Xử lý ảnh số - Nâng cao chất lượng ảnh part 1 pdf

Chu . o . ng 4 N ˆ ANG CAO CH ˆ A ´ TLU . O . . NG A ˙’ NH Nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh (image enhancement) l`a xu . ˙’ l´y a ˙’ nh d¯ˆe ˙’ cho a ˙’ nh d¯ˆa ` u ra th´ıch ho . . pho . n so v´o . ia ˙’ nh gˆo ´ c nhˇa ` mmˆo . tsˆo ´ ´u . ng du . ng d¯ˇa . cbiˆe . t. C´o hai c´ach tiˆe ´ pcˆa . nd¯ˆe ˙’ nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh l`a (1) C´ac phu . o . ng ph´ap miˆe ` n khˆong gian; v`a (2) C´ac phu . o . ng ph´ap miˆe ` ntˆa ` nsˆo ´ . Trong miˆe ` n khˆong gian, nguyˆen tˇa ´ c chung l`a su . ˙’ du . ng c´ac gi´a tri . x´am cu ˙’ a pixel trong a ˙’ nh. Xu . ˙’ l´y trong miˆe ` ntˆa ` nsˆo ´ du . . a trˆen phu . o . ng ph´ap biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ amˆo . t a ˙’ nh. K˜y thuˆa . t nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh du . . a trˆen co . so . ˙’ kˆe ´ tho . . p nhiˆe ` uphu . o . ng ph´ap cu ˙’ a hai miˆe ` n khˆong gian v`a tˆa ` nsˆo ´ . 4.1 Co . so . ˙’ cu ˙’ anˆang cao chˆa ´ tlu . o . . ng a ˙’ nh C´ac phu . o . ng ph´ap nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh trong chu . o . ng n`ay du . . a trˆen c´ac k˜y thuˆa . t miˆe ` n khˆong gian hoˇa . cmiˆe ` ntˆa ` nsˆo ´ .Mu . cd¯´ıch cu ˙’ a phˆa ` n n`ay cung cˆa ´ pnh˜u . ng ´y tu . o . ˙’ ng co . ba ˙’ nv`amˆo ´ iliˆenhˆe . gi˜u . a hai c´ach tiˆe ´ pcˆa . n n`ay. 69 4.1.1 Phu . o . ng ph´ap miˆe ` n khˆong gian C´ac phu . o . ng ph´ap miˆe ` n khˆong gian t´ac d¯ˆo . ng tru . . ctiˆe ´ plˆen tˆa . p c´ac pixel trong a ˙’ nh. C´ac h`am xu . ˙’ l´y a ˙’ nh trong miˆe ` n khˆong gian d¯u . o . . cbiˆe ˙’ udiˆe ˜ nbo . ˙’ i g(x, y):=T [f(x, y)], trong d¯´o f(x, y)l`aa ˙’ nh v`ao, g(x, y)l`aa ˙’ nh ra v`a T l`a to´an tu . ˙’ t´ac d¯ˆo . ng lˆen h`am a ˙’ nh f. To´an tu . ˙’ T c´o thˆe ˙’ t´ac d¯ˆo . ng trˆen nhiˆe ` ua ˙’ nh v`ao, chˇa ˙’ ng ha . nnhu . cˆo . ng c´ac gi´a tri . x´am cu ˙’ a c´ac pixel trong tˆa . pa ˙’ nh v`ao d¯ˆe ˙’ gia ˙’ m nhiˆe ˜ u. Ta c˜ung c´o thˆe ˙’ t´ınh hiˆe . ucu ˙’ a hai h`am a ˙’ nh f( x, y)v`ah(x, y) g(x, y):=f( x, y) −h(x, y) nhˆa . nd¯u . o . . cbˇa ` ng c´ach t´ınh hiˆe . ugi˜u . atˆa ´ tca ˙’ c´ac cˇa . p pixel tu . o . ng ´u . ng cu ˙’ a f v`a h. Tr`u . a ˙’ nh c´o mˆo . tsˆo ´ ´u . ng du . ng quan tro . ng trong phˆan d¯oa . na ˙’ nh v`a nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh. C´ach tiˆe ´ pcˆa . nch´ınh d¯u . o . . cd`ung trong lˆan cˆa . n (x´ac d¯i . nh tru . ´o . c) cu ˙’ a(x, y) l`a su . ˙’ du . ng mˆo . tv`ung a ˙’ nh con h`ınh ch˜u . nhˆa . t tˆam d¯ˇa . tta . i(x, y). Tˆam cu ˙’ aa ˙’ nh con n`ay d¯u . o . . c di chuyˆe ˙’ n theo c´ac pixel (x, y) (kho . ˙’ id¯ˆa ` ut`u . g´oc trˆen bˆen tr´ai) v`a ´ap du . ng to´an tu . ˙’ T lˆen d¯iˆe ˙’ m(x, y). Da . ng d¯o . n gia ˙’ n nhˆa ´ tcu ˙’ a T khi lˆan cˆa . n c´o k´ıch thu . ´o . c1× 1. Trong tru . `o . ng ho . . p n`ay, g ch ı ˙’ phu . thuˆo . c v`ao gi´a tri . cu ˙’ a f ta . i(x, y)v`aT tro . ˙’ th`anh ph´ep biˆe ´ nd¯ˆo ˙’ im´u . c x´am s = T (r), trong d¯´o k´y hiˆe . u r, s l`a c´ac gi´a tri . x´am cu ˙’ a f v`a g ta . ivi . tr´ı (x, y). V`ı nˆang cao chˆa ´ t lu . o . . ng a ˙’ nh ta . imˆo . td¯iˆe ˙’ m n`ao d¯´o trong a ˙’ nh chı ˙’ phu . thuˆo . cv`aom´u . c x´am ta . id¯iˆe ˙’ m d¯´o, nˆen c´ach tiˆe ´ pcˆa . n n`ay d¯u . o . . cgo . il`axu . ˙’ l´y d¯iˆe ˙’ m. V´ı d u . 4.1.1 (i) T (r) trong H`ınh 4.1(a) c´o t´ac du . ng ta . omˆo . ta ˙’ nh c´o d¯ˆo . tu . o . ng pha ˙’ n cao ho . na ˙’ nh gˆo ´ cbˇa ` ng c´ach l`am d¯en c´ac m´u . c <m,v`a l`am s´ang lˆen c´ac m´u . c >mtrong a ˙’ nh gˆo ´ c. K˜y thuˆa . t n`ay d¯u . o . . cgo . il`ad˜an d¯ˆo . tu . o . ng pha ˙’ n. (ii) T (r) trong H`ınh 4.1(b) c´o t´ac du . ng ta . omˆo . ta ˙’ nh nhi . phˆan. C´ac lˆan cˆa . nl´o . nho . nc˜ung thu . `o . ng d¯u . o . . csu . ˙’ du . ng nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh. Gi´a tri . cu ˙’ a g ta . i(x, y)d¯u . o . . c x´ac d¯i . nh thˆong qua c´ac gi´a tri . cu ˙’ a f trong lˆan cˆa . ncu ˙’ a(x, y). Mˆo . t trong nh ˜u . ng nguyˆen tˇa ´ c d¯´o du . . a trˆen co . so . ˙’ cu ˙’ a mˇa . tna . (mask) (c`on go . il`acu . ˙’ a 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s = T (r) L −1 r L − 1 m (0, 0) (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s = T (r) L − 1 r L − 1 m (0, 0) (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H`ınh 4.1: D - ˆo ` thi . c´ac h`am biˆe ´ nd¯ˆo ˙’ im´u . c x´am d¯ˆe ˙’ nˆang cao d¯ˆo . tu . o . ng pha ˙’ n. sˆo ˙’ (window) hoˇa . c lo . c (filter)). Vˆe ` co . ba ˙’ n, mˆo . tmˇa . tna . l`a mˆo . tma ˙’ ng hai chiˆe ` uc´ok´ıch thu . ´o . c nho ˙’ (chˇa ˙’ ng ha . n, k´ıch thu . ´o . c3× 3), m`a c´ac hˆe . sˆo ´ d¯ u . o . . ccho . nd¯ˆe ˙’ ph´at hiˆe . n c´ac t´ınh chˆa ´ t d¯˜a cho cu ˙’ aa ˙’ nh. Chˇa ˙’ ng ha . n, gia ˙’ su . ˙’ a ˙’ nh f c´o cu . `o . ng d¯ˆo . s´ang hˇa ` ng ch´u . amˆo . t d¯ i ˆe ˙’ m cˆo lˆa . p (cu . `o . ng d¯ˆo . s´ang ta . i d¯´o kh´ac nˆe ` n). D - iˆe ˙’ m n`ay c´o thˆe ˙’ bi . x´oa bˇa ` ng c´ach su . ˙’ du . ng mˇa . tna . W :=    −1 −1 −1 −18−1 −1 −1 −1    . Thuˆa . t to´an nhu . sau: Tˆam cu ˙’ amˇa . tna . (g´an nh˜an 8) d¯u . o . . c di chuyˆe ˙’ n xung quanh a ˙’ nh. Ta . imˆo ˜ ivi . tr´ı (x, y) trong a ˙’ nh, ta nhˆan mˆo ˜ i gi´a tri . x´am cu ˙’ a pixel d¯u . o . . cch´u . a trong v`ung mˇa . tna . v´o . i c´ac hˆe . sˆo ´ cu ˙’ amˇa . tna . ;t´u . c l`a pixel tˆam cu ˙’ amˇa . tna . d¯ u . o . . c nhˆan v´o . i8, trong khi 8 pixel lˆan cˆa . nd¯u . o . . c nhˆan v´o . i −1. D - ´a p ´u . ng cu ˙’ amˇa . tna . ta . i(x, y)bˇa ` ng tˆo ˙’ ng c´ac t´ıch n`ay. Nˆe ´ utˆa ´ tca ˙’ c´ac pixel trong v`ung c´o c`ung gi´a tri . , d¯´ap ´u . ng bˇa ` ng khˆong. Mˇa . t kh´ac, nˆe ´ u tˆam cu ˙’ amˇa . tna . d¯ ˇa . tta . id¯iˆe ˙’ m cˆo lˆa . p, d¯´ap ´u . ng s˜e kh´ac khˆong. Nˆe ´ ud¯iˆe ˙’ m cˆo lˆa . pd¯ˇa . tgˆa ` n (nhu . ng kh´ac) tˆam, d¯´ap ´u . ng c˜ung kh´ac khˆong, nhu . ng gi´a tri . tuyˆe . td¯ˆo ´ i cu ˙’ a d¯´ap ´u . ng s˜e yˆe ´ uho . n. C´ac d¯´ap ´u . ng yˆe ´ uho . n n`ay s˜e bi . khu . ˙’ bˇa ` ng c´ach so s´anh v´o . i ngu . ˜o . ng n`ao d¯´o. Nhu . trong H`ınh 4.2, nˆe ´ u w 1 ,w 2 , ,w 9 l`a c´ac hˆe . sˆo ´ cu ˙’ amˇa . tna . v`a kha ˙’ o s´at 8−lˆan cˆa . ncu ˙’ a(x, y), ta c´o thˆe ˙’ tˆo ˙’ ng qu´at ho´a thuˆa . t to´an trˆen nhu . viˆe . c thu . . chiˆe . n ph´ep to´an 71 . . . z 1 z 2 z 3 ··· z 4 z 5 z 6 ··· z 7 z 8 z 9 . . . (a) w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 (b) H`ınh 4.2: sau: T [f(x, y)] := w 1 f( x −1,y− 1) + w 2 f( x −1,y)+w 3 f( x −1,y+ 1)+ w 4 f( x, y − 1) + w 5 f( x, y)+w 6 f( x, y + 1)+ w 7 f( x +1,y−1) + w 8 f( x +1,y)+w 9 f( x +1,y+1) (4.1) trˆen lˆan cˆa . n3×3cu ˙’ a(x, y). C´ac mˇa . tna . k´ıch thu . ´o . cl´o . nho . nd¯u . o . . c ´ap du . ng tu . o . ng tu . . . Ch´u´yrˇa ` ng, trong biˆe ˙’ uth´u . c (4.1) viˆe . c thay d¯ˆo ˙’ i c´ac hˆe . sˆo ´ cu ˙’ amˇa . tna . s˜e thay d¯ˆo ˙’ i ch´u . c nˇang cu ˙’ amˇa . tna . . C´ac phu . o . ng ph´ap nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh du . . a v`ao mˇa . tna . thu . `o . ng go . il`axu . ˙’ l´y mˇa . tna . hoˇa . c lo . c. Trong c´ac phˆa ` n sau ta s˜e x´et c´ac mˇa . tna . nhˇa ` m phu . chˆo ` ia ˙’ nh, phˆan d¯oa . na ˙’ nh 4.1.2 Phu . o . ng ph´ap miˆe ` ntˆa ` nsˆo ´ Co . so . ˙’ cu ˙’ a c´ac phu . o . ng ph´ap xu . ˙’ l´y a ˙’ nh trong miˆe ` ntˆa ` nsˆo ´ du . . a trˆen d¯i . nh l´y t´ıch chˆa . p. X´et to´an tu . ˙’ bˆa ´ tbiˆe ´ nvi . tr´ı, tuyˆe ´ n t´ınh Φtu . o . ng ´u . ng v´o . i h`am phˆan t´an d¯iˆe ˙’ m h Φ (x, y) t´ac d¯ˆo . ng trˆen a ˙’ nh f. Khi d¯´o a ˙’ nh d¯ˆa ` ura g(x, y) := [Φ(f)](x, y)=h Φ (x, y) ∗ f(x, y). Do d¯´o, theo d¯i . nh l´y t´ıch chˆa . p: G(u, v)=H( u, v)F (u, v), trong d¯´o G, H, F l`a c´ac biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a g,h Φ ,f tu . o . ng ´u . ng. Vˆa ´ nd¯ˆe ` l`a v´o . i h`am a ˙’ nh f d¯˜a cho, mu . c tiˆeu l`a cho . n H d¯ ˆe ˙’ d¯ u . o . . ca ˙’ nh mong muˆo ´ n g(x, y)=F −1 [H(u, v)F (u, v)]. (4.2) 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f( x, y) g(x, y)h(x, y) (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F (u, v) G(u, v)H(u, v) (b) H`ınh 4.3: Thao t´ac cu ˙’ ahˆe . thˆo ´ ng tuyˆe ´ n t´ınh. Trong (a) t´ın hiˆe . u ra l`a t´ıch chˆa . pcu ˙’ a h(x, y)v´o . it´ınhiˆe . u v`ao. Trong (b) t´ınh hiˆe . u ra l`a t´ıch cu ˙’ a H(u, v)v´o . it´ınhiˆe . u v`ao. V´ıdu . c´ac d¯u . `o . ng biˆen trong a ˙’ nh f(x, y)d¯u . o . . c l`am nˆo ˙’ ibˇa ` ng c´ach d`ung h`am H(u, v) l`am nˆo ˙’ i c´ac th`anh phˆa ` n c´o tˆa ` nsˆo ´ cao cu ˙’ a F(u, v). Trong H`ınh 4.3(a), h`am h(x, y)d¯ˇa . c tru . ng cho hˆe . thˆo ´ ng m`a ch´u . c nˇang cu ˙’ a n´o l`a ta . o ra t´ın hiˆe . u g(x, y)t`u . t´ın hiˆe . u v`ao f(x, y). Hˆe . thˆo ´ ng thu . . chiˆe . n t´ıch chˆa . pcu ˙’ a h(x, y) v´o . ia ˙’ nh v`ao f(x, y) v`a xuˆa ´ trakˆe ´ t qua ˙’ . Theo d¯i . nh l´y t´ıch chˆa . p, c´o thˆe ˙’ thu . . chiˆe . ntiˆe ´ n tr`ınh n`ay theo c´ach kh´ac: nhˆan F (u, v)v´o . i H(u, v)d¯ˆe ˙’ c´o G(u, v) v`a sau d¯´o biˆe ´ nd¯ˆo ˙’ i Fourier ngu . o . . c. Gia ˙’ su . ˙’ rˇa ` ng h`am h(x, y)chu . abiˆe ´ tv`ach´ung ta ´ap du . ng mˆo . t h`am xung d¯o . nvi . (t ´u . cl`amˆo . td¯iˆe ˙’ m s´ang) lˆen hˆe . thˆo ´ ng. Biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a xung d¯o . nvi . bˇa ` ng 1 nˆen G(u, v)=H(u, v). Do d¯´o biˆe ´ nd¯ˆo ˙’ i ngu . o . . ccu ˙’ a G(u, v)l`ah(x, y). D - ˆay l`a mˆo . tkˆe ´ t qua ˙’ d¯˜a biˆe ´ t trong l´y thuyˆe ´ thˆe . thˆo ´ ng tuyˆe ´ n t´ınh: Mˆo . thˆe . thˆo ´ ng tuyˆe ´ n t´ınh bˆa ´ tbiˆe ´ nvi . tr´ı ho`an to`an d¯u . o . . c x´ac d¯i . nh bo . ˙’ i d¯´ap ´u . ng xung cu ˙’ ahˆe . thˆo ´ ng d¯ˆo ´ iv´o . imˆo . t xung. T´u . c l`a, biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a h`am xung d¯o . nvi . ´ap du . ng d¯ˆo ´ iv´o . ihˆe . thˆo ´ ng tuyˆe ´ n t´ınh bˆa ´ tbiˆe ´ n vi . tr´ı ch´ınh l`a h`am H( u, v). Ta c˜ung c´o thˆe ˙’ t´ac d¯ˆo . ng xung tru . . ctiˆe ´ pd¯ˆe ˙’ c´o t´ın hiˆe . u ra h(x, y). V`ı l´y do n`ay trong l´y thuyˆe ´ thˆe . thˆo ´ ng tuyˆe ´ n t´ınh, biˆe ´ nd¯ˆo ˙’ i ngu . o . . c h(x, y) cu ˙’ a h`am chuyˆe ˙’ nd¯ˆo ˙’ ihˆe . thˆo ´ ng go . il`ad¯´ap ´u . ng xung. Trong quang ho . c, biˆe ´ nd¯ˆo ˙’ i ngu . o . . c h(x, y)cu ˙’ a h`am biˆe ´ nd¯ˆo ˙’ i quang ho . cgo . il`ah`am phˆan t´an d¯iˆe ˙’ m. Viˆe . cd¯ˇa . t tˆen du . . a trˆen a ˙’ nh hu . o . ˙’ ng quang ho . co . ˙’ d¯´o xung tu . o . ng ´u . ng v´o . id¯iˆe ˙’ m s´ang v`a hˆe . thˆo ´ ng quang ho . c pha ˙’ n´u . ng l`am nho`e (phˆan t´an) d¯iˆe ˙’ m; m´u . cd¯ˆo . nho`e x´ac d¯i . nh bo . ˙’ i c´ac th`anh phˆa ` n quang ho . c. Do vˆa . y h`am biˆe ´ nd¯ˆo ˙’ i quang ho . c v`a h`am phˆan t´an d¯iˆe ˙’ m l`a c´ac biˆe ´ nd¯ˆo ˙’ i Fourier cu ˙’ a nhau. Mˆo ´ i quan hˆe . n`ay s˜e d¯u . o . . c kha ˙’ o s´at trong Phˆa ` n 4.3. Ch´u´yrˇa ` ng, biˆe ˙’ uth´u . c (4.2) ch´ınh l`a xu . ˙’ l´y miˆe ` n khˆong gian tu . o . ng tu . . viˆe . csu . ˙’ du . ng c´ac mˇa . tna . x´et trong phˆa ` n tru . ´o . c. V`ı l´y do n`ay, c´ac mˇa . tna . khˆong gian thu . `o . ng 73 . d¯ˆo . s´ang ta . i d¯´o kh´ac nˆe ` n). D - iˆe ˙’ m n`ay c´o thˆe ˙’ bi . x´oa bˇa ` ng c´ach su . ˙’ du . ng mˇa . tna . W :=    1 1 1 18 1 1 1 1    . Thuˆa . t to´an nhu . sau: Tˆam. to´an 71 . . . z 1 z 2 z 3 ··· z 4 z 5 z 6 ··· z 7 z 8 z 9 . . . (a) w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 (b) H`ınh 4.2: sau: T [f(x, y)] := w 1 f( x 1, y− 1) + w 2 f( x 1, y)+w 3 f( x 1, y+ 1) + w 4 f(. + w 2 f( x 1, y)+w 3 f( x 1, y+ 1) + w 4 f( x, y − 1) + w 5 f( x, y)+w 6 f( x, y + 1) + w 7 f( x +1, y 1) + w 8 f( x +1, y)+w 9 f( x +1, y +1) (4 .1) trˆen lˆan cˆa . n3×3cu ˙’ a(x, y). C´ac mˇa . tna . k´ıch