Robust Face Recognition Using Eigen Faces and Karhunen-Loeve Algorithm

Transcription

1 Robust Face Recognton Usng Egen Faces and Karhunen-Loeve Algorthm Parvnder S. Sandhu, Iqbaldeep Kaur, Amt Verma, Prateek Gupta Abstract The current research paper s an mplementaton of Egen Faces and Karhunen-Loeve Algorthm for face recognton. The desgned program works n a manner where a unque dentfcaton number s gven to each face under tral. These faces are kept n a database from where any partcular face can be matched and found out of the avalable test faces. The Karhunen Loeve Algorthm has been mplemented to fnd out the approprate rght face (wth same features) wth respect to gven nput mage as test data mage havng unque dentfcaton number. The procedure nvolves usage of Egen faces for the recognton of faces. Keywords Egen Faces, Karhunen-Loeve Algorthm, Face Recognton. F I. INTRODUCTION ACE recognton system based on Egen Faces Method and Karhunen-Loeve algorthm. Egen values and Egenvectors are [] defned as If v s a nonzero vector and λ s a number such that Av = λv () Then, v (Equaton ) s sad to be an egenvector of A wth egen value λ. Example: Egenvectors of Covarance Matrx The egenvectors v of ATA are Parvnder S. Sandhu s workng as Professor wth the Rayat & Bahra Insttute Of Engneerng & Bo-Technology, Mohal- Inda. Amt Verma and Iqbaldeep Kaur are Assstant Professor wth the Rayat & Bahra Insttute Of Engneerng & Bo-Technology, Mohal, Inda,.E- Mal:eramtverma@redffmal.com, er_qbaldeel@yahoo.com. Prateek Gupta s Lecturer (ECE Deptt.) at the Rayat & Bahra Insttute Of Engneerng & Bo-Technology, Mohal v v v Consder the egenvectors [2] v of ATA such that A Av = µ v (2) T Pre multplyng both sdes by A, we have AA Av ) = ( µ Av ) (3) ( T Egenvectors of Covarance Matrx (Equaton 2 and 3) µ = (4) u = Av u u u u u u Here, u resemble facal mages whch look ghostly, hence called Egenfaces A face mage can be proected nto ths face space by pk = UT(xk m) where k=,,m The test mage x s proected nto the face space to obtan a vector p (Equaton 5)

2 p = UT ( x m) (5) The dstance of p (Equaton 6) to each face class s defned by Є k 2 = p pk 2; k =, m (6) A dstance threshold [2] Ө c, (Equaton 7) s half the largest dstance between any two face mages: θ = / 2 max, k{ p p };, k =, m (7) c k Fnd the dstance Є (Equaton 8) between the orgnal mage x and ts reconstructed mage from the egen face space, xf, Є2 = x xf 2, where xf = U * x + m (8) The Recognton [2] process begns wth the condtons, f Єns Ө c Then, nput mage s not a face mage; f Є < AND Є k θ for all k (9) θ c Then, nput mage contans an unknown face; c If Є< θ c AND Єk*=mnk{ Єk} < θ c (0) then nput mage contans the face of ndvdual k* II. EIGEN FACES The Egen face method for human face recognton s remarkably clean and smple. The basc concept behnd the Egen face method s nformaton reducton. When one evaluates even a small mage, there s an ncredble amount of nformaton present. From all the possble thngs that could be represented n a gven mage, pctures of thngs that look lke faces clearly represent a small porton of ths mage space. Because of ths, we seek a method to break down pctures that wll be better equpped to represent face mages rather than mages n general. To do ths, one should generate basefaces and then represent any mage beng analyzed by the system as a lnear combnaton of these base faces. Once the base faces have been chosen we have essentally reduced the complexty of the problem from one of mage analyss to a standard classfcaton problem. Each face that we wsh to classfy can be proected nto face-space and then analyzed as a vector. A k-nearest-neghbor approach, a neural network or even a smply Eucldan dstance measure can be used for classfcaton. The technque can be broken down nto the Generate the Egen faces, Proect tranng data nto face-space to be used wth a predetermned classfcaton method and evaluaton of a proected test element by proectng t nto face space and comparng to tranng data. III. GENERATION EIGEN FACES Before any work can be done to generate the Egen faces, sample faces are needed. These mages wll be used as examples of what an mage n face-space looks lke. These mages do not necessarly need to be mages of the people the system wll later be used to dentfy (though t can help); however the mage should represent varatons one would expect to see n the data on whch the system s expected to be used, such as head tlt/angle, a varety of shadng condtons etc. Ideally these mages should contan pctures of faces at close to the same scale, although ths can be accomplshed through preprocessng f necessary. It s requred that all of the mages beng used n the system, both sample and test mages, be of the same sze. The resultng Egen faces wll also be of ths same sze once they have been calculated. It should be noted that t s assumed that all mages beng dealt wth are grayscale mages, wth pxel ntensty values rangng from 0 to 255. Suppose, there are K mages n our data set. Each sample mage wll be referred to as X where n ndcates that we are dealng wth th sample mage (<= <= K). Each X s a column vector. Generally mages are thought of as pxels, each havng (x, y) coordnates wth (0, 0) beng at the upper left corner (or one could thnk of an mage as a matrx wth y rows and x columns). Convertng ths to a column form s a matter of convenence, t can be done n ether column or row maor form, so long as t s done consstently for all sample mages t wll not affect the outcome. The sze of the resultng X column vector wll depend on the sze of the sample mages. If the sample mages are x pxels across and y pxels tall, the column vector wll be of sze (x*y) x. These orgnal mage szes must be remembered f one wshes to vew the resultng Egen faces, or proectons of test mages nto face-space. Ths s to allow a normal mage to be constructed from a column vector of mage pxels. Let X be the mean of all X ( <= <= K). Ths s the step to calculate an average face of the database. If one were to renterpret the vector as a normal mage, t would appear as one mght expect, as shown n Fg.. Fg. Addton of Faces The next step s to calculate dfference faces U such that U = X - X - (where X - s mean) and form a matrx U, such that U = [U U 2.U K ]. Our goal now s to generate the egenfaces whch s done by calculatng the egenvectors of the covarance matrx UU T.Ths cannot be done drectly as the sze of UU T s (x*y)*(x*y) whch s very large. Clearly, dong these calculatons on a resultng matrx of ths sze s gong to be taxng on all but the most specalzed, advance hardware. To avod ths problem, a trck from lnear algebra s appled. The egenvectors of the UU T matrx can actually be found by 045

3 consderng lnear combnatons of the egenvectors of the U T U matrx. Ths s extremely usefully when one realzes that the sze of the U T U matrx s K x K. For practcally all real world stuatons K << (x*y). The egenvectors w of ths matrx can be readly found through the followng formula: W k l = = U λ l E l () Where, E l s the lth value of the th egenvector of U T U and λ s the correspondng Egen value of w and E. The lnear algebra part of ths trck s gven below: Let the egenvectors of U T U be E ( <= <= K) and the correspondng Egen values be λ. Hence, t can be wrtten: t U U E = λ E (2) Multplyng both the sdes by U, T U * U U E = λ UE (3) Thus, w = UE s the th egenvector of UU T wth correspondng egenvalue λ. The fact that the egenvalues for the UU T and U T U are the same (though f we were gong to calculate all of the Egen values of the UU T matrx, we could get more values, the egenvectors of the U T U only represent the most mportant subset of the Egen values of the UU T matrx). IV. KARHUNEN LOEVE (KL) TRANSFORM The Karhunen Loeve (KL) transform s a preferred method for approxmatng a set of vectors or mages by a low dmensonal subspace. The method provdes the optmal subspace, spanned by the KL bass, whch mnmzes the MSE between the gven set of vectors and ther proectons on the subspace. The KL transform has found many applcatons n tradtonal felds such as statstcs and communcaton. In computer vson, t was used for a varety of tasks such as face recognton, obect recognton, moton estmaton, vsual learnng, and obect trackng. Typcal computer vson applcatons calculate the KL bass of hundreds or thousands mages, each of sze (Wdth Heght) n the range of 0 K to M. The bass s partal and typcally ncludes only a few dozens vectors or less. Calculatng the KL bass for mages of sze requres roughly operatons. In many applcatons, ths large computatonal demands may be prohbtve. In fact, for such applcatons, even the memory requrement may exceed the resources of common computers. Several attempts were already made to reduce the computatonal effort by usng effcent mage codng. The Karhunen-Loeve expanson s a powerful spectral technque for the analyss and synthess of dynamcal systems. It conssts n decomposng a spatal correlaton matrx, whch can be obtaned through numercal or physcal experments. The decomposton produces orthogonal Egen functons or proper orthogonal modes, and Egen values that provde a measurement of how much energy s contaned n each mode. It makes a prncpal component analyss that s a mathematcal way of determnng that lnear transformaton of a sample of ponts n L-dmensonal space whch exhbts the propertes of the sample most clearly along the coordnate axes. Along the new axes, the sample varance are extremes and uncorrelated, see Fg. 2. Usng a cutoff on the spread along each axs, a sample may be reduced n ts dmensonalty. Ths way t can be used to transform ndependent coordnates nto sgnfcant and ndependent ones. The KLC transform s a reversble lnear transform that explot the statstcal property of a vector representaton. The basc functon of KLC transform are orthogonal Egen vector of the covarance matrx of a data set. A KL transform optmally decor relates the nput data. After a KL transform most of the energy of the transform coeffcent s concentrated wthn the frst few components. Ths s compact energy property of KL transform.(as from Fg. 2) Fg. 2 Larhunen-Loeve (KL) Map V. KARHUNEN LOEVE (KL) IMPLEMENTATION The sesmc traces x(t) correspond to the rows of the named data matrx xn*m, n s the number of traces n the gather and m the number of gathers. The zero-lag covarance matrx *nn s * nm xn* m x n* m (4) Ths expresson can be decomposed as * nm Vn * p An* mv n* p (5) In equaton 2 the columns of the matrx Vn*p are the egenvectors n*n and An*n s a dagonal matrx wth the egen values on the dagonal. These egen values are ordered n descendng sequence along the prncpal dagonal of the matrx An*n, V*npt s the correspondng transpose matrx. The prncpal component of the data can be wrtten by: Mp * m Xn * mv n* p (6) If we form the matrx M p*m by selectng the upper m rows of the matrx Mp*m, m s related wth the order of the flter or prncpal row number, placng zeros n the remanng n-m rows, n s related wth the Cut of the Flter, the Ground roll s assumed to be represented by the matrx product between the egenvectors matrx Vn*p and the matrx M p*m, that nclude 046

4 the prncpal components n agreement wth the followng expresson: X ' p * m Vn * mm p * m (7) Fnally, the Ground roll X n*m s subtracted from the data matrx Xn*m, obtanng the gather wthout Ground roll X X X On m n* m n* m * (8) Where, X o s a n*m matrx that represents the fltered sesmc gather. As mentoned before Xn*m s the nput data matrx and X n*m s a matrx that represents the Ground roll extracted from the nput data. VI. PRESENT IMPLEMENTATION In the system (desgned) functons by proectng face mages onto a feature space that spans the sgnfcant varatons among known face mages. The sgnfcant features are known as "egenfaces"[4] because they are the egenvectors (prncpal components) of the set of faces. Face mages must be collected nto sets: every set (called "class") should nclude a number of mages for each person (entty), wth some varatons n expresson and n the lghtng. When a new nput mage s read and added to the tranng database, the number of class s requred. Otherwse, a new nput mage can be processed and confronted wth all classes present n database. We choose a number of egenvectors M' equal to the number of classes. Before start mage processng frst select nput mage. Ths mage can be successvely added to database (tranng) or, f a database s already present, matched wth known faces. Frst, select an nput mage clckng on "Select mage". Then add ths mage to database.if one choose to add mage to database, a postve nteger (face ID) s requred. Ths postve nteger s a progressve number whch dentfes a person (entty) (each person (entty) corresponds to a class). For example: run the GUI delete prevous database add "mage.pg" to database the ID has to be snce mage s the frst entty whch s addng to database add "mage.pg" to database the ID has to be snce you have already added a mage mage to database as frst entty add "mage2" to database the ID has to be 2 snce mage2 s the second one entty addng to database. Add "mage3" to database the ID has to be 3 snce mage3 s the thrd one entty you are addng to database add "mage2" to database the ID has to be 2 once agan snce we have already added mage2 to database and so on! The recognton[6] gves as results the ID of nearest entty present n database. For example f you select mage "mage2" the ID gven SHOULD be 2 "t should be" because errors are possble. VII. MAIN FUNCTIONS AND WORKING The man functons [6]-[8] of the algorthm and functonal descrpton (As from Table I) are as gven below. TABLE I. FUNCTION DESCRIPTION Sr.No Functon Dscrpton. Select mage: Read the nput mage 2. Add selected mage to database The nput mage s added to database and wll be used for tranng 3. Database Info Show nformaton s about the mages present n database. Images must have the same sze. If ths s not true you have to resze them. 4. Face Recognton [9-0] Face matchng. The selected nput mage s processed 5. Delete Database Remove database from the current drectory 6. Vsualzaton tool Show nformaton s 7. Ext Qut the procedure Frst, the recognton [3] and[-3] system s run n the GUI (as shown n the Fg. 3).Then nput mage s selected by pressng the select mage con n the GUI add the mage n the database (Fg. 4) an gven a unque number called ID number (postve). Fg. 3 Runnng GUI We can add as many faces and correspondngly can assgn the ID number for each mage (As shown n Fg. 5) so that each mage s represented by the Unque ID number[4-6]. We can perform face recognton[5] (As shown n Fg. 6) as 047

5 we select a partcular face whch was prevously added n the data base algorthm search for the near by face as per work flow (As shown n Fg. 7) wth reference to class [6]-[8] and dstance from the face space. Fg. 6 Performng Face Recognton Fg. 4. Selectng Input Image Fg. 5 Assgnng ID Number Fg. 7 Work Flow of Algorthm VIII. SCOPE FOR OPTIMIZATION IN FACE RECOGNITION The face s the commonly used bometrc characterstcs for person recognton. The most popular approaches to face recognton are based on shape of facal attrbutes, such as eyes, eyebrows, nose, lps, chn and the relatonshps of these attrbutes. Recognzng people by ther facal features (or vectors) s the oldest dentfcaton mechansm of all, gong back at least to our early prmate ancestors. But there can be varous factors affectng recognton of faces as t nvolves human behavor, hence dstnct characterstcs resultng n dfferent mages of the same face. Below lsted are some factors that may be mproved upon n order to acheve better face recognton[6-7]. These are llustrated by the shown mages as follows. Lmtatons of Egen faces Approach. Varatons n lghtng condtons Dfferent lghtng condtons for enrolment and query. Brght (As from Fg. 6) lght causng mage saturaton. 048

6 Fg. 8 Effect on mages due to Lghtng Dfferences n pose Head orentaton 2D feature dstances appear to dstort. Expresson -Change n feature locaton and shape. REFERENCES [] Turk, M., Pentland, A., Egenfaces for recognton, J. Cogntve Neuroscence, 3 (99), pp [2] Belhumeur, P., Hespanha, J., Kregman, D., Egenfaces vs. Fsherfaces: recognton usng class specfc lnear proecton, IEEE Transactons on Pattern Analyss and Machne Intellgence, 9 (997), pp [3] Tolba, El-Baz, El-Harby, Face Recognton A Lterature Revew, Internatonal Journal of Sgnal Processng, vol. 2, no. 2, 2005, pp [4] M. Turk and A. Pentland, Egenfaces For Recognton, Journal of Cogntve Neuroscence, vol. 3, 99, pp [5] Turk, Pentland, Face Recognton Usng Egen Faces, In Proceedngs of Computer Vson and Pattern Recognton, IEEE, June 99, pp [6] [7] [8] [9] S. Zek, The Vsual Image n Mnd and Bran, Scent. Amercan, 992, pp [0] R. Baron, Mechansms of human facal recognton, Int. J. Man- Machne Studes, Vol. 5, 98, pp [] C. Lu, Statstcal And Evolutonary Approaches For Face Recognton, Ph.D. Dssertaton, Geo. Mason Un, 999. [2] Y. Cu and J. Weng, A Learnng-Based Predcton-and-Verfcaton Segmentaton Scheme for Hand Sgn Image Sequance, IEEE Trans. on Patt. Ana. and Mach. Intell., vol. 2, 999, pp [3] S. J. McKenna, S. Gong and Y. Raa, Modellng Facal Color and Identty wth Gaussan Mxtures, Patt. Recog, vol. 3(2), 998, pp [4] M. Turk and A. Pentland, Egenfaces for Recognton, Journal of Cogntve Neuroscence, vol. 3(), 99, pp [5] T. McInery and D. Terzopoulos, Deformable Models In Medcal Image Analyss: A Survey, Medcal Image Analyss, vol., 996, pp [6] W.N. Martn and J.K. Aggarwal, Dynamc Scene Analyss: A Survey, Computer Vson, Graphcs and Image Process, vol. 7, 978, pp [7] J. K. Aggarwal and N. Nandhakumar, On The Computaton Of Moton From Sequences Of Images, Proc. IEEE, Vol. 76, 988, pp