Unified Subspace Analysis for Face Recognition

Transcription

1 Unfed Subspace Analyss for Face Recognton Xaogang Wang and Xaoou Tang Department of Informaton Engneerng The Chnese Unversty of Hong Kong Shatn, Hong Kong {xgwang,

2 Abstract PCA, LDA and Bayesan analyss are three of the most representatve subspace based face recognton approaches. We show that they can be unfed under the same framework. Startng from the framework, a unfed subspace analyss s developed usng PCA, Bayes, and LDA as three steps. It acheves better performance than the standard subspace methods.

3 Notaton Face data vector length: Tranng face mages: X = [ x f, ] Tranng sample number: Face classes: { X,, } X L Face classes number: L Class label: ( x ) N, M x M 3

4 Two Knds of Varaton Extrapersonal varaton Ω E Intrapersonal varaton ΩI 4

5 Face Dfference Model The dfference between two face mages can be decomposed nto three components. ~ I : ~ T : ~ N : = Ω Intrnsc dfference dscrmnatng face dentty Transformaton dfference arsng from all knds of transformatons, such as lghtng, expresson, changes etc. Nose I + Ω E ~ = I ~ + T ~ + N Intrapersonal varaton: Extrapersonal varaton: Ω Ω I E ~ ~ = T + N ~ ~ = I + T + ~ N Deteroratng recognton 5

6 Dagram of the Unfed Framework for Subspace Based Face Recognton Probe Face T P Subspace V D? Intrapersonal varaton Extrapersonal varaton Reference PCA subspace Class Class L Intrapersonal subspace (Bayes) LDA subspace Gallery database 6

7 Prncpal Component Analyss (PCA) PCA subspace W s computed from the egenvectors of covarance matrx of tranng set { X } C = M ( x m)( x m) = T m = M x = Theorem : The PCA subspace characterzes the dfference between any two face mages {( x x j )}, whch may belong to the same ndvdual or dfferent ndvduals M T C = ( x M M m)( x m) ( = )( ) T x x j x x j = M = j= M 7

8 Prncpal Component Analyss (PCA) PCA subspace s not deal for face recognton. In PCA subspace, both I ~ and T ~ as structured sgnals, concentratng on the small number of prncpal egenvectors. By selectng the prncpal components, most of the nose encoded on the large number of tralng egenvectors s removed. But I ~ and T ~ are stll coupled. N ~ Prncpal subspace ~ ~ T & I PCA subspace Complementary subspace Egenvectors PCA subspace drectly computed on the set {}, whch contans both ntrapersonal dfference and extrapersonal dfference. ~ ~ = I + T + ~ N 8

9 Bayesan Face Recognton The smlarty between two face mages s based on the ntrapersonal lkehood P( Ω I ) Apply PCA on the ntrapersonal dfference set { Ω I }. The mage space s decomposed to prncpal ntrapersonal subspace F and ts complementary subspace F. K = y DIFS: d F ( ) = λ DFFS: ε ( ) = y s the projecton weghts of on the ntrapersonal egenvectors, and λ s the ntrapersonal egenvalue K = y F DIFS DFFS F 9

10 Bayesan Face Recognton P( Ω I ) s computed as P ( Ω ) I = ( π ) N / exp d ( ) / F K ( N K )/ K λ ( πρ) = = λ / ( ) ε exp ρ ρ s the average egenvalue n the complementary subspace F All the parameters are fxed n recognton procedure. It s equvalent to evaluatng the dstance d K ( ) = + ε ( ) / ρ = y λ 0

11 Intrapersonal Subspace The ntrapersonal subspace s computed from PCA on the ~ ~ ntrapersonal dfference set { Ω I = T + N}. So the axes are arranged accordng to the energy dstrbuton of T ~. Most energy of the T ~ component wll concentrate on the frst few largest egenvectors, whle the I ~ & N ~ components are randomly dstrbuted over the egenvectors. K The Mahalanobs dstance y / λ n the prncpal subspace = weghts the feature vectors by the nverse of egenvalues, so t effectvely reduces the component. T ~ T ~ The complementary subspace throws away most of the component whle keep the majorty of I ~, so ε ( ) s also dstnctve for recognton.

12 Intrapersonal Subspace Intrapersonal subspace N ~ Prncpal subspace Complementary subspace T ~ I ~ Egenvectors F DFFS DIFS F Intrapersonal subspace s computed from the egenvectors of C I = ( x x )( x x ) ( x ) = ( x j ) ~ Ω = T + I j ~ N j T d K ( ) = + ε ( ) / ρ = y λ

13 Lnear Dscrmnant Analyss LDA seeks for the subspace best dscrmnatng dfferent classes. The projecton vectors W maxmze the rato between the between-class scatter matrx S b and wthnclass scatter matrx S w W can be computed from the egenvectors of S w S b In face recognton, the tranng sample number s small (M<<N). The rank of Sw s at most M-L. So Sw, the N by N matrx may become sngular. Usually, the dmensonalty of face data s frst reduced to M-C usng PCA, and then apply LDA n the reduced PCA subspace. 3

14 4 LDA Subspace Theorem : The wthn-class scatter matrx s dentcal to the covarance C I of ntrapersonal subspace n Bayes, whch characterzes the dstrbuton of face varaton for the same ndvduals. Usng the mean face mage to descrbe each ndvdual class, the between class scatter matrx characterzes the varaton between any two mean face mages. ( )( ) = = L X x T k k W k m x m x S ( )( ) = = L X x x T k k k k k k x x x x n, = C I ( )( ) = = L T b m m m m S ( )( ) = = = L L j T j j m m m m M

15 LDA Subspace Computng LDA subspace can be dvded nto three steps. PCA and Bayes can be vewed as the ntermedate steps of LDA. PCA subspace sgnfcantly reduces the nose N ~ and data dmenson. Compute the ntrapersonal subspace from the wthn-class matrx and whten the projecton data by dvdng ntrapersonal egenvalues, such that the transformaton dfference T ~ s sgnfcantly reduced. PCA s agan appled on the whtened class centers. It further reduces the nose and concentrates the energy of ntrnsc dfference onto a small number of features. I ~ 5

16 LDA Subspace Data PCA Whten PCA Subspace PCA on class centers Intrapersonal Subspace LDA Subspace PCA Bayes(ML) LDA N ~ Prncpal subspace ~ ~ T & I Complementary subspace Egenvectors PCA subspace N ~ Prncpal subspace Complementary subspace T ~ I ~ Egenvectors Intrapersonal subspace I ~ T ~ N ~ Prncpal subspace I ~ Complementary subspace Egenvectors LDA subspace Energy dstrbuton of the three components, and on egenvectors n the PCA subspace, the ntrapersonal subspace, and the LDA subspace. 6 N ~ T ~

17 Compare Dfferent Subspaces Behavor of the subspaces on characterzng the face dfference Algorthm PCA Bayes LDA Subspace PCA subspace Intrapersonal subspace LDA subspace Decompose face mage dfference Prncpal subspace Complementary subspace T ~ + I ~ N ~ T ~ ~ ~ I + N I ~ ~ ~ T + N The subspace dmenson of each method can affect the recognton performance. Conventonal LDA fals to attan the best performance wthout sgnfcant changes n each ndvdual step. It s drectly computed from the egenvectors of S w S b. In fact, t fxes the PCA and ntrapersonal subspace as M-L dmenson, and LDA subspace at L- dmenson. 7

18 Unfed Subspace Analyss D DIFS E DFFS F PCA A K LDA B L dp: PCA subspace dmenson d: Intrapersonal subspace dmenson dl dp C H G dl: LDA subspace dmenson O d N M 3D parameter space 8

19 Unfed Subspace Analyss. Project the face data to PCA subspace and adjust the PCA dmenson (dp) to reduce the nose.. Apply Bayesan analyss n the PCA subspace and adjust the dmenson (d) of ntrapersonal subspace. The PCA subspace and ntrapersonal subspace may be computed from an enlarged tranng set contanng the extra samples not n the classes to be recognzed. 3. Compute the class centers of the L ndvduals n the gallery, and project them to the ntrapersonal subspace, whtened by the ntrapersonal egenvalues. 4. Apply PCA on the whtened L class centers to compute the dscrmnant feature vector of dmenson (dl) 9

20 Unfed Subspace Analyss Advantages It provdes a new 3D parameter space to mprove the recognton performance. The optmal parameters can be found n the full 3D space, whle orgnal PCA, LDA and Bayes only occupy some local areas n ths 3D parameter space It adopts dfferent tranng data at dfferent tranng steps accordng to the specal requrement of each step. For the ntrapersonal subspace estmaton (step), we use a enlarged tranng set that contans ndvduals both nsde and outsde the gallery to effectvely estmate T ~. Then for the dscrmnant analyss step (step4), we only use the ndvduals n the gallery, so that the features extracted are specfcally tuned for the ndvduals n the gallery. 0

21 Experments Data set from FERET face database There are two face mages (FA/FB) for each ndvdual 990 face mages of 495 people for tranng Another 700 people for testng 700 face mages n gallery as reference 700 face mages for probe Examples of FA/FB par Normalzed face mage

22 Experments PCA Bayes Recognton accuracy Recognton accuracy Drect correlaton PCA (Eucld) Number of egenvectors Drect correlaton ML DIFS Number of egenvectors K DIFS: d( ) = = y λ DFFS: ε ( ) = K = y ML: DIFS+DFFS

23 Experments Bayesan analyss n the reduced PCA space Bayes on raw face data Maxmum pont Maxmum pont Bayes on raw face data PCA benchmark 800 dp Low accuracy regon Accuracy curves for Bayesan analyss n PCA subspace D Dp PCA be nchmark Hghest accuracy of Bayes analyss n each PCA subspace D 3

24 Experments Extract dscrmnant features from ntrapersonal subspace Recognton accuracy Standard LDA dp=900, d=495 dp=900, d=300 dp=50, d=50 dp=50, d=50 (unfed subspace analyss) Number of dscrmnate features (dl) Accuraces usng dfferent number of dscrmnant features extracted from ntrapersonal subspace Unfed subspace analyss (dp=50, d=50) PCA+Bayes(dp=50) Bayes ML (DIFS) Bayes MAP (DIFS) PCA Feature number Recognton accuraces usng small feature number for each step of the framework. 4