Xử lý ảnh số - Các phép biến đổi part 2 pdf

H`ınh 3.1: A ˙’ nh gˆo ´ c, a ˙’ nh cu ˙’ a ln(1 + F [u, v])v`aa ˙’ nh cu ˙’ a g´oc pha ϕ(u, v). 3.3.1 T´ınh t´ach d¯u . o . . c X´et cˇa . pbiˆe ´ nd¯ˆo ˙’ i Fourier r`o . ira . ccu ˙’ a h`am a ˙’ nh f(x, y) c´o k´ıch thu . ´o . c M × N : F (u, v)= 1 MN M−1  x=0 N−1  y=0 f(x, y)e −2πi ( ux M + vy N ) , (3.3) v`a f(x, y)= M−1  u=0 N−1  v=0 F (u, v)e 2πi ( ux M + vy N ) , (3.4) trong d¯´o x, u =0,1, ,M − 1v`ay, v =0, 1, ,N − 1. U . ud¯iˆe ˙’ ml`aF ( u, v) hoˇa . c f(x, y) c´o thˆe ˙’ nhˆa . nd¯u . o . . c theo hai bu . ´o . cbiˆe ´ nd¯ˆo ˙’ i Fourier 1D thuˆa . n hoˇa . c ngu . o . . c. D - iˆe ` u n`ay l`a hiˆe ˙’ n nhiˆen, v`ı t `u . (3.3) ta c´o F (u, v)= 1 M M−1  x=0 G(x, v)e −2πi ux M , trong d¯´o G(x, v):= 1 N N−1  y=0 f(x, y)e −2πi vy N . (3.5) D - ˆo ´ iv´o . imˆo ˜ i gi´a tri . x, vˆe ´ pha ˙’ icu ˙’ abiˆe ˙’ uth´u . c (3.5) l`a biˆe ´ nd¯ˆo ˙’ i Fourier 1D v´o . i c´ac gi´a tri . tˆa ` nsˆo ´ v =0, 1, ,N − 1. V`ıvˆa . y h`am hai chiˆe ` u F (u, v) nhˆa . nd¯u . o . . c theo c´ac bu . ´o . c sau Bu . ´o . c1. Biˆe ´ nd¯ˆo ˙’ i Fourier 1D theo t`u . ng h`ang cu ˙’ a f(x, y) ta d¯u . o . . cma ˙’ ng trung gian G(x, v); Bu . ´o . c2. Biˆe ´ nd¯ˆo ˙’ i Fourier 1D theo cˆo . tcu ˙’ a G(x, v). 48 C´o thˆe ˙’ nhˆa . nd¯u . o . . ckˆe ´ t qua ˙’ giˆo ´ ng nhu . trˆen khi biˆe ´ nd¯ˆo ˙’ i theo c´ac cˆo . tcu ˙’ a f(x, y) v`a sau d¯´o do . c theo c´ac h`ang. 3.3.2 Ti . nh tiˆe ´ n V´o . imo . i x 0 ,y 0 ,u 0 ,v 0 ∈ C ta c´o F  f(x, y)e 2πi ( u 0 x M + v 0 y N )  = F(u − u 0 ,v− v 0 ), F [f(x − x 0 ,y− y 0 )] = F ( u, v)e −2πi ( ux 0 M + vy 0 N ) . T`u . d¯´o suy ra F  (−1) x+y f(x, y)  = F(u − M/2,v−N/2). Ho . nn˜u . a, ti . nh tiˆe ´ n khˆong l`am thay d¯ˆo ˙’ i phˆo ˙’ Fourier cu ˙’ a F. 3.3.3 Chu k`y Gia ˙’ su . ˙’ h`am a ˙’ nh f tuˆa ` n ho`an theo c´ac tru . c x v`a y tu . o . ng ´u . ng v´o . ichuk`y M v`a N;t´u . c l`a f(x, y)=f(x + M,y)=f(x, y + N)=f(x + M,y + N). (3.6) Khi d¯´o F (u, v)=F (u + M,v)=F(u, v + N)=F (u + M,v + N). (3.7) Ngu . o . . cla . i, nˆe ´ ubiˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a f thoa ˙’ (3.7) th`ı h`am a ˙’ nh f thoa ˙’ m˜an (3.6). Ho . n n˜u . a, nˆe ´ u h`am f thu . . c, th`ı F (u, v)= ¯ F(−u, −v), trong d¯´o ¯ F (u, v) l`a sˆo ´ ph´u . c liˆen ho . . p cu ˙’ a F (u, v). Suy ra F (u, v) = F (−u, −v). 3.3.4 Ph´ep quay X´et ph´ep biˆe ´ nd¯ˆo ˙’ ito . ad¯ˆo . cu . . c x(r, θ)=r cos θ, y(r, θ)=r sin θ, u(ω,ϕ)=ω cos ϕ, v(ω,ϕ)=ω sin ϕ. D - ˇa . t g(r, θ):=f(x(r, θ),y(r, θ)), G(ω,ϕ):=F(u(ω,ϕ),v(ω, ϕ)). 49 Khi d¯´o v´o . imo . i θ 0 ∈ R ta c´o F[g(r, θ + θ 0 )] = G(ω, ϕ + θ 0 ), N´oi c´ach kh´ac, quay f(x, y)mˆo . t g´oc θ 0 s˜e l`am quay F(u, v)c`ung mˆo . t g´oc. Tu . o . ng tu . . , ta quay F(u, v)s˜e l`am quay f(x, y)v´o . ic`ung mˆo . t g´oc. 3.3.5 Tuyˆe ´ n t´ınh v`a co gi˜an Biˆe ´ nd¯ˆo ˙’ i Fourier l`a ´anh xa . tuyˆe ´ n t´ınh, t´u . cl`a F(af + bg)=aF(f)+bF(g)v´o . imo . i a, b ∈ C. Tuy nhiˆen, n´oi chung F(fg) = F(f) F(g). Ngo`ai ra, dˆe ˜ d`ang ch ´u . ng minh rˇa ` ng v´o . imo . i a, b ∈ C v´o . i a, b = 0 ta c´o F[f(ax, by)] = 1 ab F  u a , v b  . 3.3.6 Gi´a tri . trung b`ınh Gi´a tri . trung b`ınh cu ˙’ a h`am r`o . ira . c hai chiˆe ` u f l`a 1 MN M−1  x=0 N−1  y=0 f(x, y)=F (0, 0). 3.3.7 Biˆe ´ nd¯ˆo ˙’ i Laplace Biˆe ´ nd¯ˆo ˙’ i Laplace cu ˙’ a f x´ac d¯i . nh bo . ˙’ i ∆f(x, y):= ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 . Dˆe ˜ d`ang ch ´u . ng minh rˇa ` ng F (∆f)=−(2π) 2 (u 2 + v 2 )F (u, v). Ph´ep biˆe ´ nd¯ˆo ˙’ i Laplace thu . `o . ng d¯u . o . . cd`ung trong k˜y thuˆa . t t´ach biˆen cu ˙’ aa ˙’ nh. 50 3.3.8 T´ıch chˆa . p v`a tu . o . ng quan Nhˇa ´ cla . il`at´ıch chˆa . p (liˆen tu . c) cu ˙’ a f v`a g, k´yhiˆe . u(f ∗ g), x´ac d¯i . nh bo . ˙’ i (f ∗ g)(x, y):=  +∞ −∞  +∞ −∞ g(x −α, y −β)f(α, β)dαdβ. V´ı du . 3.3.2 Gia ˙’ su . ˙’ f(x):=    1nˆe ´ u0≤ x ≤ 1, 0nˆe ´ u ngu . o . . cla . i, g(x):=    1/2nˆe ´ u0≤ x ≤ 1, 0nˆe ´ u ngu . o . . cla . i. Khi d¯´o dˆe ˜ d`ang kiˆe ˙’ m tra rˇa ` ng (f ∗ g)(x)=          x/2nˆe ´ u0≤ x ≤ 1, 1 − x/2nˆe ´ u1≤ x ≤ 2, 0nˆe ´ u ngu . o . . cla . i. D - ˆe ˙’ d¯ i . nh ngh˜ıa t´ıch chˆa . pr`o . ira . c cu ˙’ a hai h`am a ˙’ nh f v`a g tu . o . ng ´u . ng c´ac ma ˙’ ng hai chiˆe ` uv´o . i k´ıch thu . ´o . c A ×B v`a C ×D ta cˆa ` nmo . ˙’ rˆo . ng k´ıch thu . ´o . ca ˙’ nh lˆen M ×N. D - ˆe ˙’ tr´anh hiˆe . ntu . o . . ng lˆo ˜ ibo . c, ta cho . n M,N sao cho M ≥ A + C − 1,N≥ B + D − 1. (3.8) X´et c´ac mo . ˙’ rˆo . ng cu ˙’ a f(x, y)l`a f r (x, y):=    f(x, y)nˆe ´ u0≤ x ≤ A −1, 0 ≤ y ≤ B − 1, 0nˆe ´ u A ≤ x ≤ M − 1 hoˇa . c B ≤ y ≤ N − 1, v`a mo . ˙’ rˆo . ng cu ˙’ a g(x, y)l`a g r (x, y):=    g(x, y)nˆe ´ u0≤ x ≤ C −1, 0 ≤ y ≤ D −1, 0nˆe ´ u C ≤ x ≤ M − 1 hoˇa . c D ≤ y ≤ N − 1. T´ıch chˆa . p hai chiˆe ` u (r`o . ira . c) cu ˙’ a f r v`a g r d¯ i . nh ngh˜ıa bo . ˙’ i (f r ∗g r )(x, y):= M−1  α=0 N−1  β=0 g r (x − α, y − β)f r (α, β), (3.9) 51 v´o . i x =0, 1, ,M −1,y=0, 1, ,N − 1. Trong thu . . ctˆe ´ ,viˆe . c t´ınh to´an t´ıch chˆa . pr`o . ira . c trong miˆe ` ntˆa ` nsˆo ´ hiˆe . u qua ˙’ ho . n khi ´ap du . ng cˆong th´u . c (3.9). D - i . nh l ´y 3.3.3 Gia ˙’ su . ˙’ F v`a G l`a c´ac biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a f v`a g. Khi d¯´o biˆe ´ nd¯ˆo ˙’ i Fourier ngu . o . . ccu ˙’ a FG ch´ınh l`a f ∗ g. Ch´u . ng minh. Gia ˙’ su . ˙’ H l`a biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a f ∗g. Ta cˆa ` nch´u . ng minh rˇa ` ng H = FG. Thˆa . tvˆa . y H(u, v)=  +∞ −∞  +∞ −∞ e −2πi(ux+vy)   +∞ −∞  +∞ −∞ g(x −α, y −β)f(α, β)dαdβ  dxdy =  +∞ −∞  +∞ −∞ f(α, β)   +∞ −∞  +∞ −∞ e −2πi(ux+vy) g(x − α, y −β)dxdy  dαdβ =  +∞ −∞  +∞ −∞ f(α, β)e −2πi(uα+vβ) G(u, v)dαdβ = G(u, v)  +∞ −∞  +∞ −∞ f(α, β)e −2πi(uα+vβ) dαdβ = F(u, v)G(u, v). D - i . nh l´y d¯u . o . . cch´u . ng minh. ✷ Tu . o . ng quan cu ˙’ a hai h`am liˆen tu . c f v`a g, k´yhiˆe . u f ⊗ g, x´ac d¯i . nh bo . ˙’ i (f ⊗ g)(x, y):=  +∞ −∞  +∞ −∞ g(x + α, y + β) ¯ f(α, β)dαdβ. Trong tru . `o . ng ho . . pr`o . ira . c (f r ⊗ g r )(x, y):= M−1  α=0 N−1  β=0 g r (x + α, y + β) ¯ f r (α, β), (3.10) v´o . i x =0, 1, ,M − 1,y=0, 1, ,N − 1. Nhu . trong tru . `o . ng ho . . pcu ˙’ at´ıchchˆa . pr`o . i ra . c, f r (x, y)v`ag r (x, y) l`a nh˜u . ng h`am d¯u . o . . cmo . ˙’ rˆo . ng v`a M,N d¯ u . o . . ccho . n theo (3.8) d¯ ˆe ˙’ tr´anh hiˆe . ntu . o . . ng lˆo ˜ ibo . c. Nhˆa . nx´et 3.3.4 (i) D - ˆo ´ iv´o . ica ˙’ hai tru . `o . ng ho . . pr`o . ira . c v`a liˆen tu . c, ta dˆe ˜ d`ang ch´u . ng minh c´ac quan hˆe . sau: F(f ⊗g)= ¯ FG, F( ¯ f ⊗ g)=FG. 52 . Laplace cu ˙’ a f x´ac d¯i . nh bo . ˙’ i ∆f(x, y):= ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 . Dˆe ˜ d`ang ch ´u . ng minh rˇa ` ng F (∆f)=− (2 ) 2 (u 2 + v 2 )F (u, v). Ph´ep biˆe ´ nd¯ˆo ˙’ i Laplace thu . `o . ng. ngu . o . . c. D - iˆe ` u n`ay l`a hiˆe ˙’ n nhiˆen, v`ı t `u . (3.3) ta c´o F (u, v)= 1 M M−1  x=0 G(x, v)e 2 i ux M , trong d¯´o G(x, v):= 1 N N−1  y=0 f(x, y)e 2 i vy N . (3.5) D - ˆo ´ iv´o . imˆo ˜ i. dˆe ˜ d`ang kiˆe ˙’ m tra rˇa ` ng (f ∗ g)(x)=          x/2nˆe ´ u0≤ x ≤ 1, 1 − x/2nˆe ´ u1≤ x ≤ 2, 0nˆe ´ u ngu . o . . cla . i. D - ˆe ˙’ d¯ i . nh ngh˜ıa t´ıch chˆa . pr`o . ira . c cu ˙’ a