Phu . o . ng tr`ınh (3.14) c˜ung c´o thˆe ˙’ khai triˆe ˙’ nnhu . sau F (u, 0) = M−1  x=0 G(x, 0)e −2πi ux M , F (u, 1) = M−1  x=0 G(x, 1)e −2πi ux M , . . . F (u, N −1) = M−1  x=0 G(x, N − 1)e −2πi ux M . C´ac phu . o . ng tr`ınh trˆen d¯u . ad¯ˆe ´ nthu ˙’ tu . c FFT hai chiˆe ` u: Bu . ´o . c1.Biˆe ´ nd¯ˆo ˙’ i FFT 1D mˆo ˜ i h`ang v`a lu . utr˜u . v`ao ma ˙’ ng trung gian. Bu . ´o . c2.Chuyˆe ˙’ nvi . ma ˙’ ng trung gian. Bu . ´o . c3. Biˆe ´ nd¯ˆo ˙’ i FFT 1D mˆo ˜ i h`ang cu ˙’ ama ˙’ ng trung gian. Kˆe ´ t qua ˙’ cuˆo ´ ic`ung l`a chuyˆe ˙’ nvi . cu ˙’ ama ˙’ ng FFT 2D. Ch´ung ta c˜ung c´o thˆe ˙’ viˆe ´ tla . i (3.14) nhu . sau F (u, v)= 1 MN N−1  y=0  M−1  x=0 f(x, y)e −2πi ux M  e −2πi vy N . (3.16) D - ˇa . t G(u, y):= 1 M M−1  x=0 f(x, y)e −2πi ux M . Th`ı F (u, v)= 1 N N−1  y=0 G(u, y)e −2πi vy N . D - iˆe ` u n`ay d¯u . ad¯ˆe ´ nthu ˙’ tu . c FFT hai chiˆe ` u: Bu . ´o . c1.Chuyˆe ˙’ nvi . tˆa . p tin a ˙’ nh. Bu . ´o . c2. Biˆe ´ nd¯ˆo ˙’ i FFT 1D mˆo ˜ i h`ang cu ˙’ aa ˙’ nh d¯u . o . . c chuyˆe ˙’ nvi . v`a lu . utr˜u . v`ao ma ˙’ ng trung gian. Bu . ´o . c3.Chuyˆe ˙’ nvi . ma ˙’ ng trung gian. Bu . ´o . c4.Biˆe ´ nd¯ˆo ˙’ i FFT 1D mˆo ˜ i h`ang cu ˙’ ama ˙’ ng trung gian. Kˆe ´ t qua ˙’ l`a FFT 2D. 58 3.4.2 Biˆe ´ nd¯ˆo ˙’ i FFT ngu . o . . c Thuˆa . t to´an biˆe ´ nd¯ˆo ˙’ i thuˆa . nc´othˆe ˙’ ca ˙’ i biˆen d¯ˆe ˙’ nhˆa . nd¯u . o . . cbiˆe ´ nd¯ˆo ˙’ i ngu . o . . c. Thˆa . tvˆa . y, lˆa ´ y liˆen ho . . pph´u . c hai vˆe ´ v`a chia cho N cu ˙’ a f(x)= N−1  u=0 F (u)e 2πi ux N ta d¯u . o . . c 1 N ¯ f(x)= 1 N N−1  u=0 ¯ F(u)e −2πi ux N . (3.17) So s´anh v´o . i F (u)= 1 N N−1  x=0 f(x) e −2πi ux N ta thˆa ´ yvˆe ´ pha ˙’ icu ˙’ a (3.17) c´o da . ng cu ˙’ a ph´ep biˆe ´ nd¯ˆo ˙’ i Fourier thuˆa . n. Do d¯´o su . ˙’ du . ng thuˆa . t to´an n`ay v´o . id˜u . liˆe . u nhˆa . pl`a ¯ F (u)d¯ˆe ˙’ t´ınh ¯ f(x)/N. T`u . d¯ ´o d ˆe ˜ d`ang suy ra f(x). Tu . o . ng tu . . cho 2D, t`u . (3.4) dˆe ˜ d`ang suy ra 1 MN ¯ f(x, y)= 1 MN M−1  u=0 N−1  v=0 ¯ F(u, v)e −2πi ( ux M + v y N ) . Biˆe ˙’ uth´u . cbˆenvˆe ´ pha ˙’ i c´o da . ng biˆe ´ nd¯ˆo ˙’ i Fourier 2D thuˆa . n. Do d¯´o nhˆa . p ¯ F(u, v)v`a d`ung thuˆa . t to´an biˆe ´ nd¯ˆo ˙’ i thuˆa . n ta c´o 1 MN ¯ f(x, y). T`u . d¯´o suy ra f(x, y). Ch´u´yrˇa ` ng, nˆe ´ u f thu . . c th`ı ph´ep to´an liˆen ho . . pph´u . c l`a khˆong cˆa ` n thiˆe ´ t. Nhˆa . nx´et 3.4.1 Thuˆa . t to´an FFT 2D d¯u . o . . c tr`ınh b`ay trˆen liˆen quan d¯ˆe ´ n b`ai to´an chuyˆe ˙’ nvi . ma trˆa . n. D - ˆe ˙’ gia ˙’ mb´o . t th`o . i gian thu . . chiˆe . n trong tiˆe ´ n tr`ınh n`ay, ch´ung ta c´o thˆe ˙’ su . ˙’ du . ng phu . o . ng ph´ap chuyˆe ˙’ nvi . ma trˆa . ncu ˙’ a Eklundh (xem [19]). 3.5 C´ac ph´ep biˆe ´ nd¯ˆo ˙’ ikh´ac Biˆe ´ nd¯ˆo ˙’ i Fourier mˆo . tchiˆe ` u l`a mˆo . t trong nh˜u . ng ph´ep biˆe ´ nd¯ˆo ˙’ ic´oda . ng T (u):= N−1  x=0 f(x) g(x, u), 59 trong d¯´o T (u) l`a ph´ep biˆe ´ nd¯ˆo ˙’ i, g(x, u)l`aha . t nhˆan biˆe ´ nd¯ˆo ˙’ i thuˆa . n v`a gi´a tri . u thay d¯ ˆo ˙’ i trong pha . mvi{0, 1, ,N − 1}. Tu . o . ng tu . . , ph´ep biˆe ´ nd¯ˆo ˙’ i ngu . o . . c f(x)= N−1  u=0 T (u)h(x, u), trong d¯´o h(x, u)l`aha . t nhˆan biˆe ´ nd¯ˆo ˙’ i ngu . o . . c v`a gi´a tri . x thay d¯ˆo ˙’ i trong pha . mvi {0, 1, ,N − 1}. C´ac t´ınh chˆa ´ tcu ˙’ aha . t nhˆan biˆe ´ nd¯ˆo ˙’ i x´ac d¯i . nh t´ınh chˆa ´ tcu ˙’ a ph´ep biˆe ´ nd¯ˆo ˙’ i. Cˇa . pbiˆe ´ nd¯ˆo ˙’ i hai chiˆe ` ur`o . ira . c thuˆa . n v`a ngu . o . . ctˆo ˙’ ng qu´at c´o da . ng              T (u, v):= N−1  x=0 N−1  y=0 f(x, y)g(x, y,u, v), f(x, y):= N−1  u=0 N−1  v=0 T (u, v)h(x, y,u,v), (3.18) trong d¯´o g, h d¯ u . o . . cgo . i l`a c´ac ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i thuˆa . n v`a ngu . o . . c. C´ac ha . t nhˆan n`ay chı ˙’ phu . thuˆo . c v`ao x, y, u, v, m`a khˆong phu . thuˆo . c v`ao f v`a T. Ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i thuˆa . nl`at´ach d¯u . o . . c nˆe ´ u ta c´o thˆe ˙’ viˆe ´ t g(x, y, u, v)=g 1 (x, u)g 2 (y,v). Ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i thuˆa . n t´ach d¯u . o . . cgo . il`ad¯ ˆo ´ ix´u . ng nˆe ´ u g 1 ≡ g 2 . Tu . o . ng tu . . ta c˜ung c´o c´ac kh´ai niˆe . m t´ach d¯u . o . . c, d¯ˆo ´ ix´u . ng cho ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i ngu . o . . c. Biˆe ´ nd¯ˆo ˙’ i Fourier 2D l`a tru . `o . ng ho . . pd¯ˇa . cbiˆe . t c´o ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i g(x, y, u, v):= 1 NN e −2πi ( ux N + vy N ) t´ach d¯u . o . . cv`ad¯ˆo ´ ix´u . ng. (Ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i Fourier 2D ngu . o . . cc˜ung t´ach d¯u . o . . cv`ad¯ˆo ´ i x´u . ng). Ph´ep biˆe ´ nd¯ˆo ˙’ iv´o . i nhˆan t´ach d¯u . o . . cc´othˆe ˙’ d¯ u . o . . c t´ınh thˆong qua hai bu . ´o . c, mˆo ˜ i bu . ´o . csu . ˙’ du . ng ph´ep biˆe ´ nd¯ˆo ˙’ i 1D nhu . d¯ u . o . . cchı ˙’ ra du . ´o . i d¯ˆay. Thuˆa . t to´an Bu . ´o . c1.D - ˆa ` u tiˆen biˆe ´ nd¯ˆo ˙’ i 1D do . c theo mˆo ˜ i h`ang cu ˙’ aa ˙’ nh f : T (x, v):= N−1  y=0 f(x, y)g 2 (y,v), v´o . i x =0, 1, ,N −1,v =0, 1, ,N −1. 60 Bu . ´o . c2. Kˆe ´ tiˆe ´ p, lˆa ´ ybiˆe ´ nd¯ˆo ˙’ i1Ddo . c theo mˆo ˜ icˆo . tcu ˙’ a T (x, v) (d¯˜a d¯u . o . . c t´ınh theo Bu . ´o . c 1): T (u, v):= N−1  x=0 T (x, v)g 1 (x, u), v´o . i u =0, 1, ,N −1,v =0, 1, ,N −1. Ta c˜ung c´o thˆe ˙’ x´ac d¯i . nh T(u, v)bˇa ` ng c´ach: (1) biˆe ´ nd¯ˆo ˙’ imˆo ˜ icˆo . tcu ˙’ a f d¯ ˆe ˙’ nhˆa . n d¯ u . o . . c T (y, u); (2) sau d¯´o biˆe ´ nd¯ˆo ˙’ ido . c theo mˆo ˜ i h`ang. Tu . o . ng tu . . cho biˆe ´ nd¯ˆo ˙’ i ngu . o . . c nˆe ´ u h t´ach d¯u . o . . c. Nˆe ´ uha . t nhˆan g t´ach d¯u . o . . cv`ad¯ˆo ´ ix´u . ng, th`ı (3.18) c´o thˆe ˙’ viˆe ´ tla . idu . ´o . ida . ng ma trˆa . n T = AfA, (3.19) trong d¯´o f l`a ma trˆa . na ˙’ nh N ×N; A := (a ij ) l`a ma trˆa . nbiˆe ´ nd¯ˆo ˙’ id¯ˆo ´ ix´u . ng cˆa ´ p N ×N, v´o . i a ij := g 1 (i, j); v`a T l`a ma trˆa . nkˆe ´ t qua ˙’ . D - ˆe ˙’ c´o biˆe ´ nd¯ˆo ˙’ i ngu . o . . c, nhˆan tru . ´o . c v`a sau (3.19) v´o . i ma trˆa . nbiˆe ´ nd¯ˆo ˙’ i ngu . o . . c B, ta d¯u . o . . c BTB = BAFAB. (3.20) Nˆe ´ u B = A −1 , th`ı f = BTB (3.21) cho biˆe ´ ta ˙’ nh sˆo ´ f c´o thˆe ˙’ khˆoi phu . c ho`an to`an t`u . biˆe ´ nd¯ˆo ˙’ i n`ay. Nˆe ´ u B = A −1 , ta d`ung (3.20) d¯ˆe ˙’ c´o mˆo . txˆa ´ pxı ˙’ v´o . i f : ˆ f = BAfAB. Nhiˆe ` u loa . ibiˆe ´ nd¯ˆo ˙’ i (Fourier, Walsh, Hadamard, cosin r`o . ira . c, Haar, Slant) c´o thˆe ˙’ biˆe ˙’ u diˆe ˜ ndu . ´o . ida . ng (3.19) v`a (3.21). Mˆo . t t´ınh chˆa ´ t quan tro . ng cu ˙’ a c´ac ma trˆa . nbiˆe ´ nd¯ˆo ˙’ i l`a ch´ung c´o thˆe ˙’ t´ach d¯u . o . . c th`anh t´ıch c´ac ma trˆa . nv´o . i ´ıt phˆa ` ntu . ˙’ kh´ac khˆong ho . nma trˆa . ngˆo ´ c. Kˆe ´ t qua ˙’ n`ay l`am gia ˙’ md¯ˆo . du . th `u . av`asˆo ´ ph´ep t´ınh cˆa ` n thiˆe ´ t cho biˆe ´ nd¯ˆo ˙’ i 2D. 3.5.1 Biˆe ´ nd¯ˆo ˙’ i Walsh Gia ˙’ su . ˙’ N =2 n ,n ∈ N. Biˆe ´ nd¯ˆo ˙’ i Walsh cu ˙’ a h`am r`o . ira . c f(x, y) c´o ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i thuˆa . n ngu . o . . cchobo . ˙’ i 61            g(x, y, u,v):= 1 N n−1  i=0 (−1) [b i (x)b n−1−i (u)+b i (y)b n−1−i (v)] , h(x, y, u, v):= 1 N n−1  i=0 (−1) [b i (x)b n−1−i (u)+b i (y)b n−1−i (v)] , trong d¯´o b k (z) l`a bit th´u . k trong biˆe ˙’ udiˆe ˜ n nhi . phˆan cu ˙’ a z. Chˇa ˙’ ng ha . n, nˆe ´ u n =3v`a z = 6 = (110) 2 th`ı b 0 (z)=0,b 1 (z)=1,b 2 (z)=1. Biˆe ´ nd¯ˆo ˙’ i Walsh thuˆa . n ngu . o . . c              W (u, v):= 1 N N−1  x=0 N−1  y=0 f(x, y) n−1  i=0 (−1) [b i (x)b n −1−i (u)+b i (y)b n −1−i (v)] , f(x, y):= 1 N N−1  u=0 N−1  v=0 W (u, v) n−1  i=0 (−1) [b i (x)b n−1−i (u)+b i (y)b n−1−i (v)] . Nhˆa . nx´et rˇa ` ng, ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i Walsh l`a t´ach d¯u . o . . c v`a d¯ˆo ´ ix´u . ng. Do d¯´o W (u, v) v`a biˆe ´ nd¯ˆo ˙’ i ngu . o . . ccu ˙’ a n´o c´o thˆe ˙’ d¯ u . o . . c t´ınh bˇa ` ng thuˆa . t to´an tr`ınh b`ay trˆen. Ho . nn˜u . a, biˆe ´ nd¯ˆo ˙’ i Walsh c´o thˆe ˙’ d¯ u . o . . ct´ınh bˇa ` ng thuˆa . t to´an nhanh nhu . FFT. Chˆo ˜ kh´ac biˆe . t duy nhˆa ´ t l`a tˆa ´ tca ˙’ c´ac l ˜uy th`u . a W N d¯ u . o . . cd¯ˇa . tbˇa ` ng 1 trong tru . `o . ng ho . . p FWT. 3.5.2 Biˆe ´ nd¯ˆo ˙’ i Hadamard Ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i thuˆa . n ngu . o . . ccu ˙’ a biˆe ´ nd¯ˆo ˙’ i Hadamard cho bo . ˙’ i      g(x, y, u,v):= 1 N (−1)  n−1 i=0 [b i (x)b i (u)+b i (y)b i (v)] % 2 , h(x, y, u, v):= 1 N (−1)  n−1 i=0 [b i (x)b i (u)+b i (y)b i (v)] % 2 . Ta c´o cˇa . pbiˆe ´ nd¯ˆo ˙’ i Hadamard 2D              H(u, v):= 1 N N−1  x=0 N−1  y=0 f(x, y)(−1)  n−1 i=0 [b i (x)b i (u)+b i (y)b i (v)] , f(x, y):= 1 N N−1  u=0 N−1  v=0 H(u, v)(−1)  n−1 i=0 [b i (x)b i (u)+b i (y)b i (v)] . T`u . d¯ i . nh ngh˜ıa dˆe ˜ d`ang suy ra ha . t nhˆan biˆe ´ nd¯ˆo ˙’ i Hadamard t´ach d¯u . o . . c v`a d¯ˆo ´ ix´u . ng. Nhˆa . nx´et 3.5.1 Nˆe ´ u N =2 n th`ı sau mˆo . tsˆo ´ ho´an vi . h`ang v`a cˆo . t, ta c´o thˆe ˙’ chuyˆe ˙’ nma trˆa . nbiˆe ´ nd¯ˆo ˙’ i Walsh vˆe ` ma trˆa . nbiˆe ´ nd¯ˆo ˙’ i Hadamard. Nˆe ´ u N =2 n th`ı c´o su . . kh´ac nhau 62 . 2D. Ch´ung ta c˜ung c´o thˆe ˙’ viˆe ´ tla . i (3. 14) nhu . sau F (u, v)= 1 MN N−1  y=0  M−1  x=0 f(x, y)e −2πi ux M  e −2πi vy N . (3.16) D - ˇa . t G(u, y):= 1 M M−1  x=0 f(x, y)e −2πi ux M . Th`ı F. gian. Bu . ´o . c3.Chuyˆe ˙’ nvi . ma ˙’ ng trung gian. Bu . ´o . c4.Biˆe ´ nd¯ˆo ˙’ i FFT 1D mˆo ˜ i h`ang cu ˙’ ama ˙’ ng trung gian. Kˆe ´ t qua ˙’ l`a FFT 2D. 58 3 .4. 2 Biˆe ´ nd¯ˆo ˙’ i FFT ngu . o . . c Thuˆa . t. khˆong cˆa ` n thiˆe ´ t. Nhˆa . nx´et 3 .4. 1 Thuˆa . t to´an FFT 2D d¯u . o . . c tr`ınh b`ay trˆen liˆen quan d¯ˆe ´ n b`ai to´an chuyˆe ˙’ nvi . ma trˆa . n. D - ˆe ˙’ gia ˙’ mb´o . t th`o . i gian