Xử lý ảnh số - Khôi phục ảnh part 2 doc

v`a g l`a c´ac vector cˆo . t trong R 6 v`a H l`a ma trˆa . n vuˆong cˆa ´ p6: H =          h e (0) h e (5) h e (4) ··· h e (1) h e (1) h e (0) h e (5) ··· h e (2) h e (2) h e (1) h e (0) ··· h e (3) . . . h e (5) h e (4) h e (3) ··· h e (0)          . Nhu . ng h e (x)=0v´o . i x =3, 4, 5v`ah e (x)=h(x)v´o . i x =0, 1, 2nˆen H =            h(0) h(2) h(1) h(1) h(0) h(2) h(2) h(1) h(0) h(2) h(1) h(0) h(2) h(1) h(0) h(2) h(1) h(0)            trong d¯´o tˆa ´ tca ˙’ c´ac phˆa ` ntu . ˙’ khˆong d¯u . o . . cviˆe ´ t ra bˇa ` ng khˆong. Bˆay gi`o . x´et tru . `o . ng ho . . p hai chiˆe ` u: hai a ˙’ nh sˆo ´ f(x, y)v`ah(x, y)c´ok´ıch thu . ´o . c A ×B v`a C × D tu . o . ng ´u . ng. Cho . n M,N thoa ˙’ m˜an M ≥ A + B −1,N≥ C + D − 1. Ta mo . ˙’ rˆo . ng k´ıch thu . ´o . c c´ac a ˙’ nh bˇa ` ng c´ach d¯ˇa . t f e (x, y):=    f(x, y)nˆe ´ u0≤ x ≤ A − 1, v`a 0 ≤ y ≤ B −1, 0nˆe ´ u A ≤ x ≤ M −1, hoˇa . c B ≤ y ≤ N − 1, v`a h e (x, y):=    h(x, y)nˆe ´ u0≤ x ≤ C − 1, v`a 0 ≤ y ≤ D −1, 0nˆe ´ u C ≤ x ≤ M −1, hoˇa . c D ≤ y ≤ N − 1, Ch´ung ta c˜ung gia ˙’ thiˆe ´ t c´ac h`am f e (x, y)v`ah e (x, y) tuˆa ` n ho`an v´o . ichuk`yM v`a N theo hai hu . ´o . ng x v`a y tu . o . ng ´u . ng. Nhˇa ´ cla . i g e (x, y)= M−1  m=0 N−1  n=0 f e (m, n)h e (x −m, y −n), v´o . i x =0, 1, ,M − 1, v`a y =0, 1, ,N − 1. H`am g e (x, y) tuˆa ` n ho`an v´o . ic`ung chu k`ynhu . f e (x, y)v`ah e (x, y) . D - ˆe ˙’ ho`an thiˆe . nmˆoh`ınh suy gia ˙’ mchˆa ´ tlu . o . . ng r`o . ira . c, cˆa ` n thˆem nhiˆe ˜ u η e (x, y) v`ao h`am g e (x, y); t ´u . cl`a g e (x, y)= M−1  m=0 N−1  n=0 f e (m, n)h e (x −m, y −n)+η e (x, y) , (5.4) 116 v´o . i x =0, 1, ,M −1, v`a y =0, 1, ,N − 1. K´yhiˆe . u f, g v`a n l`a c´ac vector cˆo . t k´ıch thu . ´o . c MN nhˆa . nd¯u . o . . cbˇa ` ng c´ach sˇa ´ pxˆe ´ p la . i c´ac h`ang cu ˙’ a c´ac ma ˙’ ng tu . o . ng ´u . ng f e ,g e v`a η e v´o . i k´ıch thu . ´o . c M × N. Chˇa ˙’ ng ha . n, N phˆa ` ntu . ˙’ d¯ ˆa ` u tiˆen cu ˙’ a f tu . o . ng ´u . ng c´ac phˆa ` ntu . ˙’ trong h`ang d¯ˆa ` u tiˆen cu ˙’ a f e (x, y); N phˆa ` ntu . ˙’ kˆe ´ tiˆe ´ ptu . o . ng ´u . ng h`ang th´u . hai, v`a vˆan vˆan. Khi d¯´o biˆe ˙’ uth´u . c (5.4) c´o thˆe ˙’ viˆe ´ tdu . ´o . ida . ng ma trˆa . n g=Hf+n, (5.5) trong d¯´o H l`a ma trˆa . n vuˆong k´ıch thu . ´o . c MN. Ma trˆa . n n`ay c´o thˆe ˙’ phˆan hoa . ch th`anh M 2 ma trˆa . n con v´o . imˆo ˜ i ma trˆa . nconk´ıch thu . ´o . c N × N v`a d¯u . o . . csˇa ´ p theo th´u . tu . . H =            H 0 H M−1 H M−2 H 1 H 1 H 0 H M−1 H 2 H M−1 H M−2 H M−3 H 0            , v´o . imˆo ˜ i phˆa ` ntu . ˙’ H j cu ˙’ a h`ang th´u . j x´ac d¯i . nh bo . ˙’ i h`am mo . ˙’ rˆo . ng h e (x, y): H j =              h e (j, 0) h e (j, N − 1) h e (j, N − 2) h e (j, 1) h e (j, 1) h e (j, 0) h e (j, N − 1) h e (j, 2) h e (j, 2) h e (j, 1) h e (j, N − 2) h e (j, 3) h e (j, N −1) h e (j, N − 2) h e (j, N − 3) h e (j, 0)              . Nhˆa . n x´et rˇa ` ng, H j l`a ma trˆa . n chu tr`ınh v`a c´ac khˆo ´ icu ˙’ a H d¯ u . o . . c d¯´anh chı ˙’ sˆo ´ theo ´y ngh˜ıa chu tr`ınh. Do d¯´o ma trˆa . n H go . il`ama trˆa . n chu tr`ınh khˆo ´ i. Hˆa ` uhˆe ´ t c´ac kˆe ´ t qua ˙’ trong phˆa ` n sau tˆa . p trung v`ao mˆo h`ınh suy gia ˙’ mchˆa ´ tlu . o . . ng da . ng (5.5). Ch´u´yrˇa ` ng biˆe ˙’ uth´u . c n`ay du . . a trˆen gia ˙’ thiˆe ´ t tuyˆe ´ n t´ınh v`a bˆa ´ tbiˆe ´ nvi . tr´ı cu ˙’ amˆoh`ınh xu . ˙’ l´y. Mu . c tiˆeu l`a t`ım u . ´o . clu . o . . ng f(x, y)du . . a trˆen h`am g(x, y)v´o . isu . . hiˆe ˙’ ubiˆe ´ tvˆe ` h(x, y)v`aη(x, y). N´oi c´ach kh´ac cˆa ` nt`ımu . ´o . clu . o . . ng f du . . a trˆen g, n v`a H. Mˇa . cd`ud¯o . n gia ˙’ n, nhu . ng viˆe . c x´ac d¯i . nh f t`u . Phu . o . ng tr`ınh (5.5) l`a rˆa ´ tph´u . cta . p trong tru . `o . ng ho . . p k´ıch thu . ´o . cl´o . n. Chˇa ˙’ ng ha . n, nˆe ´ u M = N = 512 th`ı H c´o k´ıch thu . ´o . c 262144. Do d¯´o d¯ˆe ˙’ x´ac d¯i . nh f ta cˆa ` n gia ˙’ ihˆe . 262144 phu . o . ng tr`ınh tuyˆe ´ nt´ınh. Tuy nhiˆen, su . ˙’ du . ng t´ınh chˆa ´ t chu tr`ınh cu ˙’ a ma trˆa . n H c´o thˆe ˙’ gia ˙’ m d¯´ang kˆe ˙’ khˆo ´ ilu . o . . ng t´ınh to´an. 117 5.2 Ch´eo ho´a ma trˆa . n chu tr`ınh v`a ma trˆa . n khˆo ´ i chu tr`ınh Phˆa ` n n`ay tr`ınh b`ay phu . o . ng ph´ap hiˆe . u qua ˙’ gia ˙’ iPhu . o . ng tr`ınh (5.5) bˇa ` ng c´ach ch´eo ho´a ma trˆa . n H.D - ˆe ˙’ d¯ o . n gia ˙’ n, ch´ung ta bˇa ´ td¯ˆa ` uv´o . i ma trˆa . n chu tr`ınh v`a sau d¯´o s˜e x´et ma trˆa . n khˆo ´ i chu tr`ınh. 5.2.1 Ma trˆa . n chu tr`ınh X´et ma trˆa . n chu tr`ınh cˆa ´ p M × M da . ng H =              h e (0) h e (M − 1) h e (M − 2) ··· h e (1) h e (1) h e (0) h e (M − 1) ··· h e (2) h e (2) h e (1) h e (0) ··· h e (3) . . . . . . . . . . . . h e (M − 1) h e (M − 2) h e (M − 3) ··· h e (0)              . D - ˇa . t λ(k):= M  j=1 h e (M − j) exp  2πi M jk  v`a w(k):=          1 exp  2πi M k  exp  2πi M 2k  . . . exp  2πi M (M − 1)k           v´o . i k =0, 1, ,M − 1. Dˆe ˜ d`ang kiˆe ˙’ m tra rˇa ` ng Hw(k)=λ(k)w(k),k=0, 1, ,M −1. N´oi c´ach kh´ac, ma trˆa . n chu tr`ınh H c´o M gi´a tri . riˆeng λ(k)tu . o . ng ´u . ng v´o . i c´ac vector riˆeng w(k),k =0, 1, ,M − 1. Ho . nn˜u . a, c´ac vector riˆeng n`ay l`a tru . . c giao, t´u . cl`a w(k),w(k  ) =0, v´o . imo . i k = k  ,k,k  =0, 1, ,M − 1. 118 Gia ˙’ su . ˙’ W := (W (k,j)) l`a ma trˆa . n vuˆong cˆa ´ p M v´o . i c´ac cˆo . t l`a c´ac vector riˆeng cu ˙’ a ma trˆa . n chu tr`ınh H;t´u . cl`a W(k,j) := exp  2πi M kj  , v´o . i k,j =0, 1, ,M − 1. Dˆe ˜ thˆa ´ yrˇa ` ng W l`a ma trˆa . n tru . . c giao v`a do d¯´o tˆo ` nta . ima trˆa . n nghi . ch d¯a ˙’ o W −1 := (W −1 (k,j)) v´o . i W −1 (k,j)= 1 M exp  − 2πi M kj  . Suy ra H = WDW −1 , (5.6) trong d¯´o D l`a ma trˆa . n vuˆong cˆa ´ p M da . ng d¯u . `o . ng ch´eo, c´o c´ac phˆa ` ntu . ˙’ trˆen d¯u . `o . ng ch´eo D(k, k)=λ(k) . 5.2.2 Ma trˆa . n chu tr`ınh khˆo ´ i Ma trˆa . nbiˆe ´ nd¯ˆo ˙’ id¯ˆe ˙’ ch´eo ho´a c´ac khˆo ´ i chu tr`ınh d¯u . o . . c xˆay du . . ng nhu . sau. D - ˇa . t w M (k,j) := exp  2πi M kj  v`a w N (k,j) := exp  2πi N kj  . Ta d¯i . nh ngh˜ıa W l`a mˆo . t ma trˆa . n vuˆong cˆa ´ p MN × MN gˆo ` m M 2 khˆo ´ i, mˆo ˜ i khˆo ´ il`a mˆo . t ma trˆa . n vuˆong cˆa ´ p N, khˆo ´ inˇa ` m trˆen h`ang m cˆo . t n x´ac d¯i . nh bo . ˙’ i W(k,m):=w M (k,m)W N , v´o . i k,m =0, 1, ,M − 1v`aW N l`a ma trˆa . n vuˆong cˆa ´ p N v´o . i c´ac phˆa ` ntu . ˙’ W N (k,n)=w N (k,n) v´o . i k,n =0, 1, 2, ,N − 1. Ma trˆa . n nghi . ch d¯a ˙’ o W −1 c˜ung l`a ma trˆa . n vuˆong k´ıch thu . ´o . c MN ×MN gˆo ` m M 2 khˆo ´ i, mˆo ˜ i khˆo ´ i l`a ma trˆa . n vuˆong cˆa ´ p N. Khˆo ´ io . ˙’ h`ang m cˆo . t n cu ˙’ a W −1 =(W −1 (m, n)) x´ac d¯i . nh bo . ˙’ i W −1 (m, n):= 1 M w −1 M (m, n)W −1 N , 119 trong d¯´o w −1 M (m, n) = exp  − 2πi M mn  , v`a W −1 N := 1 N  w −1 N (k,j)  k,j=0,1, ,N−1 l`a ma trˆa . n vuˆong cˆa ´ p N v´o . i w −1 N (k,j) = exp  − 2πi N kj  . T`u . c´ac kˆe ´ t qua ˙’ cu ˙’ a Phˆa ` n 5.2.1 v`a nˆe ´ u H l`a ma trˆa . n khˆo ´ i chu tr`ınh th`ı c´o thˆe ˙’ chı ˙’ ra rˇa ` ng D = W −1 HW l`a ma trˆa . n ch´eo ho´a cu ˙’ a H trong d¯´o c´ac phˆa ` ntu . ˙’ trˆen d¯u . `o . ng ch´eo D(k,k) c´o liˆen quan d¯ ˆe ´ nbiˆe ´ nd¯ˆo ˙’ i Fourier r`o . ira . ccu ˙’ a h`am th´ac triˆe ˙’ n h e (x, y) trong Phˆa ` n 5.1.3. Suy ra H = WDW −1 . (5.7) Ho . nn˜u . a, ma trˆa . n chuyˆe ˙’ nvi . cu ˙’ a H l`a H t = W ¯ DW −1 trong d¯´o ¯ D l`a ma trˆa . n liˆen ho . . pph´u . ccu ˙’ a D. 5.2.3 Hiˆe . u qua ˙’ cu ˙’ a ch´eo ho´a ma trˆa . n trong mˆo h`ınh suy gia ˙’ m chˆa ´ tlu . o . . ng Ma trˆa . n H trong mˆo h`ınh 1D r`o . ira . c (5.3) l`a ma trˆa . n chu tr`ınh. V`ıvˆa . y n´o c´o thˆe ˙’ biˆe ˙’ u diˆe ˜ nda . ng (5.6). Khi d¯´o (5.3) tro . ˙’ th`anh g = WDW −1 f. Suy ra W −1 g = DW −1 f. (5.8) Nhu . ng dˆe ˜ d`ang kiˆe ˙’ m tra rˇa ` ng vector cˆo . t W −1 f thuˆo . c R M c´o phˆa ` ntu . ˙’ o . ˙’ h`ang th´u . k F (k):= 1 M M−1  j=0 f e (j) exp  2πi M kj  ,k=0, 1, ,M − 1, 120 . 5v`ah e (x)=h(x)v´o . i x =0, 1, 2nˆen H =            h(0) h (2) h(1) h(1) h(0) h (2) h (2) h(1) h(0) h (2) h(1) h(0) h (2) h(1) h(0) h (2) h(1) h(0)            trong d¯´o tˆa ´ tca ˙’ c´ac. h e (0)              . D - ˇa . t λ(k):= M  j=1 h e (M − j) exp  2 i M jk  v`a w(k):=          1 exp  2 i M k  exp  2 i M 2k  . . . exp  2 i M (M − 1)k           v´o . i. thu . ´o . cl´o . n. Chˇa ˙’ ng ha . n, nˆe ´ u M = N = 5 12 th`ı H c´o k´ıch thu . ´o . c 26 2144. Do d¯´o d¯ˆe ˙’ x´ac d¯i . nh f ta cˆa ` n gia ˙’ ihˆe . 26 2144 phu . o . ng tr`ınh tuyˆe ´ nt´ınh. Tuy nhiˆen,