Face Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction

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1 F Dtto Roto Lr Alr F Roto C Y I Ursty O solto: tto o l trs s s ys os ot. Dlt to t to ltpl ws. F Roto Aotr ppro: ort y rry s tor o so E > pot stol sp A st o s t ps to ollto o pots ts sp. F Roto Is o s slr orll orto wll ot roly strt ts sp. Pttrs t r to sol sp. y sr y rltly low sol ssp. Dso rto Ftr slto: ro so sos so tt w work o ssp.

2 Vtors Vtor Ato Strto ltplto y Slr ε ε Vtor rspos Lr Cotos β α β α β α β α Vtor Ir Prot y y σ σ L or / / tr Ato strto ltplto y Slr tr rspos tr: ol tors j j j A A ] [ ] [ ] [ ;

3 Lr Ip wo tors r ot pt ty l lo t s l. ; y tr ltplto ltply tr y tor s spl s o tr ltplto wr y A s wrtt s: Altrtly w s t trsorto s lr oto o t ols o A y y k kj j Coort Systs tors spl trprtto s oort syst or ss or ltsol sp. For pl t trtol ss tr sos s ss s ortool s or ll j s tt j y j y y y ss tor llow y to wrtt s Otr Bss A o-ortool ss wol lso work. A wol stll llow y to rprst lto t ots wol o ors rt. Howr t tr A wol ot work s tr s o wy o rprst t tr opot. Lr Ip Cosr two lrly pt tors tr tor w ot prss s lr oto o t t st {w} s lrly pt. Lr Ip o rprst -sol tors t ss st sp t sp. A rl oto or ts s tt t ols o A st lrly pt. Forlly ts s tt t oly wy yo ol wrt wol t s tt

4 4 Lr Ip r o lrly pt tors s ll t rk o t tr. W t rk r s lss t t so t tors r s to sp r-sol ssp. C o Bss C o ss or oort trsor: I w t tt s rlt to so ss w r r to r-p tt t to w ss Hr A s or w ss. W lwys ort k to t orl ss : A * * A Etors For y tr W tr r spl tors s tt: s rsl y ostt. rto o s ot. tors r kow s tors t ssot slrs r kow s ls. W Epl 4 F Etors lr qto A oly s solto o-trl t ols o A r lrly pt. ols o A r lrly pt t trt o A s ql to zro A. Rr: t trt o tr s y A - A F Etors For two-sol s: Or For ts qto to solto t ols o t tr st lrly pt ts W. s

5 F Etors Ssttt 4 to t qto rslts Oly o sl qto two kows. Pk V. V. s t tor ssot wt 4 s Slr trs Sppos tt oort trsorto s y * A y* Ay For y tr W s tt y W r s orrspo tr W* s tt y* W * * Wt s t rlto tw W W*? G W A ow w W*? W W* r slr A * * y W y* Ay y* AWA W AWA * Dolzto Lt s oos t tors o W s t ss st. For tor Wy y I Y s tr wos ols r t tors t WY YΛ Hr Λ s tr wos oly ozro opots r t ol lts. Y WY Λ Cor Vr or slr-l ro rl X X X X X r X Cor s sr o ow t sos ry ro t wt rspt to otr. Y Y X X o X Y Cor Cor t? Zro? oxy oyx? How ot > sos? 5

6 Cor tr o X X Σ o X X o X X o X X o X X o X X K o X X K o X X O K o X X. rk A. Ptl 99. ``Es or roto''. Jorl o Cot ros. D H. Bllr 999. ``A Itroto to trl Coptto Copl Apt Systs'' Cptr 4 pp 7-94 I Prss. F Roto Es Cosr o to pot ry sol sp W kow opt ts st to ll o t st pots ts o kow s. H so s. Dst tr Sprs PCA tors o t or tr orrspo to t rtos o r t t orrspo ls t t ot o r. ror w trsor t w sp y t rtos o rto l lo t s. Stp : Rprst s s tors 6

7 7 Stp : Rprst s s tors Stp : opt t Stp : strt t ro Stp : Bl tr ] [ A Cor tr AA F t tors or tr s too lr t opttol ort s too. rk E ppr Ist o t tors o t lrr syst osr t tors o t syst ltply ot ss y A A A AA A A µ µ

8 F t tors AA A µ A I s tor o tor o AA. A A t A s ls o t sllr syst r t s s tos o t lrr syst. It trs ot tt ts r t lrst ls. Es Es w oort syst. W projt ll t s to t sp. U A Ω U ; Ω U ; Ω U ; o t tors o t lrr syst rst t tors o t sllr syst t ltply t tors y A. F Dtto Roto G w r r r Ω U r r r r r Wtr t s wtr t s kow Coos t Dso K ls How y s to s? K Look t t y o t ls t l tlls yo t ot o r t rto o tt or s wt low r 8

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