Modular PCA Face Recognition Based on Weighted Average

Transcription

1 odern Appled Scence odular PCA Face Recognton Based on Weghted Average Chengmao Han (Correspondng author) Department of athematcs, Lny Normal Unversty Lny 76005, Chna E-mal: Abstract Ths paper presents an mproved modular PCA approach, that s, modular PCA algorthm based on weghted average. Ths algorthm extracts weghted average for every sub-block of every tranng sample n each type of tranng sample, and normally operates the correspondng sub-block n tranng sample usng weghted average, then all standardzed sub-blocks consttute the overall scatter matrx, and thus the optmal projectve matrx s obtaned; From the mddle value of sub-blocks n tranng set, and normally projectng sub-blocks of tranng samples and test samples to the projectve matrx, then we can get dentfed characterstcs; At last, use the recent dstance classfer to class. The test results n the ORL face database show that the proposed method n dentfyng performance s superor to ordnary modular PCA approach. Keywords: Face recognton, Prncpal component analyss, Weghted average Face recognton s an actve subject n the current pattern recognton feld, whch has broad applcaton prospects[valentn D, 1994-Zhang Cupng, 1995] and has a lot of algorthms[j.lu, 003-Gan Junyng, 007]. n the human face mage recognton, prncpal component analyss (PCA)[Krby., 1990] also known as KL transformaton, s consdered one of the most successful lnear dscrmnant analyss methods, whch s stll wdely used n mage recognton feld such as human face, etc. PCA method can not only effectvely reduce the dmenson of human face mages, but also retan ts key dentfyng nformaton. However, ths method reures the matrx of human face mage pre-converted nto one-dmensonal vector, and then takes vector as the orgnal characterstc to feature extracton. For the dmenson of the converted one-dmensonal vector s generally hgher, extractng the subseuent feature causes dffculty, whch makes the followng algorthm has hgher computatonal complexty. n addton, n face recognton when the facal expresson and llumnaton condtons change largely, for the ordnary PCA method extracts the global features of mage, ts dentfcaton results are unsatsfactory. n fact, when facal expresson and llumnaton condtons change, only some face sgnfcants varate, but lttle changes n other parts, or even no change. Dscrmnant analyss for the blocked sub-mages can capture local nformaton characterstcs of human face, thus help dentfy. Chen Fubng[Chen Fubng, 007] proposed blocked PCA algorthm based on tradtonal PCA method. Ths method, frst blocks an mage, then dscrmnant analyses the blocked sub-mages usng PCA. ts characterstcs are able to effectvely extract the local characterstcs of mages, especally for the mages whose facal expressons and llumnaton condtons change largely. Compared wth the PCA method, blockng the orgnal dgtal mage, can not only easly reduce the dmenson of mage vector by two powers, but also ncrease the sub-mages number of tranng samples by two parts, whch converts the small sample problem nto large sample problem to deal wth and can reduce complexty of the problem average face method proposed by He Guohu[He Guohu,006] effectvely ncreases the dstance between the samples of dfferent categores, whle narrowng the dstance between the samples, that s, makng the dstance between classes larger and dstance smaller, whch s conducve to dentfcaton, and mproves the correct recognton rate of human face. However, n small sample cases, average cannot guarantee that average of varous types of tranng samples s the center of ths sample dstrbuton. And the projectve matrx taken from the average of tranng sample as the center of ths class samples can not guarantee to be optmal. n order to further mprove the recognton performance of PCA method and reduce the nfluence of takng the optmal projectve matrx by average dervaton center of tranng samples. Ths paper presents an mproved modular PCA approach based on the above method and by the adaptve weghted average dea n paper [Yn Hongtao,006], that s modular PCA approach based on weghted average. Ths algorthm extracts weghted average for every sub-block of 64

2 odern Appled Scence November, 009 every tranng sample n each type of tranng sample, and normally operates the correspondng sub-block n tranng sample usng weghted average. The test results n the ORL face database show that the proposed method n dentfyng performance s superor to ordnary modular PCA approach. 1. PCA Algorthm PCA method s a statstcal analyss method based on Karhunen-Loeve (KL) transformaton, whose prncple s that hgh-dmensonal vector projects to low-dmensonal vector space by a specfc feature vector matrx. Through the vector of low-dmensonal representaton and the feature vector matrx, we can reconstruct the correspondng orgnal hgh-dmensonal vector. n the face recognton process, after the KL transformaton, we can get a set of feature vectors to form a lower dmensonal subspace. Any human face mage can project t and get a set of coordnates factors. Ths group of coeffcents shows that the mage locaton n the sub-space can be used as a bass for face recognton. n ths method, generatng matrx s the total scatter matrx of tranng samples,.e: ( )( ) S 1 X X X X T 1 XX T, (1) 1 Where X s the mage vector of the -th tranng sample; Vector dmenson s n ; X s the average fgure vector for tranng sample; s the total number of tranng samples. Accordng to the general scatter matrx, we can derve a set of orthogonal egenvectors u 1, u, L, un, and ts correspondng characterstc values are λ1 λ L λ. Through choosng the correspondng egenvectors of the prevous,,, n m ( m < n) non-zero egenvalues as the orthogonal bass, n the new orthogonal sub-space U, the face sample X can be expressed as: T Y U X (). Adaptve weghted average [Yn Hongtao, 006] When usng PCA algorthm, we frst spread the mage matrx by row (column) as a vector. Suppose the vector spread by all the mage matrx be: Where 1,,, c ( (,1), (,), (, )) ( ) ( ) ( ) ( ) T j j j j m L, (3) X X X X L, c s the type number of tranng samples; j 1,, L N, s the number of the -th tranng sample; m s the vector dmenson. Because the mean vector of several vectors s taken averagely from the scalar vector of the correspondng dmenson, we explan the determned method of weghted value by takng the frst dmenson as example. ( Frst of all, calculate the dstance sum ) ( ) ( d j 1,, L, N d ) ( j 1,, L, N ) of every sample n the -th sample and other j,1 j,1 sample, then fnd the mnmum of them arg mn ( j ) d d d. ( ) ( ) ( ) 1 (,1) 1 j We beleve that the sample whose dstance sum wth the same type sample s larger. t devates greater from the class center. n calculatng class average, t should be gven a smaller weght and the weghted value n the frst dmenson of the j -th sample n the -th class s d ( ) ( ) 1 ( j,1) 1 ( ) d ( j,1) μ +β, (4) whereβ s a constant greater than or eual to zero. For regulatng the weght extent, when β 0, the algorthm becomes the tradtonal averagng method. The average of the frst dmenson n the -th sample can be modfed to ( ) 1 N ( ) ( ) μ 1 ( j,1) X j ( j,1) N ( ) μ j 1 ( j,1) X% (5) Smlarly, we can fnd the means of other dmensons of tranng samples. 65

3 odern Appled Scence 3. odular PCA algorthm based on the weghted average The basc dea of modular PCA algorthm based on the weghted average s as follows: block m n mage matrx nto p blocked mage matrx, namely, 11 1 L 1 1 L O p1 L p (6) Where each sub-mage matrx s a m 1 n 1 matrx, pm1 m, n1 n, then takng the sub-mage matrx of all tranng samples as the mage vector of tranng sample to purpose PCA method. The dfference from the tradtonal PCA algorthm s that we derve all scatter matrx not usng the sub-block average of all tranng samples, but usng weghted average of sub-blocks. Ths can reduce the mpact of dervng the optmal projectve matrx from the mean devaton n tranng samples, thus mprovng the recognton rate. Algorthm steps are as follows: For convenence, we frst ntroduce the concept of uantzaton matrx. m n Defnton: Suppose A ( A, A, L, A ) R, mn 1 vector s defned as 1 n Vec( A) A1 A A n, (7) Where the vector s arranged n turn by column vector of the matrx A, whch s called the uantfcaton of matrx A. 1) Suppose the model category s C, the mage matrx n the -th class tranng sample s n ( ; ) 1 A, N ( ) A, n C n N s the total number of tranng samples, and each sample mage s m n matrx. The p blocked matrx of tranng sample mage A s expressed as: A ( A ) ( A ) ( ) p1 L A p ) Reure overall scatter matrx of sub-mage matrx n all the tranng sample mages. Let ( ) Vec( A ) η, k 1,, L, p, l 1,,, sub-blocks s: m1 n1 L, then ( ) η R. So the overall scatter matrx of all tranng sample 1 (8) S n ( ) p 1 j 1 k 1 l 1 (( ) ( ) ) ( ) ( ) ( ) C 1 T η η η η. (9) Where 1,,, C L ; j 1,, n ( ) L ; k 1,, L p ; l 1,,, L, n ( ) s the number of each class of tranng samples; C n p Np 1 s the total number of tranng sample sub-mages matrx.. ( ) (10) ( η ) s the weghted average mage between the -th sample mage and the -th block. The specfc calculaton method s to spread all sub-blocks by row to column vector, then calculates ts weghted mean vector by euatons (4) and (5), and reverts the mddle measures to a matrx. 66

4 odern Appled Scence November, 009 Easly, we can prove S s a mn 1 1 mn 1 1 non-negatve defnte matrx. 3) Seek optmal projectve matrx Take correspondng orthonormal egenvectors (dscrmnant vectors) Z1 Z L Z of the r largest egenvalue of S to consttute [ ] Q Z1, Z, L, Zr,,,, r 4) Reure weghted average vector of all tranng sample sub-blocks matrx n order that test samples and tranng samples are comparable, standardze them by the same weghted average matrx. So we must calculate weghted average matrx η of all tranng sub-block samples. 5) Feature extracton of tranng samples. Each block of tranng samples obtan the characterstcs matrx of 6) Feature extracton of test samples Each block of test sample mage A ( A ) ( A ) ( ) p1 L A p A after projectng to Q [ Z Z Z ] x B s uantfed by euaton (7) and normalzed, then 1,, L, r : ( ) ( ) (( ) ) (( ) ) L ( ) (( ) ) (( ) ) L ( ) Q η η Q η 11 η Q η 1 η 1 Q η η Q η 1 η Q η η Q (( η ) η) Q (( η ) ) (( ) p1 η L Q η η p ) 11 1 L 1 1 L p1 L p characterstcs matrx of test samples after projectng to Q [ Z Z Z ] where Vec( ) 7) Sort η, l 1,, L,. ( ) ( ) ( ) Suppose B Y, Y,, Y 1 L ( x) ( x) ( x), Bx Y 1, Y,, Y L B x pr. (11) s uantfed by euaton (7) and normalzed, then obtan the 1,, L, r : ( η11 η) ( η1 η) L ( η1 η) ( η1 η) ( η η) L ( η η) Q Q Q Q Q Q Q ( ηp1 η) Q ( η η) Q ( ηp η) L pr, carryng the most recent method to sort:, (1) 1,, L, C ; j 1,, n ( ) ( ) ( x) (, x ) m m d B B Y Y (13) m 1 L ; x s dentfed the x -th sample under test. f d ( Bnj, B x ) mn (, x ) d B B, the sample x belongs to the -th category. 67

5 odern Appled Scence 4. Experment and result analyss Test the method of ths paper n ORL (olvett research laboratory) face database. Ths face database contans 40 ndvduals, and each person has 10 mages. The mage s a postve mage of sngle dark background that contans a certan amount of llumnaton changes, facal changes (open eyes and closed eyes, laughng or not laughng), facal detals changes(wearng glasses or not wearng glasses), and the depth rotaton wthn a certan range. The szes of these mages are 11 9 pxels. Other part faces are showed n Fgure 1. For each person, randomly selecte fve mages as tranng samples and the rest fve mages are used to test the dentfcaton method performance. The expermental results are shown n Fgure and Fgure 3. Fgure shows expermental result of tradtonal PCA method, modular PCA method and blocked PCA method based on weghted average. From the fgure, we can see that recognton rate of tradtonal PCA method s lower, that s up to 77 %. odule PCA method mproves the recognton rate, whle the blocked PCA method based on weghted average s superor to ordnary blocked PCA method. Fgure 3 respectvely shows the test results of 4 sub-blocks and 4 4 sub-blocks condtons. From the fgures we can see that n the 4 block case, modular PCA method based on weghted average has a hgher recognton rate and a more robust than ordnary blocked PCA method; n addton, test result also shows that 4 blocked approach s superor to blocked approach. n the blocked mode, the correct recognton rate s greatly decreased. The cause s that the more blocks number of each mage s, the more reduced the dstngushed nformaton contaned n each sub-block. So there wll be more smlar sub-blocks and t s not conducve to classfcaton, thus correct dentfcaton rate has dropped. n ths case, modular PCA method based on weghted average s stll better than ordnary blocked PCA method. At the same tme, we fnd n experments the recognton performance of 4 sub-blocks s far better than that of 4 sub-blocks, whch s shown n Table 1. The cause s that the dfference between dfferent people faces focus on eyes, nose, mouth, chn and other parts, so the vertcal mult-block s not conducve to dentfcaton. 5. Concluson The promnent advantage of face recognton method based on modular PCA s the ablty to extract the local features of mage, whch better reflects the dfference between mages. We can easly use dscrmnant analyss method n the smaller mage for the process s smple. To further mprove the recognton rate, ths paper mproves face recognton method based on modular PCA and proposes modular PCA algorthm based on weghted average. The experment on ORL face database shows that ths method s superor to the tradtonal PCA method and ordnary PCA method. For the same database, f the orgnal mage has dfferent sub-blocks, the obtaned hghest recognton rate s generally dfferent. How to fnd the best sub-blocks acured hghest recognton rate and how to smplfy the sub-blocks PCA algorthm have yet to be further studed. References Chen Fubng, Yang Jngyu. (007). odular PCA and ts applcaton n human face recognton. Computer Enneerng and Desgn,8(8): Gan Junyng, L Chunzh. (007). DCA based on wavelet transformaton and applcaton. Journal of System Smulaton,19(3): He Guohu, Gan Junyng. (006). Study for class average face method based on PCA n face recognton. Applcaton Reseach of Computer, 3: J. Lu, K. Platanots, A.Venetsanopoulos.(003). Face Recognton usng LDA-based Algorthms[J]. EEE Trans. Neural Networks, 14 (1): Jan Yang, Davd Zhang. (004). Two-Dmensonal PCA: A New Approach to Appearance-Based Face Representaton and Recognton [J]. EEE Trans. Pattern Analyss and achne ntellgence, 6(1): Keun-Chang Kwak, Wtold Pedrycz. (007). Face Recognton usng an Enhanced ndependent Component Analyss Approach[J]. EEE Trans. Neural Networks, 18(): Krby, Srovch L.(1990). Applcaton of the KL Procedure for the Characterzaton of Human Faces[J].EEE Trans. Pattern Analyss and achne ntellgence, 1(1): Rama Chellappa, et al.(1995). Human and achne Recognton of Faces: A Survey[J]. Proceedngs of the EEE, 83(5): Valentn D, Abd H, OToole A J. (1994). Connectonst odel of Face Processng: A Survey [J]. Pattern Recognton, 7( 9 ): Yn Hongtao, Fu Png, eng Shengwe. (006). Face recognton based on adaptvely weghted Fsherface. Journal of Optoelectroncs. Laser,17(11): Zhang Cupng, Sun Guangda. (005). A survey on face recognton. Journal of mage and Graphcs, 5(11):

6 odern Appled Scence November, 009 Table 1. recognton rate of 4 blocks and 4 blocks of method n ths paper (%) recognton rate number 4 blocks blocks Fgure 1. mage n ORL face database Fgure. Expermental result of sub-blocks 69

7 odern Appled Scence Fgure 3. Experment result of 4 blocks and 4 4 blocks 70