EIGENFACESCrystal Pepper and Chris WilsonNov. 13, 2009Table of ContentsIEigenfacesIFacesIMean FaceIDifferenceICovariance MatrixIEigenvalues and EigenvectorsIBasisIFace RecognitionIProjectionIRepresented DifferenceIError MinimumIFace RecognizedIBibliography=⇒ Ii=a11a12. . . a1Ma21a22. . . a2M............aN1aN2. . . aNM=⇒ Γi=a11...aN1...a12...aN2...a1M...aNMγ1,1γ1,2. . . γ1,20γ2,1γ2,2. . . γ2,20............γNM, 1γNM,2. . . γNM, 20=⇒ S = [Γ1, Γ2, · · · , Γ20]Ψ =12020Xi=1Γi=⇒120[Γ1+ Γ2+ · · · + Γ20]Φi= Γi− ΨCommon elements removed.φ1,1φ1,2. . . φ1,20φ2,1φ2,2. . . φ2,20............φNM, 1φNM, 2. . . φNM, 20=⇒ [Φ1, Φ2, . . . , Φ20] = AC =12020Xn=1ΦnΦTnC =120AATA dim (19200 by 20)ATdim (20 by 19200)A = UΣVTAA = UΣVTUΣVTAAT= UΣVTV ΣTUTAAT= UΣ2UTA = UΣVTAA = UΣVTUΣVTATA = V ΣTUTUΣVTATA = V Σ2VTAATand ATA have the same non-zero eigenvalues σ and theireigenvectors are related as follows:Avi= σiui.Φi=KXj=1wjuj,wj= uTjΦiΩi=wi1wi2...wiK, i = 1, 2, · · · , MˆΦ =KXi=1wiui,wi= uTiΦiΩ =w1w2...wKer= minlkΩ − ΩlkTurk, M., Pentland, A. (1991) Eigenfaces for
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