), Yang e al. (2003), Zhao e al. (2003), and heir varians. LDA usually ouperforms PCA for classificaion asks since LDA akes class informaion ino accoun while PCA does no. However, in face recogniion, LDA ofen faces he so-called small sample size (SSS) problem due o he relaively small number of raining images per individual compared o he dimensionaliy of he image space, and would resul in he singulariy of he wihin-class scaer marix. A number of approaches have been proposed o address he SSS problem. One of he mos successful approaches is he subspace LDA, which is a wo-phase framework: firs projecing he original images ino a subspace, where dimensionaliy is reduced; nex preceding LDA-based mehods on he lower 126-1

2 dimensional subspace. This paper repors he experimenal resuls on 3 daabases of four LDA-based mehods: Fisherface (PCA plus LDA) by Belhumeur e al.(1997), Complee PCA plus LDA by Yang e al. (2003), an improved discriminae analysis (PCA plus IDA) as shown by Zhuang e al.(2007), and he Bi- Direcional PCA plus LDA ( plus LDA) as shown by Zuo e al.(2006). The res of his paper is o give he mahemaical foundaion of PCA and LDA, review he four subspace LDA mehods and show he resuls of comparison on hree daabases followed by a conclusion. 2. Background Review Noaions Le{X 1, X 2,, X N } be a raining se of N gray level face images of R rows and C columns, and X n (r, c) {0, 1,, 255}, n = 1, 2,, N, r = 0, 1,, R 1, c = 0, 1,, C 1. Le X = 1 N N n=1 X n R R C be he mean image of all samples. Le x n R RC be a column-vecor represenaion of X n such ha x n (r R + c) = X n (r, c), n = 1, 2,, N, r = 0, 1,, R 1, c = 0, 1,, C 1. Assume ha each face image belongs o one of K classes, and here are N k face images in class k, of which he mean image is x k = 1 N x class k x R RC, k k = 1, 2,, K. Le x = 1 N N n=1 x n R RC be he mean image of all samples in a column vecor form Scaer Marices The oal scaer marix S T R RC RC, wihin-class scaer marix S W R RC RC and beween-class scaer marix S B R RC RC can be defined as S T = N n=1 (x n x )(x n x ) (1) K k=1 x class k (2) S W = (x x k)(x x k) S B = K k=1 N k (x k x )(x k x ) (3) I s easy o verify ha S T = S W + S B, and S W and S B are nonnegaive definie. The rank of S T R RC RC is he dimensionaliy of he range space of S T, which is denoed as rank(s T ) and equal o he number of nonzero eigenvalues of S T. If rank(s T ) = RC, S T is called full-rank and i is inverible; if rank(s T ) < RC, S T is singular and have zero eigenvalues. Noe ha rank(s T ) N 1, rank(s W ) N K and rank(s B ) K Principal Componen Analysis (PCA) PCA aims o find a se of projecion vecors ha map he original RC-dimensional image space ino a D PCA -dimensional feaure space (D PCA RC) such ha he projecion of he daa ono he firs projecion vecor has he larges scaer, and he projecion ono he second projecion vecor, which is orhogonal o he firs one, has he second larges scaer, and so on. So PCA aemps o find a se of orhonormal eigenvecors, {w 1, w 2,, w DPCA } of S T, o form he projecion marix W PCA = [w 1, w 2,, w DPCA ] corresponding o he D PCA larges eigenvalues, where D PCA is considered as he smalles number such ha D PCA d=1 λ d RC d=1 λ d > 85%. The new feaure vecor y n R D PCA can be obained by y n = W PCA x n, n = 1, 2,, N (4) 126-2

3 2. 4. Linear Discriminan Analysis (LDA) LDA as shown by Fisher e al. (1936) is a linear dimensionaliy reducion echnique used for face recogniion [Belh1997]. I aims o find he opimal se of discriminan vecors ha maps he original RCdimensional image space ino a D LDA -dimensional feaure space (D LDA RC) such ha images from differen classes are more separaed and images of he same class are more compac. LDA hen aims o find a se of orhonormal projecion vecors such ha he firs projecion vecor maximizes he crierion J 0 defined as J 0 (w) = w S B w w S W w (5) The goal of LDA becomes o find a projecion marix W LDA = [w 1, w 2,, w DLDA ], where S B w d = λ d S W w d, d = 1, 2,, D LDA (6) D LDA is considered as he smalles number such ha R D LDA can be obained by D LDA d=1 λ d RC d=1 λ d > 85%. The new feaure vecor y n y n = W LDA x n, n = 1, 2,., N (7) Discussion LDA is usually more suiable for solving he classificaion ask han PCA because LDA aemps o model he difference beween he classes while PCA is good for dimensionaliy reducion bu does no ake class informaion ino accoun. The major drawback of applying LDA for he face recogniion ask is ha i may encouner he so-called small sample size (SSS) problem. I is because S W R RC RC and rank(s W ) N K. If N < RC + K, hen rank(s W ) < RC, S W is singular. For example, if he size of an image is , hen RC = bu i is no pracical o collec Kface images. 3. Subspace LDA Mehods In recen years, many researchers have noiced he SSS problem and ried o ackle i using differen approaches. The following four subspace LDA mehods begin wih he projecion wih y n = W PCA x n by PCA as defined in (4) for he firs sage of a wo-sage framework. Second, i applies he LDA-based algorihm in he reduced subspace o ge he opimal projecion marix which will be discussed as follows Fisherface Among all of he subspace LDA mehods proposed over he las few decades, he Fisherface proposed by Belhumeur e al. (1997) has araced much aenion. The projecion marix is compued according o he noaions given in Secion 2. We compue S B = W PCA S B W PCA R D PCA D PCA and he wihin-class scaer marix S W = W PCA S W W PCA R D PCA D PCA in he reduced subspace. Consruc he projecion marix W LDA = [w LDA 1, w LDA 2,, w LDA DLDA ] by solving he generalized eigenvalue problem: S B w d LDA = λ d S W w d LDA, d = 1, 2,, D LDA (8) where w d LDA denoes he eigenvecor corresponding o he d-h larges eigenvalue λ d. The opimal projecion marix W OPT R RC D LDA and a reduced face feaure vecor y n R D LDA are given as follows. W OPT = W LDA W PCA (9) 126-3

4 y n = W OPT x n, n = 1, 2,, N. (10) Complee PCA plus LDA The Complee PCA plus LDA as shown by Yang e al. (2003) was suggesed o give a heoreically opimal and more efficien algorihm ha can overcome he weaknesses of he Fisherface menioned above. Le D com = rank(s T ). We firs find he orhonormal eigenvecors of S T o consruc he projecion marix W com = [w com 1, w com 2,, w com Dcom ], hen compue he beween-class scaer marix S B = W com S B W com R D com D com, he wihin-class scaer marix S W = W com S W W com R D com D com, and he oal scaer marix S T = W com S T W com R D com D com in he reduced subspace. Le D iso = rank(s iso W ). In he D com -dimensional ransformed space, le {w 1, w iso 2,, w iso Dcom } be a se of all orhonormal eigenvecors of S W. Denoe W iso = [w iso 1,, w iso iso Diso ] and W iso = [w Diso +1,, w iso Dcom ]. Le D = rank(s B ).Le S B = W iso S B W iso R D iso D iso, S W = W iso S W W iso R D iso D iso. Consruc he projecion marixw = [w 1, w 2,, w D ] by solving he generalized eigenvalue problem S Bw d = λ d S Ww d, d = 1, 2,, D (11) Le S B = W iso T S B W iso R (D com D iso ) (D com D iso ). Find he orhonormal eigenvecors of S B o consruc he orhogonal marix W = [w 1, w 2,, w D ]. The projecion marix W OPT R RC 2D and a reduced feaure vecor y n R 2D are obained by W OPT = W com [W iso W, W iso W ] (12) y n = W OPT x n, n = 1, 2,, N. (13) IDAface The improved discriminae analysis (IDAface) proposed by Zhuang e al. (2007) is a wo-sage framework. Firs, a modified PCA, called PCA wih selecion (PCA_S), is used for dimensionaliy reducion. Second, he algorihm uses inverse Fisher discriminan analysis (IFDA) which inroduces a new crierion o derive he discriminaive informaion from boh range space and null space of wihinclass scaer marix. The inverse Fisher discriminan crierion is defined as W w = arg min S W W W W S B W (14) Le p = rank(s T ). Selec he orhonormal eigenvecors of S T corresponding o p eigenvalues, which saisfy he inequaliy w S B w > w S W o form he projecion marix W PCA_S = [w 1, w 2, w DPCA_S ] R RC D PCA_S, where D PCA_S K 1. Calculae he beween-class scaer marix S B R D PCA_S D PCA_S by S B = W PCA_S S B W PCA_S. Le D proj = rank(s B ). Work ou he orhonormal eigenvecors of S B o consruc he projecion proj proj proj marix W proj = [w 1, w2,, wdproj] R D PCA_S D proj proj, where w d denoes he eigenvecor corresponding o he d-h larges posiive eigenvalue of S B, d = 1, 2,, D proj. Calculae he beweenclass scaer marix S B R D proj D proj and he wihin-class scaer marix S W R D proj D proj by S B = W proj S B W proj (15) S W = W proj W PCAS S W W PCAS W proj (16) 126-4

5 Adop he opimizaion problem using he inverse Fisher discriminan crierion, we have W IFDA = arg min W W W proj W W proj W PCAS S W W PCAS W proj W W PCAS S B W PCAS W proj W = arg min W W W proj S W W proj W W W proj S B W proj W W = arg min S W W W W S B W (17) Obain W IFDA = [w 1, w 2,, w Dproj ] R D proj D proj by solving he generalized eigenvalue problem S W w d = λ d S B w d, d = 1, 2,, D proj (18) where w d is he generalized eigenvecor corresponding o he d-h smalles generalized eigenvalue λ d. Compue he opimal projecion marix W OPT R RC D proj and a reduced vecor y n R D proj as follows W OPT = W IFDA W proj W PCA_S (19) y n = W OPT x n, n = 1, 2,, N. (20) plus LDA Bi-Direcional PCA () proposed by Zuo e al. (2006) is a generalizaion of wo-dimensional principal componen analysis (2DPCA) as shown by Yang e al. (2004), which direcly compues eigenvecors of he scaer marix wihou convering 2D marices ino 1D vecors. Raher han working in he row direcion of an image like 2DPCA, can reflec informaion beween boh rows and columns of an image. The plus LDA performs LDA in he low-dimensional subspace. The deailed algorihm and furher discussion is given as follows Algorihm (S1) (): Consruc he row oal scaer marix S T row R C C and he column oal scaer marix S T col R R R by S T row = 1 N NR n=1 (X n X ) (X n X ), S col T = 1 N NC n=1 (X n X ) (X n X ) (21) Find he projecion marix W row = [w row 1, w row 2, w row Drow ] and W col = [w col 1, w col 2, w col Dcol ] according o he larges eigenvalues of S row T and S col T, respecively. (S2) ( feaure exracion): Obain feaure marix Y n R D col D row of image marix X n (n = 1, 2,, N) by Y n = W col X n W row (22) Le y n R D cold row be a column-vecor represenaion of Y n such ha y n (r D row + c) = Y n (r, c), n = 1, 2,, N, r = 0, 1,, D row 1, c = 0, 1,, D col 1. Compue he mean image of class k : y k = 1 y, k = 1, 2,, K, and he mean image of all N k samples: y = 1 N N n=1 y n. y class k 126-5

6 (S3) (LDA): Consruche beween-class scaer marix S B R D cold row D col D row and he wihinclass scaer marixs W R D cold row D col D row in he subspace by S B = K k=1 N k (y k y )(y k y ) (23) K S W = k=1 (y y k )(y y k ) (24) y class k Le D LDA = rank(s B ), find he projecion marix W OPT = [w 1, w 2,, w DLDA ] R D cold row D LDA by solving he generalized eigenvalue problem S B w d = λ d S W w d, d = 1, 2,, D LDA (25) (S4) (Feaure represenaion): The final feaure vecor y n R D LDA is defined by y n = W OPT y n, n = 1, 2,, N. (26) 4. Experimens The experimens were carried ou using hree face daabases: he JAFFE daabase (Web-1) as shown by Lyons e al. (1998), he ORL daabase (Web-2) as shown by Samaria e al. (1994) and he FEI daabase (Web-3). The specificaions for each of he face daabases are furher described below The JAFFE Daabase The JAFFE daabase is a Japanese face daa base which consiss of 213 face images from 10 females. There are seven differen facial expressions. Each individual has wo o four images for each expression. All individuals are in an uprigh, fron posiion. The original size of each image is wih 256 possible gray levels for each pixel. In our experimens, we cropped and resized hem o pixels. Only one image per facial expression was seleced for one individual, so here are 7 images per individual and 70 images in oal. The seven images of one individual are shown in Fig. 1. Fig.1. Sample images from he JAFFE daabase The ORL Daabase The ORL daabase consiss of 400 face images from 40 individuals, including 36 males and 4 females. Each conribues 10 differen images. Some images were aken a differen sessions, and here are variaions in facial expressions and facial deails. All he images were aken agains a dark homogenous background wih he individuals in an uprigh, fron posiion (wih a olerance for iling and roaion of he face up o 20 degrees).the size of each image is wih 256 possible gray levels for each pixel, we resized hem o pixels. The en images of one individual from he ORL daabase are shown in Fig. 2. Fig. 2.Sample images from he ORL daabase 126-6

7 The FEI Daabase The FEI face daabase is a Brazilian face daabase ha conains 2800 face images from 200 individuals, including 100 males and 100 females. Each conribues 14 images. All images are colorful and aken agains a whie homogenous background in an uprigh fronal posiion wih an individual having roaion of up o 180 degrees. Scale migh vary abou 10% and he original size of each image is 640x480. In our experimens, we cropped and resized hem o pixels, and convered hem o 256 gray levels. Only firs 11 images in differen roaion degrees were seleced from each individual, so here are 2200 images in oal. The eleven images of one individual used in he experimens from he FEI are shown in Fig. 3. Fig. 3. Sample images from he FEI daabase The informaion of images used in he experimens from hree face daabases are summarized in Table 1. Table 1. A summary of he face daabases Image Number Variaions Name Size Toal Images/in Individ Male/ Expres Facial Illumina (pixels) images dividual uals Female sion deails ion Pose JAFFE / 7 Yes No No No ORL / 4 Yes Yes No Yes FEI / 100 No No No Yes Performance Evaluaion We evaluaed he performance of he subspace LDA mehods on he aforemenioned daabases. The Euclidean disance and he neares neighbour classifier is adoped. The recogniion rae is calculaed as he raio of number of successful recogniion and he oal number of es images. We esed under differen number of images per individual by spliing all images of each class ino wo subses, such ha here are k imagesper individual in raining ses and he remaining N k images per individual are in he K esing ses, k = 1,, N 1. We ried all possible splis and repor he average recogniion rae for each K daabase. All he experimens were carried ou on an Inel Core i compuer wih an 8GB RAM and esed on a Malab plaform (version: R2012b). The resuls are given in Tables 2 o 4. Table 2. Recogniion raes (%) of he subspace LDA mehods on he JAFFE daabase Subspace LDA mehods Size of he images Number of raining images per individual a. Fisherface 112 x 92 N/A b. Complee PCA plus LDA 112 x c. IDAface 112 x d. plus LDA 112 x

8 Table 3. Recogniion raes (%) of LDA and he subspace LDA mehods on he ORL daabase Mehods Number of raining images per individual Fisherface N/A Complee PCA plus LDA IDAface plus LDA LDA N/A Table 4. Recogniion raes (%) of LDA and he subspace LDA mehods on he FEI daabase Subspace LDA mehods Number of raining images per individual Fisherface N/A Complee PCA plus LDA IDAface plus LDA LDA N/A Conclusion We have discussed he LDA and four subspace LDA mehods for solving he small sample size problem encounered in face recogniion. Experimenal resuls, by esing on hree well-known face image daabases: JAFFE(Web-1), ORL(Web-2), and FEI(Web-3), show ha he LDA mehod direcly convers a 2d image ino a larger column vecor perform he wors. On he oher hand, he plus LDA mehod applies boh row and column projecion o reduce he size of marix achieves he overall bes resuls, whereas, he size selecion of subspace projecion by eiher PCA or LDA meris furher sudies. Acknowledgemens This work is suppored by Taiwan Naional Science Council Gran NSC E MY3. References Belhumeur P.N., Hespanha J.P., Kriegman, D. (1997). Eigenfaces vs. Fisherfaces: Recogniion Using Class Specific Linear Projecion, IEEE Transacions on Paern Analysis and Machine Inelligence, vol. 19, no. 7, pp Chen C.C., Chu H.T., Shieh Y.S. (2008). Face Image Rerieval By Projecion-Based Feaures, The 3rd In. Conf. on Image Media Qualiy and Is Applicaions, Kyoo, Japan, Sep.5-6, pp Chen L.F., Liao H.Y.M., Ko M.T., Lin J, C., Yu G.J. (2000). A New LDA-Based Face Recogniion Sysem which can Solve he Small Sample Size Problem, Paern Recogniion, vol. 33, no. 10, pp Fisher R.A. (1936). The Use of Muliple Measuremens in Taxonomic Problems, Annals of Eugenics, vol. 7, no. 2, pp Huang C.T. (2014). A Sudy on he Subspace LDA Mehods for Solving he Small Sample Size Problem in Face Recogniion, M.S. Thesis, Naional Tsing Hua Universiy, January Kirby M., Sirovich L. (1990). Applicaion of he Karhunen-Loeve Procedure for he Characerizaion of Human Faces, IEEE Trans. on Paern Analysis and Machine Inelligence, vol. 12, no. 1, pp Li S.Z., Jain A.K. (2011). Handbook of Face Recogniion, Springer

9 Lyons M., Akamasu S., Kamachi M., Gyoba J. (1998). Coding Facial Expressions wih Gabor Waveles, The hird IEEE In. Conf. on Auomaic Face and Gesure Recogniion, pp Marínez A.M., Kak A.C. (2001). PCA versus LDA, IEEE Trans. on Paern Analysis and Machine Inelligence, vol. 23, no. 2, pp Samal A., Iyengar P.A. (1992). Auomaic Recogniion and Analysis of Human Faces and Facial Expressions: A Survey, Paern Recogniion, vol. 25, no. 1, pp Samaria F.S., Harer A.C. (1994). Parameerisaion of a Sochasic Model for Human Face Idenificaion, The Second IEEE Workshop on Applicaions of Compuer Vision, pp Sirovich L., Kirby M. (1987). Low-Dimensional Procedure for he Characerizaion of Human Faces, The Journal of he Opical Sociey of America A, vol. 4, no. 3, pp Turk M., Penland A. (1991). Eigenfaces for recogniion, Journal of Cogniive Neuroscience, vol. 3, no. 1, pp Yang J., Yang J.Y. (2003). Why Can LDA Be Performed in PCA Transformed Space? Paern Recogniion, vol. 36, no. 2, pp Yang J., Zhang D., Frangi A.F., Yang J.Y. (2004). Two-dimensional PCA: a New Approach o Appearance-based Face Represenaion and Recogniion, IEEE Trans. on Paern Analysis and Machine Inelligence, vol. 26, no. 1, pp Yu H., Yang J. (2001). A Direc LDA Algorihm for High-dimensional Daa - wih Applicaion o Face Recogniion, Paern Recogniion, vol. 34, pp Zhao W., Chellappa R., Phillips P.J., Rosenfeld A. (2003). Face Recogniion: A Lieraure Survey, ACM Compuing Surveys, vol. 35, no. 4, pp Zhuang X.S., Dai D.Q. (2007). Improved Discriminan Analysis for High-dimensional Daa and Is Applicaion o Face Recogniion, Paern Recogniion, vol. 40, no. 5, pp Zuo W., Zhang D., Yang, J., Wang K. (2006). Plus LDA: A Novel Fas Feaure Exracion Technique for Face Recogniion, IEEE Trans. on Sysems, Man, and Cyberneics, Par B: Cyberneics, vol. 36, no. 4, pp Web sies: Web-1: hp:// consuled 1 March, Web-2: hp:// consuled 1 March, Web-3: hp://fei.edu.br/~ce/facedaabase.hml, consuled 1 March,