Xử lý ảnh số - Các phép biến đổi part 5 pot

gi˜u . a hai biˆe ´ nd¯ˆo ˙’ i n`ay: c´o thˆe ˙’ thu . . chiˆe . nbiˆe ´ nd¯ˆo ˙’ i Walsh d¯ˆo ´ iv´o . i N ∈ N t`uy ´y trong khi d¯´o chı ˙’ c´o thˆe ˙’ thu . . chiˆe . nbiˆe ´ nd¯ˆo ˙’ i Hadamard d¯ˆo ´ iv´o . i N ≤ 200. Hˆa ` uhˆe ´ t c´ac ´u . ng du . ng biˆe ´ nd¯ˆo ˙’ i trong xu . ˙’ l´y a ˙’ nh v´o . i N =2 n , nˆen c´ac biˆe ´ nd¯ˆo ˙’ i Walsh v`a Hadamard thu . `o . ng d¯ u . o . . ckˆe ´ tho . . pv´o . i nhau trong xu . ˙’ l´y, nˆen go . i chung l`a biˆe ´ nd¯ˆo ˙’ i Walsh-Hadamard. 3.5.3 Biˆe ´ nd¯ˆo ˙’ i cosin r`o . ira . c Cˇa . pbiˆe ´ nd¯ˆo ˙’ i cosin r`o . ira . cmˆo . tchiˆe ` u thuˆa . n ngu . o . . c cho bo . ˙’ i C(u)= 2 u N N−1  x=0 f(x) cos  πu(2x +1) 2N  ,u=0, 1, ,N − 1, v`a f(x)= N−1  u=0  u c(u) cos  πu(2x +1) 2N  ,x=0, 1, ,N − 1, trong d¯´o  u :=    1 √ 2 nˆe ´ u u =0, 1nˆe ´ u ngu . o . . cla . i. Thuˆa . t to´an hiˆe . u qua ˙’ nhˆa ´ td¯ˆe ˙’ thu . . chiˆe . nbiˆe ´ nd¯ˆo ˙’ i cosin nhanh (FCT) cu ˙’ a Chan v`a Ho (xem [17]). Cˇa . pbiˆe ´ nd¯ˆo ˙’ i cosin r`o . ira . c hai chiˆe ` u thuˆa . n ngu . o . . c x´ac d¯i . nh bo . ˙’ i C(u, v)= 4 u  v N 2 N−1  x=0 N−1  y=0 f(x, y) cos  πu(2x +1) 2N  cos  πv(2y +1) 2N  , v´o . i u =0, 1, ,N − 1,v =0, 1, ,N −1, v`a f(x, y)= N−1  u=0 N−1  v=0  u  v C(u, v) cos  πu(2x +1) 2N  cos  πv(2y +1) 2N  , v´o . i x =0, 1, ,N − 1,y =0, 1, ,N −1. Trong Chu . o . ng 6 ch´ung ta s˜e su . ˙’ du . ng ph´ep biˆe ´ nd¯ˆo ˙’ i cosin r`o . ira . c hai chiˆe ` u trong n´en a ˙’ nh khˆong ba ˙’ o to`an thˆong tin. 3.5.4 Biˆe ´ nd¯ˆo ˙’ i Haar Biˆe ´ nd¯ˆo ˙’ i Haar ´ıt d¯u . o . . csu . ˙’ du . ng trong thu . . ctˆe ´ . N´o du . . a trˆen co . so . ˙’ c´ac h`am Haar h k (z),k =0, 1, ,N−1(N =2 n ), x´ac d¯i . nh v`a liˆen tu . c trˆen d¯oa . n[0, 1]. D - ˆa ` u tiˆen, v´o . i 63 mˆo ˜ i k =0, 1, ,N −1, ta x´ac d¯i . nh p, q sao cho k =2 p + q − 1, trong d¯´o 0 ≤ p ≤ n −1, v`a q = 0 hoˇa . c1nˆe ´ u p =0, v`a 1 ≤ q ≤ 2 p nˆe ´ u p>0. V´ı du . 3.5.2 V´o . i N =4, c´ac gi´a tri . p, q d¯ u . o . . c cho trong ba ˙’ ng sau: k p q 0 0 0 1 0 1 2 1 1 3 1 2 C´ac h`am Haar d¯i . nh ngh˜ıa theo quy na . pnhu . sau (v´o . i z ∈ [0, 1]) : h 0 (z)=h 00 (z):= 1 √ N , v`a v´o . i k>0, h k (z)=h pq (z):= 1 √ N          2 p/2 nˆe ´ u q−1 2 p ≤ z< q−1/2 2 p , −2 p/2 nˆe ´ u q−1/2 2 p ≤ z< q 2 p , 0nˆe ´ u ngu . o . . cla . i. Ma trˆa . n vuˆong A N cˆa ´ p N cu ˙’ abiˆe ´ nd¯ˆo ˙’ i Haar c´o h`ang th´u . i, i =0, 1, ,N − 1, x´ac d¯ i . nh bo . ˙’ i c´ac phˆa ` ntu . ˙’ h i (z),z =0/N, 1/N, ,(N − 1)/N. V´ı du . 3.5.3 A 2 = 1 √ 2  11 1 −1  ,A 4 = 1 √ 4       1111 11−1 −1 √ 2 − √ 20 0 00 √ 2 − √ 2       . C´ac ma trˆa . n Haar l`a tru . . c giao v`a c´o c´ac t´ınh chˆa ´ tcˆa ` n thiˆe ´ td¯ˆe ˙’ c´o thˆe ˙’ thu . . chiˆe . n biˆe ´ nd¯ˆo ˙’ i Haar nhanh. 64 3.5.5 Biˆe ´ nd¯ˆo ˙’ i Slant Ma trˆa . nbiˆe ´ nd¯ˆo ˙’ i Slant l`a ma trˆa . n vuˆong S N cˆa ´ p N (N>1), d¯ u . o . . cd¯i . nh ngh˜ıa d¯ˆe . qui nhu . sau S 2 = 1 √ 2  11 1 −1  , S N = 1 √ 2                              10 10 00 a N b N −a N b N 0IN 2 −2 0IN 2 −2 01 0−1 00 −b N a N b N a N 0IN 2 −2 0IN 2 −2                                                           S N 2 0 0S N 2                              , trong d¯´o I M l`a ma trˆa . nd¯o . nvi . cˆa ´ p M v`a a N :=  3N 2 4(N 2 − 1)  1 2 , b N :=  N 2 − 4 4(N 2 − 1)  1 2 . Chˇa ˙’ ng ha . n S 4 = 1 √ 4       11 1 1 3 √ 5 1 √ 5 − 1 √ 5 − 3 √ 5 1 −1 −11 1 √ 5 − 3 √ 5 3 √ 5 − 1 √ 5       . C´ac ma trˆa . n Slant l`a tru . . c giao v`a c´o c´ac t´ınh chˆa ´ tcˆa ` n thiˆe ´ td¯ˆe ˙’ c´o thˆe ˙’ thu . . chiˆe . nbiˆe ´ n d¯ ˆo ˙’ i Slant nhanh. 65 3.6 Biˆe ´ nd¯ˆo ˙’ i Hotelling Kh´ac v´o . inh˜u . ng ph´ep biˆe ´ nd¯ˆo ˙’ i tru . ´o . c, biˆe ´ nd¯ˆo ˙’ i Hoteling (c`on go . il`abiˆe ´ nd¯ˆo ˙’ i Karhunen- Lo`eve) du . . a trˆen c´ac t´ınh chˆa ´ t thˆo ´ ng kˆecu ˙’ aa ˙’ nh. Ph´ep biˆe ´ nd¯ˆo ˙’ i n`ay thu . `o . ng d`ung trong xu . ˙’ l´y a ˙’ nh. X´et vector ngˆa ˜ u nhiˆen x :=       x 1 x 2 . . . x n       . Vector trung b`ınh (hay k`yvo . ng)v`ama trˆa . nhiˆe . pbiˆe ´ n cu ˙’ a x x´ac d¯i . nh tu . o . ng ´u . ng bo . ˙’ i m x := E{x}, C x := E{(x − m x )(x −m x ) t }, trong d¯´o E{.} l`a k`y vo . ng. Nhˇa ´ cla . irˇa ` ng, k`y vo . ng cu ˙’ amˆo . t vector hay ma trˆa . n nhˆa . n d¯ u . o . . cbˇa ` ng c´ach lˆa ´ yk`yvo . ng cu ˙’ amˆo ˜ i phˆa ` ntu . ˙’ . V`ı x ∈ R n nˆen ma trˆa . n C x := (c ij ) c´o k´ıch thu . ´o . c n × n. Phˆa ` ntu . ˙’ c ii l`a phu . o . ng sai cu ˙’ a x i ;v`ac ij l`a hiˆe . pbiˆe ´ ncu ˙’ a c´ac phˆa ` ntu . ˙’ x i v`a x j . Dˆe ˜ d`ang ch´u . ng minh rˇa ` ng ma trˆa . n C x l`a thu . . c v`a d¯ˆo ´ ix´u . ng. Nˆe ´ u c´ac phˆa ` ntu . ˙’ x i v`a x j khˆong tu . o . ng quan th`ı hiˆe . p biˆe ´ ncu ˙’ ach´ung bˇa ` ng 0 v`a do d¯´o c ij = c ji =0. V´o . i M vector mˆa ˜ u x 1 , x 2 , ,x M , vector trung b`ınh v`a ma trˆa . nhiˆe . pbiˆe ´ nd¯u . o . . c xˆa ´ pxı ˙’ bo . ˙’ i m x := 1 M M  k=1 x k , C x := 1 M 2 M  k=1 x k x t k −m x m t x . V´ı du . 3.6.1 X´et c´ac vector x 1 :=    0 0 0    , x 2 =    1 0 0    , x 3 =    1 1 0    , x 4 =    1 0 1    . 66 Ta c´o m x = 1 4    3 1 1    , C x = 1 16    31 1 13−1 1 −13    . V`ı C x l`a ma trˆa . n thu . . c v`a d¯ˆo ´ ix´u . ng nˆen c´ac gi´a tri . riˆeng cu ˙’ a ma trˆa . n n`ay l`a thu . . c. Khˆong mˆa ´ t t´ınh tˆo ˙’ ng qu´at c´o thˆe ˙’ gia ˙’ su . ˙’ rˇa ` ng e i v`a λ i ,i =1, 2, ,n, l`a c´ac vector riˆeng v`a gi´a tri . riˆeng tu . o . ng ´u . ng cu ˙’ a C x sao cho λ i ≥ λ i+1 , v´o . i i =1, 2, ,n− 1. K´yhiˆe . u A l`a ma trˆa . n m`a c´ac h`ang l`a c´ac vector riˆeng cu ˙’ a ma trˆa . n C x v`a d¯u . o . . c sˇa ´ pth´u . tu . . sao cho h`ang th´u . nhˆa ´ ttu . o . ng ´u . ng gi´a tri . riˆeng l´o . n nhˆa ´ t v`a h`ang cuˆo ´ itu . o . ng ´u . ng gi´a tri . riˆeng nho ˙’ nhˆa ´ t. Khi d¯´o y = A(x − m x ) d¯ u . o . . cgo . il`aph´ep biˆe ´ nd¯ˆo ˙’ i Hotelling. Dˆe ˜ d`ang thˆa ´ yrˇa ` ng vector trung b`ınh v`a ma trˆa . nhiˆe . pbiˆe ´ ntu . o . ng ´u . ng v´o . i y l`a m y =0, C y = AC x A t . Ho . nn˜u . a C y l`a ma trˆa . nd¯u . `o . ng ch´eo trong d¯´o c´ac phˆa ` ntu . ˙’ thuˆo . cd¯u . `o . ng ch´eo ch´ınh l`a c´ac gi´a tri . riˆeng cu ˙’ a C x ;t´u . cl`a C y =       λ 1 0 0 0 λ 2 0 . . . 00 λ n       . V`ı c´ac phˆa ` ntu . ˙’ ngo`ai d¯u . `o . ng ch´eo cu ˙’ a ma trˆa . n C y bˇa ` ng 0, nˆen c´ac phˆa ` ntu . ˙’ cu ˙’ a vector y khˆong tu . o . ng quan. Nhˆa . nx´et 3.6.2 (i) Ph´ep biˆe ´ nd¯ˆo ˙’ i Hotelling d¯´ong vai tr`o quan tro . ng trong phˆan t´ıch a ˙’ nh. Sau khi d¯ˆo ´ itu . o . . ng d¯u . o . . c t´ach ra kho ˙’ ia ˙’ nh, c´ac k˜y thuˆa . td¯ˆe ˙’ nhˆa . nda . ng d¯ˆo ´ itu . o . . ng thu . `o . ng c´o liˆen quan mˆa . t thiˆe ´ td¯ˆe ´ n ph´ep quay d¯ˆo ´ itu . o . . ng. V`ıd¯ˇa . c t´ınh cu ˙’ ad¯ˆo ´ itu . o . . ng thu . `o . ng khˆong biˆe ´ t tru . ´o . c khi nhˆa . nda . ng, ch´ung ta cˆa ` nsˇa ´ pd¯ˆo ´ itu . o . . ng theo c´ac tru . c 67 . 4 4(N 2 − 1)  1 2 . Chˇa ˙’ ng ha . n S 4 = 1 √ 4       11 1 1 3 √ 5 1 √ 5 − 1 √ 5 − 3 √ 5 1 −1 −11 1 √ 5 − 3 √ 5 3 √ 5 − 1 √ 5       . C´ac ma trˆa . n Slant l`a tru . . c giao v`a c´o. chˆa ´ tcˆa ` n thiˆe ´ td¯ˆe ˙’ c´o thˆe ˙’ thu . . chiˆe . n biˆe ´ nd¯ˆo ˙’ i Haar nhanh. 64 3 .5. 5 Biˆe ´ nd¯ˆo ˙’ i Slant Ma trˆa . nbiˆe ´ nd¯ˆo ˙’ i Slant l`a ma trˆa . n vuˆong S N cˆa ´ p. u . o . . ckˆe ´ tho . . pv´o . i nhau trong xu . ˙’ l´y, nˆen go . i chung l`a biˆe ´ nd¯ˆo ˙’ i Walsh-Hadamard. 3 .5. 3 Biˆe ´ nd¯ˆo ˙’ i cosin r`o . ira . c Cˇa . pbiˆe ´ nd¯ˆo ˙’ i cosin r`o . ira . cmˆo . tchiˆe ` u