Xử lý ảnh số - Các phép biến đổi part 1 ppsx

Chu . o . ng 3 C ´ AC PH ´ EP BI ˆ E ´ ND - ˆ O ˙’ I Mu . cd¯´ıch ch´ınh cu ˙’ a chu . o . ng n`ay l`a tr`ınh b`ay c´ac ph´ep biˆe ´ nd¯ˆo ˙’ i hai chiˆe ` u v`a c´ac t´ınh chˆa ´ tcu ˙’ ach´ung. C´ac ph´ep biˆe ´ nd¯ˆo ˙’ i d¯´ong vai tr`o quan tro . ng trong xu . ˙’ l´y a ˙’ nh ca ˙’ vˆe ` mˇa . t l´y thuyˆe ´ tc˜ung nhu . ´u . ng du . ng. C´ac ph´ep biˆe ´ nd¯ˆo ˙’ i hai chiˆe ` us˜ed¯u . o . . csu . ˙’ du . ng trong nh˜u . ng chu . o . ng sau d¯ˆe ˙’ nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh, phu . chˆo ` ia ˙’ nh, m˜a ho´a v`a miˆeu ta ˙’ a ˙’ nh. Mˇa . cd`u c´ac ph´ep biˆe ´ nd¯ˆo ˙’ i kh´ac c˜ung d¯u . o . . cd¯ˆe ` cˆa . p, tuy nhiˆen ch´ung ta vˆa ˜ n nhˆa ´ n ma . nh v`ao ph´ep biˆe ´ nd¯ˆo ˙’ i Fourier v`ın´od¯u . o . . csu . ˙’ du . ng rˆo . ng r˜ai trong c´ac b`ai to´an xu . ˙’ l´y a ˙’ nh. 3.1 Biˆe ´ nd¯ˆo ˙’ i Fourier liˆen tu . c 3.1.1 Biˆe ´ nd¯ˆo ˙’ i Fourier mˆo . tchiˆe ` u Gia ˙’ su . ˙’ f(x) l`a h`am liˆen tu . c theo biˆe ´ n thu . . c x. Biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a f(x), k´yhiˆe . u F(f) hoˇa . c F, x´ac d¯i . nh bo . ˙’ i F(f)(u)=F (u):=  +∞ −∞ f(x)e −2πiux dx, trong d¯´o i := √ −1. Cho tru . ´o . c F (u) ta c´o thˆe ˙’ nhˆa . nd¯u . o . . c f(x)bˇa ` ng c´ach su . ˙’ du . ng biˆe ´ nd¯ˆo ˙’ i Fourier 43 ngu . o . . c F −1 (F )(x)=f(x)=  +∞ −∞ F (u)e 2πiux du. C´o thˆe ˙’ ch´u . ng minh su . . tˆo ` nta . icu ˙’ a cˇa . pbiˆe ´ nd¯ˆo ˙’ i Fourier, nˆe ´ u f(x)liˆen tu . c, kha ˙’ t´ıch v`a F (u) kha ˙’ t´ıch. C´ac d¯iˆe ` ukiˆe . n n`ay thu . `o . ng thoa ˙’ m˜an trong thu . . ctˆe ´ . Trong gi´ao tr`ınh n`ay, ta luˆon gia ˙’ su . ˙’ f l`a h`am thu . . c. N´oi chung biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a h`am thu . . c f l`a mˆo . t h`am ph´u . c; t´u . cl`a F (u)=R( u)+iI(u), trong d¯´o R(u) (tu . o . ng ´u . ng, I(u)) l`a phˆa ` n thu . . c (tu . o . ng ´u . ng, phˆa ` na ˙’ o) cu ˙’ a F(u): R(u)=  +∞ −∞ f(x) cos[2πux]dx, I(u)=  +∞ −∞ f(x) sin[2 πux]dx. C´ac h`am n`ay d¯ˆoi khi c`on go . il`abiˆe ´ nd¯ˆo ˙’ i Fourier cosin v`a biˆe ´ nd¯ˆo ˙’ i Fourier sin cu ˙’ a f. Ta thu . `o . ng biˆe ˙’ udiˆe ˜ n h`am F(u)du . ´o . ida . ng F (u)=F (u)e iϕ(u) , trong d¯´o F (u) :=  R 2 (u)+I 2 (u), tan[ϕ( u)] :=  I(u) R(u)  . H`am F (u) d¯ u . o . . cgo . il`aphˆo ˙’ Fourier cu ˙’ a f, v`a ϕ(u)l`ag´oc pha.B`ınh phu . o . ng cu ˙’ a phˆo ˙’ Fourier go . il`aphˆo ˙’ cˆong suˆa ´ t.Biˆe ´ n u thu . `o . ng d¯u . o . . cgo . il`abiˆe ´ n tˆa ` nsˆo ´ . V´ı du . 3.1.1 Biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a h`am f(x):=    A nˆe ´ u0≤ x ≤ α, 0nˆe ´ u ngu . o . . cla . i, l`a F (u)=  +∞ −∞ f(x)e −2πixu dx =  α 0 Ae −2πixu dx 44 = A πu sin(πuα)e −πiαu . Suy ra phˆo ˙’ Fourier F (u) =     A πu     |sin(πuα)||e −πiαu | = |A|α     sin(πuα) (πuα)     . 3.1.2 Biˆe ´ nd¯ˆo ˙’ i Fourier hai chiˆe ` u Dˆe ˜ d`ang mo . ˙’ rˆo . ng biˆe ´ nd¯ˆo ˙’ i Fourier trong tru . `o . ng ho . . p hai chiˆe ` u. Nˆe ´ u f(x, y) l`a h`am liˆen tu . c v`a kha ˙’ t´ıch, v`a F(u, v) kha ˙’ t´ıch, th`ı tˆo ` nta . icˇa . pbiˆe ´ nd¯ˆo ˙’ i Fourier: F(f)(u, v)=F (u, v):=  +∞ −∞  +∞ −∞ f(x, y)e −2πi(ux+vy) dxdy v`a F −1 (F )(x, y)=f(x, y)=  +∞ −∞  +∞ −∞ F (u, v)e 2πi(ux+vy) dudv, trong d¯´o u, v l`a c´ac biˆe ´ ntˆa ` nsˆo ´ . Tu . o . ng tu . . trong tru . `o . ng ho . . pmˆo . tchiˆe ` u, phˆo ˙’ Fourier, g´oc pha v`a phˆo ˙’ cˆong suˆa ´ t x´ac d¯i . nh tu . o . ng ´u . ng bo . ˙’ i F (u, v) :=  R 2 (u, v)+I 2 (u, v), ϕ(u, v) := tan −1  I(u, v) R(u, v)  v`a P (u, v):=F (u, v) 2 = R 2 (u, v)+I 2 (u, v). V´ı du . 3.1.2 Biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a h`am f(x, y):=    A nˆe ´ u0≤ x ≤ α, 0 ≤ y ≤ β, 0nˆe ´ u ngu . o . . cla . i, l`a F (u, v)=  +∞ −∞  +∞ −∞ f(x, y)e −2πi(ux+vy) dxdy = Aαβ  sin(πuα)e −πiαu (πuα)  sin(πuβ)e −πiβu (πuβ)  . Do d¯´o phˆo ˙’ Fourier l`a F (u, v) = |A|αβ     sin(πuα) (πuα)         sin(πuβ) (πuβ)     . 45 3.2 Biˆe ´ nd¯ˆo ˙’ i Fourier r`o . ira . c X´et d˜ay f(x),x=0, 1, ,N−1. Biˆe ´ nd¯ˆo ˙’ i Fourier r`o . ira . c thuˆa . n ngu . o . . c mˆo . tchiˆe ` u x´ac d¯ i . nh bo . ˙’ i F(f)(u)=F (u):= 1 N N−1  x=0 f(x)e −2πi ux N , (3.1) v´o . i u =0, 1, ,N − 1, v`a F −1 (F )(x)=f(x):= N−1  u=0 F (u)e 2πi ux N , (3.2) trong d¯´o x =0, 1, ,N − 1. V´ı du . 3.2.1 Gia ˙’ su . ˙’ f(0) = 2,f(1) = 3,f(2) = f(3) = 4. Tac´obiˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a f l`a F (0) = 1 4 3  x=0 f(x)e 0 =3.25; F (1) = 1 4 3  x=0 f(x)e −2πix/4 = 1 4 (−2+i); F (2) = 1 4 3  x=0 f(x)e −2πi2x/4 = − 1 4 ; F (3) = 1 4 3  x=0 f(x)e −2πi3x/4 = − 1 4 (2 + i). Trong tru . `o . ng ho . . p hai biˆe ´ n, cˇa . pbiˆe ´ nd¯ˆo ˙’ i Fourier r`o . ira . cchobo . ˙’ i F(f)(u, v):= 1 MN M−1  x=0 N−1  y=0 f(x, y)e −2πi ( ux M + vy N ) , v´o . i u =0, 1, ,M − 1, v`a v =0, 1, ,N − 1, v`a F −1 (F )(x, y):= M−1  u=0 N−1  v=0 F (u, v)e 2πi ( ux M + vy N ) , trong d¯´o x =0, 1, ,M − 1, v`a y =0, 1, ,N − 1. 46 Khi c´ac a ˙’ nh c´o k´ıch thu . ´o . c vuˆong, t´u . cl`aM = N, d¯ ˆe ˙’ thuˆa . ntiˆe . n trong c´ac t´ınh to´an ta thu . `o . ng su . ˙’ du . ng cˆong th´u . cbiˆe ´ nd¯ˆo ˙’ i Fourier thuˆa . n ngu . o . . c sau              F (u, v)= 1 N N−1  x=0 N−1  y=0 f(x, y)e −2πi ( ux N + vy N ) , f(x, y)= 1 N N−1  u=0 N−1  v=0 F (u, v)e 2πi ( ux N + v y N ) , trong d¯´o u, x =0, 1, ,N − 1, v`a y,v =0, 1, ,N − 1. Ho`an to`an tu . o . ng tu . . ta c ˜ung c´o c´ac kh´ai niˆe . m phˆo ˙’ Fourier, g´oc pha, phˆo ˙’ cˆong suˆa ´ tcu ˙’ a h`am r`o . ira . c f. Kh´ac v´o . i tru . `o . ng ho . . p liˆen tu . c, dˆe ˜ d`ang ch ´u . ng minh tˆo ` nta . icu ˙’ abiˆe ´ nd¯ˆo ˙’ i Fourier r`o . ira . c. Chˇa ˙’ ng ha . n trong tru . `o . ng ho . . pmˆo . t chiˆe ` u, ta c´o thˆe ˙’ ch´u . ng minh bˇa ` ng c´ach thay tru . . ctiˆe ´ p (3.2) v`ao (3.1): F (u)= 1 N N−1  x=0  N−1  r=0 F (r)e 2πirx/N e −2πiux/N  = 1 N N−1  r=0 F (r)  N−1  x=0 e 2πirx/N e −2πiux/N  = F(u). D - ˆo ` ng nhˆa ´ tth´u . c trˆen suy t `u . d¯ i ˆe ` ukiˆe . n tru . . c giao N−1  x=0 e 2πirx/N e −2πiux/N =    N nˆe ´ u r = u, 0nˆe ´ u ngu . o . . cla . i. V´ı du . 3.2.2 H`ınh 3.1 l`a 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 C´ac t´ınh chˆa ´ t Tru . ´o . chˆe ´ t ta c´o D - i . nh l´y 3.3.1 [Raleigh] Gia ˙’ su . ˙’ F l`a biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a f. Khi d¯´o  +∞ −∞  +∞ −∞ f 2 (x, y)dxdy =  +∞ −∞  +∞ −∞ F (u, v) 2 dudv. 47 . r`o . ira . cchobo . ˙’ i F(f)(u, v):= 1 MN M 1  x=0 N 1  y=0 f(x, y)e −2πi ( ux M + vy N ) , v´o . i u =0, 1, ,M − 1, v`a v =0, 1, ,N − 1, v`a F 1 (F )(x, y):= M 1  u=0 N 1  v=0 F (u, v)e 2πi ( ux M + vy N ) , trong. sau              F (u, v)= 1 N N 1  x=0 N 1  y=0 f(x, y)e −2πi ( ux N + vy N ) , f(x, y)= 1 N N 1  u=0 N 1  v=0 F (u, v)e 2πi ( ux N + v y N ) , trong d¯´o u, x =0, 1, ,N − 1, v`a y,v =0, 1, ,N − 1. Ho`an. cu ˙’ a f l`a F (0) = 1 4 3  x=0 f(x)e 0 =3.25; F (1) = 1 4 3  x=0 f(x)e −2πix/4 = 1 4 (−2+i); F (2) = 1 4 3  x=0 f(x)e −2πi2x/4 = − 1 4 ; F (3) = 1 4 3  x=0 f(x)e −2πi3x/4 = − 1 4 (2 + i). Trong