Cˆong th´u . c n`ay tr `ung v´o . id¯i . nh ngh˜ıa (9.1) cu ˙’ a h`am biˆe . ttˆa . p. Trong tru . `o . ng ho . . p n`ay, biˆen gi˜u . a hai l´o . p ω i v`a ω j gˆo ` m c´ac vector mˆa ˜ u x thoa ˙’ m˜an phu . o . ng tr`ınh d ij (x)=d i (x) − d j (x) = x, m i − m j − 1 2 m i −m j , m i − m j =0. (9.3) Phu . o . ng tr`ınh n`ay x´ac d¯i . nh mˆo . t siˆeu phˇa ˙’ ng (v´o . i ph´ap vector m i − m j ) trong khˆong gian Euclid n chiˆe ` u. Trong thu . . ctˆe ´ , phˆan loa . i theo khoa ˙’ ng c´ach nho ˙’ nhˆa ´ tchokˆe ´ t qua ˙’ tˆo ´ t khi khoa ˙’ ng c´ach gi˜u . a c´ac vector trung b`ınh cu ˙’ a c´ac l´o . pl´o . nho . nsov´o . im´u . cd¯ˆo . phˆan t´an hoˇa . c t´ınh ngˆa ˜ u nhiˆen cu ˙’ amˆo ˜ il´o . pd¯ˆo ´ iv´o . i vector trung b`ınh cu ˙’ a n´o. Trong Phˆa ` n 9.3.2 ch´ung ta s˜e chı ˙’ ra phˆan loa . i theo khoa ˙’ ng c´ach nho ˙’ nhˆa ´ ts˜etˆo ´ iu . u(m´u . cd¯ˆo . phˆan loa . i sai ´ıt nhˆa ´ t) khi phˆan bˆo ´ cu ˙’ amˆo ˜ il´o . p xung quanh vector trung b`ınh cu ˙’ a n´o “t´ıch l˜uy” da . ng cˆa ` u trong khˆong gian n chiˆe ` u. Su . . xuˆa ´ thiˆe . nd¯ˆo ` ng th`o . i t´ınh chˆa ´ t t´ach gi˜u . a c´ac vector trung b`ınh v`a phˆan t´an tu . o . ng d¯ˆo ´ i ´ıt cu ˙’ a c´ac l´o . phiˆe ´ m khi xa ˙’ y ra trong thu . . ctˆe ´ tr `u . khi ngu . `o . i thiˆe ´ tkˆe ´ hˆe . thˆo ´ ng d¯ i ˆe ` u khiˆe ˙’ n c´ac t´ın hiˆe . u v`ao. V´ıdu . x´et hˆe . thˆo ´ ng d¯u . o . . c thiˆe ´ tkˆe ´ d¯ ˆe ˙’ d¯ o . c c´ac font k´ytu . . d¯ ˇa . cbiˆe . tnhu . tˆa . pk´ytu . . E-13B cu ˙’ aHiˆe . phˆo . i c´ac Ngˆan h`ang M˜y. Nhu . trong H`ınh 9.3 chı ˙’ ra, font k´y tu . . n`ay gˆo ` m14k´ytu . . d¯ u . o . . c thiˆe ´ tkˆe ´ v´o . imˆa . td¯ˆo . k´ytu . . l`a 9 × 7d¯ˆe ˙’ dˆe ˜ d¯ o . c. C´ac k´ytu . . thu . `o . ng d¯u . o . . cinsu . ˙’ du . ng mu . . c in c´o pha chˆa ´ tliˆe . ut`u . . Khi qu´et trang t`ai liˆe . u, c´ac k´ytu . . n`ay s˜e d¯u . o . . c l`am nˆo ˙’ ibˆa . t. N´oi c´ach kh´ac, b`ai to´an phˆan d¯oa . na ˙’ nh d¯ u . o . . c gia ˙’ i quyˆe ´ t nhˆan ta . obˇa ` ng c´ach l`am nˆo ˙’ ic´ack´ytu . . . C´ac k´y tu . . d¯ u . o . . cqu´et theo hu . ´o . ng ngang v´o . imˆo . td¯ˆa ` ud¯o . che . pnhu . ng d`ai ho . nd¯ˆo . cao cu ˙’ ac´ack´ytu . . . Khi d¯ˆa ` ud¯o . c di chuyˆe ˙’ ndo . c qua mˆo . tk´ytu . . , n´o s˜e ta . o ra mˆo . t t´ın hiˆe . u d¯ i ˆe . ntu . ˙’ 1D, t´u . c l`a h`am mˆo . tbiˆe ´ n f(t). T´ın hiˆe . u n`ay tı ˙’ lˆe . v´o . im´u . cd¯ˆo . nhiˆe ` u hoˇa . c ´ıt cu ˙’ a diˆe . nt´ıchk´ytu . . du . ´o . id¯ˆa ` ud¯o . c. Chˇa ˙’ ng ha . n, x´et d¯ˆo ` thi . da . ng s´ong cu ˙’ a h`am f(t)tu . o . ng ´u . ng v´o . isˆo ´ 0 trong H`ınh 9.3. Khi d¯ˆa ` ud¯o . cdichuyˆe ˙’ nt`u . tr´ai sang pha ˙’ i, diˆe . n t´ıch k´y tu . . du . ´o . id¯ˆa ` ud¯o . cbˇa ´ td¯ˆa ` u tˇang (h`am f c´o d¯a . o h`am du . o . ng trong v`ung n`ay). Khi d¯ˆa ` ud¯o . c di chuyˆe ˙’ n kho ˙’ in´etd¯´u . ng bˆen tr´ai cu ˙’ ak´ytu . . 0 th`ı diˆe . n t´ıch du . ´o . id¯ˆa ` ud¯o . cbˇa ´ td¯ˆa ` u gia ˙’ m (h`am f c´o d¯a . o h`am ˆam trong v`ung n`ay). Trong v`ung gi˜u . acu ˙’ ak´ytu . . ,diˆe . n t´ıch du . ´o . i d¯ ˆa ` ud¯o . c khˆong d¯ˆo ˙’ i (h`am f c´o d¯a . o h`am bˇa ` ng khˆong trong v`ung n`ay). Qu´a tr`ınh n`ay tiˆe ´ ptu . clˇa . pla . i khi d¯ˆa ` ud¯o . c di chuyˆe ˙’ n qua kho ˙’ ik´ytu . . .Viˆe . c thiˆe ´ tkˆe ´ font ch˜u . ba ˙’ od¯a ˙’ m rˇa ` ng d¯ˆo ` thi . da . ng s´ong cu ˙’ a c´ac k´ytu . . l`a ho`an to`an kh´ac nhau. Ngo`ai ra viˆe . c thiˆe ´ tkˆe ´ c˜ung d¯a ˙’ mba ˙’ o c´ac d¯iˆe ˙’ mcu . . c tri . c˜ung nhu . khˆong d¯iˆe ˙’ mcu ˙’ a h`am f nˇa ` m trˆen c´ac d¯u . `o . ng 292 H`ınh 9.3: Tˆa . p font k´y tu . . E-13B cu ˙’ aHiˆe . phˆo . i c´ac Ngˆan h`ang M˜y v`a c´ac da . ng s´ong tu . o . ng ´u . ng. 293 thˇa ˙’ ng d¯´u . ng cu ˙’ alu . ´o . i khi hiˆe ˙’ n thi . d¯ ˆo ` thi . h`am f nhu . trong H`ınh 9.3. Tˆa . pk´ytu . . E-13B d¯ u . o . . c thiˆe ´ tkˆe ´ c´o t´ınh chˆa ´ t khi lˆa ´ ymˆa ˜ u c´ac da . ng s´ong chı ˙’ ta . inh˜u . ng d¯iˆe ˙’ m n`ay c˜ung d¯ u ˙’ thˆong tin d¯ˆe ˙’ phˆan loa . ich´ung. Su . ˙’ du . ng mu . . cinc´ot`u . t´ınh l`am cho da . ng s´ong d¯u . o . . c r˜o r`ang do d¯´o gia ˙’ m thiˆe ˙’ u kha ˙’ nˇang bi . nhiˆe ˜ u. V´o . i´u . ng du . ng n`ay ch´ung ta dˆe ˜ d`ang thiˆe ´ tkˆe ´ bˆo . phˆan loa . i theo khoa ˙’ ng c´ach nho ˙’ nhˆa ´ t. Ch ´ung ta chı ˙’ cˆa ` nlu . utr˜u . tˆa . p c´ac gi´a tri . mˆa ˜ ucu ˙’ amˆo ˜ ida . ng s´ong v`a v´o . imˆo ˜ itˆa . p mˆa ˜ u ta thiˆe ´ tlˆa . ptu . o . ng ´u . ng mˆo . t vector m i ,i=1, 2, ,14. Khi nhˆa . nda . ng mˆo . tk´ytu . . , ta qu´et n´o nhu . d¯˜a mˆo ta ˙’ trˆen, t`u . da . ng s´ong cu ˙’ ak´ytu . . n`ay ta d¯u . o . . c vector mˆa ˜ u x. Du . . a v`ao Phu . o . ng tr`ınh (9.2) dˆe ˜ d`ang x´ac d¯i . nh l´o . ptu . o . ng ´u . ng v´o . i vector x. Su . ˙’ du . ng c´ac ma . ch d¯iˆe . ntu . ˙’ d¯ u . o . . c thiˆe ´ tkˆe ´ chuyˆen du . ng ch´ung ta c´o thˆe ˙’ phˆan loa . iv´o . itˆo ´ cd¯ˆo . cao. D - ˆo ´ i s´anh theo tu . o . ng quan Phˆa ` n n`ay ´ap du . ng kh´ai niˆe . mtu . o . ng quan (xem Phˆa ` n 3.3.8) d¯ˆe ˙’ t`ım c´ac d¯ˆo ´ i s´anh cu ˙’ a mˆa ˜ u(a ˙’ nh con) w(x, y) (k´ıch thu . ´o . c J × K v´o . ia ˙’ nh f( x, y) (k´ıch thu . ´o . c M × N trong d¯ ´o J ≤ M v`a K ≤ N). Nhˇa ´ cla . irˇa ` ng, tu . o . ng quan gi˜u . a f( x, y)v`aw(x, y) x´ac d¯i . nh bo . ˙’ i c(s, t)= x y f(x, y)w(x −s, y −t) (9.4) trong d¯´o s =0, 1, ,M − 1,t =0, 1, ,N − 1, v`a gia ˙’ su . ˙’ tˆo ˙’ ng d¯u . o . . clˆa ´ ytrˆenv`ung a ˙’ nh f v`a w phu ˙’ nhau. H`ınh 9.4 minh ho . a c´ach t´ınh, trong d¯´o gia ˙’ thiˆe ´ tgˆo ´ ccu ˙’ a f(x, y) ta . ivi . tr´ı ph´ıa trˆen bˆen tr´ai cu ˙’ aa ˙’ nh v`a gˆo ´ ccu ˙’ a w(x, y)ta . i tˆam cu ˙’ a n´o. V´o . i(s, t)bˆa ´ t k`y trong a ˙’ nh f(x, y), ´ap du . ng Phu . o . ng tr`ınh (9.4) ta d¯u . o . . c gi´a tri . c(s, t)tu . o . ng ´u . ng. Gi´a tri . c(s, t)chobiˆe ´ tvi . tr´ı m`a mˆa ˜ u w(x, y)d¯ˆo ´ i s´anh tˆo ´ t nhˆa ´ tv´o . i f(x, y). Ch´u´yrˇa ` ng d¯ ˆo . ch´ınh x´ac gia ˙’ m d¯i khi s v`a t tiˆe ´ ngˆa ` nd¯ˆe ´ n c´ac d¯u . `o . ng biˆen cu ˙’ aa ˙’ nh f(x, y). H`am tu . o . ng quan x´ac d¯i . nh theo Phu . o . ng tr`ınh (9.4) c´o nhu . o . . cd¯iˆe ˙’ m l`a nha . yv´o . i su . . thay d¯ˆo ˙’ i biˆen d¯ˆo . cu ˙’ a f(x, y)v`aw(x, y). Chˇa ˙’ ng ha . n, nhˆan hai tˆa ´ tca ˙’ c´ac gi´a tri . cu ˙’ a f(x, y) s˜e tˇang d¯ˆoi c´ac gi´a tri . c(x, y). D - ˆe ˙’ tr´anh tro . ˙’ nga . i n`ay, ngu . `o . i ta thu . `o . ng d¯ˆo ´ i s´anh thˆong qua hˆe . sˆo ´ tu . o . ng quan: γ(s, t)= x y [f(x, y) − ¯ f(x, y)][w(x − s, y − t) − ¯w] x y [f(x, y) − ¯ f(x, y)] 2 x y [w(x − s, y − t) − ¯w] 2 1/2 , (9.5) trong d¯´o s =0, 1, ,M −1,t =0, 1, ,N −1, ¯w l`a gi´a tri . trung b`ınh cu ˙’ a c´ac pixel trong w (chı ˙’ cˆa ` n t´ınh mˆo . tlˆa ` n), ¯ f(x, y) l`a gi´a tri . trung b`ınh cu ˙’ a f(x, y) trong v`ung a ˙’ nh 294 ←−−−−−−−−−−−− N −−−−−−−−−−−−→ ↑ | | | | | | | M | | | | | | | ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ←−− K −−→ ↑ | | J | | ↓ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •s t (s, t) y . . . . . . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . gˆo ´ c f(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w(x − s, y −t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H`ınh 9.4: Sˇa ´ pxˆe ´ pmˆa ˜ uv`aa ˙’ nh d¯ˆe ˙’ t´ınh tu . o . ng quan cu ˙’ a f(x, y)v`aw(x, y)ta . id¯iˆe ˙’ m (s, t). tr `ung v´o . ivi . tr´ı hiˆe . n h`anh cu ˙’ a w v`a tˆo ˙’ ng lˆa ´ y trˆen c´ac to . ad¯ˆo . chung cu ˙’ a f v`a w. Hˆe . sˆo ´ tu . o . ng quan γ(s, t)d¯u . o . . cchuˆa ˙’ n ho´a trong d¯oa . n[−1, 1] v`a khˆong phu . thuˆo . c khi biˆen d¯ ˆo . cu ˙’ a f (hoˇa . c w) thay d¯ˆo ˙’ i theo c`ung mˆo . thˆe . sˆo ´ . D - ˆe ˙’ kˆe ´ t qua ˙’ d¯ ˆo ´ i s´anh khˆong phu . thuˆo . c v`ao a ˙’ nh d¯u . o . . c l`am s´ang hoˇa . c l`am tˆo ´ ita chuˆa ˙’ n ho´a h`am tu . o . ng quan. Tuy nhiˆen c´ach n`ay kh´o thu . . chiˆe . n khi thay d¯ˆo ˙’ ik´ıch thu . ´o . c hoˇa . c quay a ˙’ nh. Chuˆa ˙’ nho´ak´ıch thu . ´o . cliˆen quan d¯ˆe ´ n co gi˜an khˆong gian v`a do d¯´o d¯`oi ho ˙’ i thˆem d¯´ang kˆe ˙’ khˆo ´ ilu . o . . ng t´ınh to´an. Chuˆa ˙’ n ho´a d¯ˆo ´ iv´o . i ph´ep quay thˆa . mch´ı c`on kh´o ho . n. Nˆe ´ u ta biˆe ´ t c´ac thˆong sˆo ´ cu ˙’ a ph´ep quay t`u . a ˙’ nh f( x, y) th`ı chı ˙’ cˆa ` n´apdu . ng ph´ep quay n`ay cho mˆa ˜ u w(x, y). Tuy nhiˆen thu . . ctˆe ´ thu . `o . ng khˆong biˆe ´ tph´ep quay d¯˜a thu . . chiˆe . ntrˆena ˙’ nh f( x, y)nˆen d¯ˆe ˙’ x´ac d¯i . nh n´o cˆa ` n pha ˙’ ithu . ˙’ tˆa ´ tca ˙’ c´ac kha ˙’ nˇang cu ˙’ a w(x, y)d¯ˆe ˙’ t`ım d¯u . o . . cd¯ˆo ´ i s´anh tˆo ´ t nhˆa ´ t. D - iˆe ` u n`ay khˆong thu . . ctˆe ´ v`a do d¯´o ph´ep d¯ˆo ´ i s´anh tu . o . ng quan ´ıt khi d¯u . o . . csu . ˙’ du . ng trong tru . `o . ng ho . . pa ˙’ nh d¯u . o . . c quay tu`y ´y. Trong Phˆa ` n 3.3.8 ch´ung ta d¯˜a d¯ˆe ` cˆa . pd¯ˆe ´ nbiˆe ´ nd¯ˆo ˙’ iFFTd¯ˆe ˙’ t´ınh tu . o . ng quan trong tru . `o . ng ho . . p f( x, y)v`aw(x, y)c´oc`ung k´ıch thu . ´o . c. Nˆe ´ usu . ˙’ du . ng Phu . o . ng tr`ınh (9.4) th`ı w thu . `o . ng c´o k´ıch thu . ´o . c nho ˙’ ho . n nhiˆe ` u so v´o . i k´ıch thu . ´o . ccu ˙’ a f. Campbell d¯˜a chı ˙’ ra rˇa ` ng, nˆe ´ usˆo ´ c´ac phˆa ` ntu . ˙’ cu ˙’ a w nho ˙’ ho . n 132 (a ˙’ nh con c´o k´ıch thu . ´o . cxˆa ´ pxı ˙’ 13 ×13 pixel) th`ı t´ınh to´an tru . . ctiˆe ´ p theo Phu . o . ng tr`ınh (9.4) s˜e hiˆe . u qua ˙’ ho . nphu . o . ng 295 ph´ap FFT. D˜ı nhiˆen k´ıch thu . ´o . ccu ˙’ amˆa ˜ u w phu . thuˆo . c v`ao m´ay v`a c´ac thuˆa . t to´an su . ˙’ du . ng; do vˆa . y thao t´ac trˆen miˆe ` n khˆong gian hoˇa . cmiˆe ` ntˆa ` nsˆo ´ s˜e d¯u . o . . c´apdu . ng tu`y t`u . ng tru . `o . ng ho . . pcu . thˆe ˙’ . C´ac hˆe . sˆo ´ tu . o . ng quan t´ınh theo Phu . o . ng tr`ınh (9.5) s˜e dˆe ˜ d`ang ho . nphu . o . ng ph´ap miˆe ` ntˆa ` nsˆo ´ . 9.3.2 Phu . o . ng ph´ap thˆo ´ ng kˆe Co . so . ˙’ Phˆa ` n n`ay tr`ınh b`ay phu . o . ng ph´ap thˆo ´ ng kˆed¯ˆe ˙’ nhˆa . nda . ng. Thˆo ´ ng kˆe d¯´ong mˆo . t vai tr`o quan tro . ng trong b`ai to´an nhˆa . nda . ng do c´ac l´o . pmˆa ˜ uthu . `o . ng d¯u . o . . cta . o ra ngˆa ˜ u nhiˆen. K´yhiˆe . u p(ω i |x) l`a x´ac suˆa ´ tmˆa ˜ u x thuˆo . cl´o . p ω i v`a lˆo ˜ i khi phˆan loa . i nhˆa ` mmˆa ˜ u x ∈ ω i v`ao l´o . p ω j l`a L ij . V`ımˆa ˜ u x c´o thˆe ˙’ thuˆo . c v`ao mˆo . t trong M l´o . pnˆenlˆo ˜ i trung b`ınh khi g´an x v`ao l´o . p ω j l`a r j (x)= M k=1 L kj p(ω k |x). (9.6) Trong l´y thuyˆe ´ t quyˆe ´ td¯i . nh, Phu . o . ng tr`ınh (9.6) thu . `o . ng go . il`ad¯ ˆo . ru ˙’ i ro (tˆo ˙’ n thˆa ´ thay mˆa ´ t m´at) trung b`ınh c´o d¯iˆe ` ukiˆe . n. Theo l´y thuyˆe ´ t x´ac suˆa ´ tth`ıp(a|b)=[p(a)p(b|a)]/p(b). Do d¯´o Phu . o . ng tr`ınh (9.6) c´o thˆe ˙’ viˆe ´ tla . ida . ng r j (x)= 1 p(x) M k=1 L kj p(x|ω k )P (ω k ), (9.7) trong d¯´o p(x|ω k ) l`a h`am mˆa . td¯ˆo . x´ac suˆa ´ tcu ˙’ a c´ac mˆa ˜ u thuˆo . cl´o . p ω k v`a P (ω k ) l`a x´ac suˆa ´ t xuˆa ´ thiˆe . nl´o . p ω k . Do mˆa ˜ usˆo ´ p(x)du . o . ng v`a chung cho tˆa ´ tca ˙’ c´ac h`am tˆo ˙’ n thˆa ´ t r j (x),j =1, 2, ,M, nˆen ta c´o thˆe ˙’ bo ˙’ d¯i trong Phu . o . ng tr`ınh (9.7) m`a khˆong a ˙’ nh hu . o . ˙’ ng khi so s´anh th´u . tu . . cu ˙’ a c´ac h`am n`ay. Khi d¯´o ta c´o thˆe ˙’ viˆe ´ t r j (x)= M k=1 L kj p(x|ω k )P (ω k ). (9.8) V´o . imˆa ˜ u x chu . abiˆe ´ t, ta cˆa ` n ta cˆa ` n t`ım l´o . p ω i trong M l´o . pd¯ˆe ˙’ xˆe ´ p x ∈ ω i . Tru . ´o . c hˆe ´ t x´ac d¯i . nh r j (x),j =1, 2, ,M, v`a gia ˙’ su . ˙’ r i (x) = min{r j (x),j =1, 2, ,M}. 296 Khi d¯´o ta phˆan loa . imˆa ˜ u x thuˆo . cl´o . p ω i . N´oi c´ach kh´ac ta cho . n sao cho m´u . cd¯ˆo . tˆo ˙’ n thˆa ´ t trung b`ınh theo tˆa ´ tca ˙’ c´ac quyˆe ´ td¯i . nh l`a nho ˙’ nhˆa ´ t. Phu . o . ng ph´ap phˆan loa . i sao cho cu . . ctiˆe ˙’ u ho´a m´u . cd¯ˆo . tˆo ˙’ n thˆa ´ t trung b`ınh go . il`aphˆan loa . i Bayes. Trong nhiˆe ` u b`ai to´an nhˆa . nda . ng, m ´u . cd¯ˆo . tˆo ˙’ n thˆa ´ t khi quyˆe ´ td¯i . nh d¯´ung bˇa ` ng khˆong v`a c´o gi´a tri . hˇa ` ng sˆo ´ kh´ac khˆong (chˇa ˙’ ng ha . n 1) khi quyˆe ´ td¯i . nh sai. V´o . i gia ˙’ thiˆe ´ t n`ay ta c´o L ij =1−δ ij , (9.9) trong d¯´o δ ij = 1nˆe ´ u i = j, 0nˆe ´ u ngu . o . . cla . i. Phu . o . ng tr`ınh (9.9) chı ˙’ ra m´u . cd¯ˆo . tˆo ˙’ n thˆa ´ tbˇa ` ng 1 khi quyˆe ´ td¯i . nh sai v`a khˆong tˆo ˙’ n thˆa ´ t khi quyˆe ´ td¯i . nh d¯´ung. Thay L ij trong Phu . o . ng tr`ınh (9.9) v`ao Phu . o . ng tr`ınh (9.8) ta d¯u . o . . c r j (x)= M k=1 (1 − δ kj )p(x|ω k )P (ω k ) = p(x) −p(x|ω j )P (ω j ). (9.10) Suy ra phˆan loa . i Bayes g´an mˆa ˜ u x thuˆo . cl´o . p ω i nˆe ´ u p(x) −p(x|ω i )P (ω i ) <p(x) − p(x|ω j )P (ω j ), hay tu . o . ng d¯u . o . ng p(x|ω i )P (ω i ) >p(x|ω j )P (ω j ). Dˆe ˜ thˆa ´ yrˇa ` ng trong tru . `o . ng ho . . p h`am tˆo ˙’ n thˆa ´ t L ij =1− δ ij phˆan loa . i Bayes su . ˙’ du . ng h`am biˆe . ttˆa . p d j (x)=p(x|ω j )P (ω j ),j=1, 2, ,M. (9.11) H`am biˆe . ttˆa . p cho trong Phu . o . ng tr`ınh (9.11) l`a tˆo ´ iu . u theo ngh˜ıa cu . . ctiˆe ˙’ u ho´a tˆo ˙’ n thˆa ´ t trung b`ınh khi phˆan loa . i sai. D - ˆe ˙’ x´ac d¯i . nh c´ac h`am n`ay ch´ung ta cˆa ` nbiˆe ´ t c´ac h`am mˆa . td¯ˆo . x´ac suˆa ´ tcu ˙’ a c´ac mˆa ˜ u trong mˆo ˜ il´o . p v`a x´ac suˆa ´ t xuˆa ´ thiˆe . ncu ˙’ amˆo ˜ il´o . p. Yˆeu cˆa ` u sau dˆe ˜ d`ang thoa ˙’ m˜an. Chˇa ˙’ ng ha . n, nˆe ´ utˆa ´ tca ˙’ c´ac l´o . p xuˆa ´ thiˆe . nv´o . i x´ac suˆa ´ t bˇa ` ng nhau th`ı P (ω j )=1/M. Thˆa . m ch´ı nˆe ´ u gia ˙’ thiˆe ´ t n`ay khˆong d¯´ung, c´ac x´ac suˆa ´ t n`ay thu . `o . ng c´o thˆe ˙’ suy t`u . c´ac gia ˙’ thiˆe ´ t (tri th ´u . c) cu ˙’ a b`ai to´an d¯ˇa . t ra. Kh´o khˇan ch´ınh l`a x´ac d¯i . nh c´ac h`am mˆa . td¯ˆo . x´ac suˆa ´ t p(x|ω j ). Nˆe ´ u c´ac vector mˆa ˜ u x thuˆo . c khˆong gian n chiˆe ` uv`ap(x|ω j ) l`a h`am n biˆe ´ n chu . abiˆe ´ t th`ı cˆa ` n pha ˙’ isu . ˙’ du . ng phu . o . ng ph´ap trong l´y thuyˆe ´ t x´ac suˆa ´ td¯ˆe ˙’ xˆa ´ pxı ˙’ n´o. C´ac phu . o . ng ph´ap n`ay kh´o ´ap du . ng trong thu . . ctˆe ´ , 297 d¯ ˇa . cbiˆe . t trong tru . `o . ng ho . . pnˆe ´ usˆo ´ c´ac mˆa ˜ ubiˆe ˙’ udiˆe ˜ n trong mˆo ˜ il´o . p khˆong nhiˆe ` u hoˇa . c khi h`ınh da . ng cu ˙’ a c´ac h`am mˆa . td¯ˆo . x´ac suˆa ´ tchu . abiˆe ´ t. V`ı l´y do n`ay, phˆan loa . i Bayes thu . `o . ng du . . a trˆen gia ˙’ thiˆe ´ t cho tru . ´o . cmˆo . tbiˆe ˙’ uth´u . c gia ˙’ i t´ıch d¯ˆo ´ iv´o . i c´ac h`am mˆa . td¯ˆo . x´ac suˆa ´ t v`a sau d¯´o x´ac d¯i . nh c´ac tham sˆo ´ cu ˙’ abiˆe ˙’ uth´u . ct`u . c´ac vector mˆa ˜ ucu ˙’ amˆo ˜ i l´o . p. Da . ng phˆo ˙’ biˆe ´ n nhˆa ´ td¯ˆo ´ iv´o . i p(x|ω j ) l`a h`am mˆa . td¯ˆo . x´ac suˆa ´ t Gauss. Khi gia ˙’ thiˆe ´ t n`ay c`ang gˆa ` nv´o . i thu . . ctˆe ´ th`ı phu . o . ng ph´ap nhˆa . nda . ng theo Bayes s˜e c`ang th`anh cˆong ho . n(m´u . cd¯ˆo . sai lˆa ` m trung b`ınh trong phˆan loa . i ´ıt nhˆa ´ t). Phˆan loa . i Bayes trong tru . `o . ng ho . . p h`am mˆa . td¯ˆo . x´ac suˆa ´ t Gauss Tru . ´o . chˆe ´ t x´et tru . `o . ng ho . . pmˆo . tchiˆe ` u(n = 1) v`a hai l´o . pmˆa ˜ u(M = 2) chi . ua ˙’ nh hu . o . ˙’ ng cu ˙’ a c´ac h`am mˆa . td¯ˆo . Gauss v´o . i c´ac gi´a tri . trung b`ınh m 1 v`a m 2 v`a c´ac phu . o . ng sai σ 1 ,σ 2 tu . o . ng ´u . ng. T`u . Phu . o . ng tr`ınh (9.11) c´ac h`am biˆe . ttˆa . p Bayes c´o da . ng d j (x)=p(x|ω j )P (ω j ), = 1 √ 2πσ j exp − (x − m j ) 2 2σ 2 j P (ω j ),j=1, 2, (9.12) v´o . imˆa ˜ u trong tru . `o . ng ho . . p n`ay l`a d¯a . ilu . o . . ng vˆo hu . ´o . ng v`a k´y hiˆe . ubo . ˙’ i x. H`ınh 9.5 l`a d¯ ˆo ` thi . cu ˙’ a c´ac h`am mˆa . td¯ˆo . x´ac suˆa ´ tcu ˙’ a hai l´o . p. Biˆen gi˜u . a hai l´o . pgˆo ` mmˆo . td¯iˆe ˙’ m x 0 x´ac d¯i . nh bo . ˙’ i d 1 (x 0 )=d 2 (x 0 ). Nˆe ´ u x´ac suˆa ´ t xuˆa ´ thiˆe . ncu ˙’ a hai l´o . pbˇa ` ng nhau th`ı P (ω 1 )=P (ω 2 )= 1 2 v`a biˆen gi˜u . a hai l´o . p l`a gi´a tri . x 0 tho ˙’ a p(x 0 |ω 1 )=p(x 0 |ω 2 ). D - iˆe ˙’ m n`ay l`a giao d¯iˆe ˙’ md¯ˆo ` thi . cu ˙’ a hai h`am mˆa . td¯ˆo . x´ac suˆa ´ t (xem H`ınh 9.5). C´ac mˆa ˜ u (d¯iˆe ˙’ m) bˆen pha ˙’ i x 0 d¯ u . o . . c g´an thuˆo . cl´o . p ω 1 v`a bˆen tr´ai d¯iˆe ˙’ m x 0 d¯ u . o . . c g´an thuˆo . cl´o . p ω 2 . Khi c´ac l´o . p xuˆa ´ thiˆe . nv´o . i x´ac suˆa ´ t kh´ac nhau th`ı x 0 di chuyˆe ˙’ n sang bˆen tr´ai nˆe ´ u P (ω 1 ) >P(ω 2 ) v`a x 0 di chuyˆe ˙’ n sang bˆen pha ˙’ inˆe ´ u P(ω 1 ) <P(ω 2 ). Kˆe ´ t qua ˙’ n`ay ph`uho . . pv´o . i thu . . ctˆe ´ v`ıviˆe . c phˆan loa . icˆa ` ncu . . ctiˆe ˙’ um´u . cd¯ˆo . phˆa ` n loa . i sai. Chˇa ˙’ ng ha . n, trong tru . `o . ng ho . . pd¯ˇa . c biˆe . t, nˆe ´ ul´o . p ω 2 khˆong bao gi`o . xuˆa ´ thiˆe . n th`ı phˆan loa . id¯´ung cˆa ` n g´an c´ac l´o . pmˆa ˜ ucho l´o . p ω 1 (t ´u . cl`ax 0 di chuyˆe ˙’ nra−∞). Trong tru . `o . ng ho . . p n chiˆe ` u, h`am mˆa . td¯ˆo . Gauss cu ˙’ a vector thuˆo . cl´o . pmˆa ˜ uth´u . j c´o da . ng p(x|ω j )= 1 (2π) n/2 det C j exp − 1 2 (x − m j ) t C −1 j (x − m j ) , (9.13) trong d¯´o vector trung b`ınh m j v`a ma trˆa . nhiˆe . pphu . o . ng sai C j x´ac d¯i . nh bo . ˙’ i m j = E j {x}, (9.14) v`a C j = E j {(x − m j )(x − m j ) t }, (9.15) 298 . biˆe . ttˆa . p Bayes c´o da . ng d j (x)=p(x|ω j )P (ω j ), = 1 √ 2 σ j exp − (x − m j ) 2 2σ 2 j P (ω j ),j=1, 2, (9. 12) v´o . imˆa ˜ u trong tru . `o . ng ho . . p n`ay l`a d¯a . ilu . o . . ng. nˇa ` m trˆen c´ac d¯u . `o . ng 29 2 H`ınh 9.3: Tˆa . p font k´y tu . . E-13B cu ˙’ aHiˆe . phˆo . i c´ac Ngˆan h`ang M˜y v`a c´ac da . ng s´ong tu . o . ng ´u . ng. 29 3 thˇa ˙’ ng d¯´u . ng cu ˙’ alu . ´o . i. d 1 (x 0 )=d 2 (x 0 ). Nˆe ´ u x´ac suˆa ´ t xuˆa ´ thiˆe . ncu ˙’ a hai l´o . pbˇa ` ng nhau th`ı P (ω 1 )=P (ω 2 )= 1 2 v`a biˆen gi˜u . a hai l´o . p l`a gi´a tri . x 0 tho ˙’ a p(x 0 |ω 1 )=p(x 0 |ω 2 ).