Xử lý ảnh số - Nâng cao chất lượng ảnh part 2 docx

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go . i l`a mˇa . tna . t´ıch chˆa . p. Tuy nhiˆen cˆa ` nch´u´yrˇa ` ng, d¯i . nh l´y t´ıch chˆa . p d¯`oi ho ˙’ i H(u, v) v`a h(x, y)c´oc`ung k´ıch thu . ´o . c. Do d¯´o, nˆe ´ u, chˇa ˙’ ng ha . n, H(u, v)c´ok´ıch thu . ´o . c 512 ×512 th`ı phu . o . ng tr`ınh (4.2) cho kˆe ´ t qua ˙’ tu . o . ng d¯u . o . ng chı ˙’ nˆe ´ u h(x, y)c˜ung c´o k´ıch thu . ´o . c 512 ×512. Nhu . d¯ u . o . . cchı ˙’ ra trong c´ac Phˆa ` n 3.3, 3.4, t´ıch chˆa . pr`o . ira . cv´o . i c´ac ma ˙’ ng l´o . n d¯ u . o . . cxu . ˙’ l´y hiˆe . u qua ˙’ ho . nnˆe ´ ud`ung phu . o . ng ph´ap miˆe ` ntˆa ` nsˆo ´ thˆong qua biˆe ´ nd¯ˆo ˙’ i ph´ep Fourier nhanh (FFT). Trong Phˆa ` n4.5ch´ung ta s˜e ta . o c´ac mˇa . tna . t´ıch chˆa . p khˆong gian k´ıch thu . ´o . c nho ˙’ m`a xˆa ´ pxı ˙’ v´o . i H(u, v) cho tru . ´o . c theo ngh˜ıa sai sˆo ´ b`ınh phu . o . ng nho ˙’ nhˆa ´ t v`a c´o thˆe ˙’ d¯ u . o . . csu . ˙’ du . ng trong xu . ˙’ l´y miˆe ` n khˆong gian. C˜ung ch´u´yrˇa ` ng, lˆo ˜ ibo . c s˜e xuˆa ´ thiˆe . n khi xu . ˙’ l´y t´ıch chˆa . pr`o . ira . cd¯ˆo ´ iv´o . i c´ac h`am khˆong tuˆa ` n ho`an c´o c´ac chu k`yd¯u . o . . ccho . n theo (3.8). Trong mˆo . ta ˙’ nh, lˆo ˜ ibo . c l`am biˆe ´ nda . ng c´ac biˆen. Tuy nhiˆen, trong thu . . ctˆe ´ c´ac lˆo ˜ i loa . i n`ay c´o thˆe ˙’ bo ˙’ qua thˆa . mch´ı khi c´ac a ˙’ nh khˆong d¯u . o . . cmo . ˙’ rˆo . ng k´ıch thu . ´o . cnhu . trong Phˆa ` n 4.5. Chˇa ˙’ ng ha . n, c´ac kˆe ´ t qua ˙’ cho trong chu . o . ng n`ay nhˆa . n d¯ u . o . . cbˇa ` ng c´ach t´ınh to´an tru . . ctiˆe ´ p FFT trˆen c´ac a ˙’ nh khˆong cˆa ` nmo . ˙’ rˆo . ng k´ıch thu . ´o . c. Cuˆo ´ ic`ung cˆa ` n nhˆa ´ nma . nh rˇa ` ng khˆong c´o l´y thuyˆe ´ ttˆo ˙’ ng qu´at vˆe ` nˆang cao chˆa ´ t lu . o . . ng a ˙’ nh. Khi mˆo . ta ˙’ nh d¯u . o . . cxu . ˙’ l´y d¯ˆe ˙’ gia ˙’ i th´ıch bˇa ` ng thi . gi´ac th`ı ngu . `o . i quan s´at l`a ngu . `o . i x´ac d¯i . nh phu . o . ng ph´ap xu . ˙’ l´y c´o tˆo ´ t hay khˆong. D - ´anh gi´a chˆa ´ tlu . o . . ng cu ˙’ aa ˙’ nh bˇa ` ng thi . gi´ac l`a rˆa ´ tchu ˙’ quan, d´o d¯´o rˆa ´ t kh´o d¯i . nh ngh˜ıa mˆo . t“a ˙’ nh tˆo ´ t”. Qu´a tr`ınh d¯´anh gi´a s˜e dˆe ˜ d`ang ho . n khi b`ai to´an xu . ˙’ l´y a ˙’ nh d¯ˆe ˙’ m´ay t´ınh c´o thˆe ˙’ hiˆe ˙’ ud¯u . o . . c. Chˇa ˙’ ng ha . n, trong b`ai to´an nhˆa . nda . ng ch˜u . viˆe ´ t, phu . o . ng ph´ap xu . ˙’ l´y a ˙’ nh tˆo ´ t nhˆa ´ t l`a phu . o . ng ph´ap d¯ˆe ˙’ cho m´ay cho kˆe ´ t qua ˙’ nhˆa . nda . ng tˆo ´ t nhˆa ´ t. Tuy nhiˆen, ngay ca ˙’ khi d¯˜a c´o mˆo . t tiˆeu chuˆa ˙’ n r˜o r`ang d¯u . o . . c ´ap du . ng trˆen b`ai to´an, ngu . `o . i phˆan t´ıch vˆa ˜ n pha ˙’ id¯ˆo ´ idiˆe . nv´o . i mˆo . tsˆo ´ thu . ˙’ nghiˆe . md¯´ung sai tru . ´o . c khi c´o thˆe ˙’ su . ˙’ du . ng mˆo . t c´ach tiˆe ´ pcˆa . nxu . ˙’ l´y a ˙’ nh n`ao d¯´o. 4.2 Phu . o . ng ph´ap xu . ˙’ l´yd¯iˆe ˙’ m Trong mu . c n`ay ch´ung ta nghiˆen c´u . u c´ac k˜y thuˆa . t nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh chı ˙’ du . . a trˆen cu . `o . ng d¯ˆo . s´ang cu ˙’ a c´ac pixel. Nhu . d¯˜a biˆe ´ t, k˜y thuˆa . txu . ˙’ l´y d¯iˆe ˙’ m l`a c´ach tiˆe ´ pcˆa . p d¯ o . n gia ˙’ n nhˆa ´ t. Ta k´y hiˆe . u r, s tu . o . ng ´u . ng l`a cu . `o . ng d¯ˆo . s´ang cu ˙’ a c´ac pixel tru . ´o . c v`a sau khi xu . ˙’ l´y. 74 4.2.1 Mˆo . tsˆo ´ ph´ep biˆe ´ nd¯ˆo ˙’ icu . `o . ng d¯ˆo . s´ang A ˙’ nh ˆam ba ˙’ n C´ac a ˙’ nh ˆam ba ˙’ nh˜u . u ´ıch trong mˆo . tsˆo ´ ´u . ng du . ng, chˇa ˙’ ng ha . nnhu . yho . c. A ˙’ nh ˆam ba ˙’ n d¯ u . o . . c thu . . chiˆe . nbˇa ` ng ph´ep biˆe ´ nd¯ˆo ˙’ i (xem H`ınh 4.4) s := T (r)=(L − 1) −r, trong d¯´o L l`a sˆo ´ m´u . c x´am. Cu . `o . ng d¯ˆo . s´ang cu ˙’ aa ˙’ nh d¯ˆa ` u ra gia ˙’ m, trong khi cu . `o . ng d¯ˆo . s´ang cu ˙’ aa ˙’ nh d¯ˆa ` u v`ao tˇang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s = T (r) L −1 r L − 1 (0, 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H`ınh 4.4: H`am biˆe ´ nd¯ˆo ˙’ ia ˙’ nh ˆam ba ˙’ n. D˜an d¯ˆo . tu . o . ng pha ˙’ n A ˙’ nh c´o d¯ˆo . tu . o . ng pha ˙’ n thˆa ´ p do chiˆe ´ u s´ang yˆe ´ u, thiˆe ´ u pha . m vi d¯ˆo . ng trong bˆo . ca ˙’ m biˆe ´ n (imaging sensor), hoˇa . cd¯ˆe ˙’ d¯ ˆo . mo . ˙’ ˆo ´ ng k´ınh m´ay chu . pa ˙’ nh sai. Mu . cd¯´ıch cu ˙’ a d˜an d¯ ˆo . tu . o . ng pha ˙’ n l`a tˇang pha . m vi d¯ˆo . ng cu ˙’ a c´ac m´u . c x´am trong a ˙’ nh d¯u . o . . cxu . ˙’ l´y. H`ınh 4.5 minh ho . amˆo . tph´ep biˆe ´ nd¯ˆo ˙’ id¯u . o . . csu . ˙’ du . ng nhˇa ` m d˜an d¯ˆo . tu . o . ng pha ˙’ n, trong d¯´o (r 1 ,s 1 )v`a(r 2 ,s 2 ) l`a hai d¯iˆe ˙’ md¯iˆe ` u khiˆe ˙’ ncu ˙’ a ph´ep biˆe ´ nd¯ˆo ˙’ i tuyˆe ´ nt´ınht`u . ng kh´uc T. V´o . i r 1 = r 2 ,s 1 =0,s 2 = L − 1, ta c´o h`am ngu . ˜o . ng ta . o ra mˆo . ta ˙’ nh nhi . phˆan. C´ac gi´a tri . trung gian cu ˙’ a(r 1 ,s 1 )v`a(r 2 ,s 2 )ta . o ra c´ac m´u . cd¯ˆo . phˆan t´an trong gi´a tri . x´am kh´ac nhau cu ˙’ aa ˙’ nh ra, do d¯´o a ˙’ nh hu . o . ˙’ ng d¯ˆe ´ nd¯ˆo . tu . o . ng pha ˙’ ncu ˙’ aa ˙’ nh. N´oi chung, ta thu . `o . ng gia ˙’ thiˆe ´ t r 1 ≤ r 2 v`a s 1 ≤ s 2 d¯ ˆe ˙’ h`am biˆe ´ nd¯ˆo ˙’ id¯o . n tri . v`a d¯o . nd¯iˆe . u tˇang. D - iˆe ` u kiˆe . n n`ay ba ˙’ o to`an th´u . tu . . cu ˙’ a c´ac m´u . c x´am v`a do d¯´o ngˇan ng`u . anh˜u . ng gi´a tri . sai ta . o ra trong xu . ˙’ l´y a ˙’ nh. 75 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s = T (r) L −1 r L − 1 (0, 0) • (r 1 ,s 1 ) • (r 2 ,s 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H`ınh 4.5: H`am d˜an d¯ˆo . tu . o . ng pha ˙’ n. D˜an m´u . c x´am c´o thˆe ˙’ thu . . chiˆe . nbˇa ` ng c´ach biˆe ´ nd¯ˆo ˙’ i s := r − r min r max − r min (L −1), trong d¯´o r min v`a r max l`a c´ac gi´a tri . x´am cu . . ctiˆe ˙’ u v`a cu . . cd¯a . i trong a ˙’ nh gˆo ´ c. A ˙’ nh ra s˜e c´o c´ac m´u . c x´am d¯u . o . . c gi˜an gi˜u . a0v`aL −1. Thao t´ac n`ay c´o thˆe ˙’ nˆang cao chˆa ´ tlu . o . . ng a ˙’ nh, v`a thu . `o . ng d¯u . o . . cd`ung trong giai d¯oa . n cuˆo ´ icu ˙’ a qu´a tr`ınh xu . ˙’ l´y a ˙’ nh. N´en da ˙’ id¯ˆo . ng D - ˆoi khi da ˙’ id¯ˆo . ng cu ˙’ aa ˙’ nh vu . o . . t qu´a kha ˙’ nˇang cu ˙’ a thiˆe ´ tbi . hiˆe ˙’ n thi . , trong d¯´o chı ˙’ c´o phˆa ` n s´ang nhˆa ´ tcu ˙’ aa ˙’ nh l`a kha ˙’ kiˆe ´ n trˆen m`an h`ınh. D - iˆe ` un`ayc˜ung thu . `o . ng xa ˙’ y ra khi ta lˆa ´ ya ˙’ nh trˆen film. Mˆo . tphu . o . ng ph´ap hiˆe . u qua ˙’ d¯ ˆe ˙’ n´en da ˙’ id¯ˆo . ng cu ˙’ a c´ac gi´a tri . pixel l`a thu . . chiˆe . n ph´ep biˆe ´ nd¯ˆo ˙’ i s := c ln(1 + |r|), trong d¯´o c l`a hˇa ` ng sˆo ´ n`ao d¯´o. Nˆe ´ u r ∈ [0,R],s∈ [0,L− 1] ta cho . n c := L−1 ln(1+R) . H`ınh 4.6 minh ho . a h`ınh da . ng cu ˙’ a ph´ep biˆe ´ nd¯ˆo ˙’ i n`ay. H`ınh 4.7 l`a a ˙’ nh gˆo ´ cv`aa ˙’ nh sau khi d˜an d¯ˆo . tu . o . ng pha ˙’ nsu . ˙’ du . ng h`am logarithm L`am mo ˙’ ng m´u . c x´am Tˇang d¯ˆo . s´ang cho mˆo . tda ˙’ icˆa ` n quan tˆam c´ac m´u . c x´am trong a ˙’ nh l`a viˆe . cthu . `o . ng xa ˙’ y ra. C´ac ´u . ng du . ng bao gˆo ` m nh´om c´ac nu . ´o . c trong mˆo . ta ˙’ nh cu ˙’ avˆe . tinh hoˇa . c l`am nˆo ˙’ i 76 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s = T(r) L −1 r R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H`ınh 4.6: H`am biˆe ´ nd¯ˆo ˙’ i logarithm. d¯ u . `o . ng biˆen trong a ˙’ nh x-quang. C´o nhiˆe ` u c´ach d¯ˆe ˙’ thu . . chiˆe . n l`am mo ˙’ ng m´u . c x´am, tuy nhiˆen hˆa ` uhˆe ´ t l`a ca ˙’ i biˆen cu ˙’ a hai phu . o . ng ph´ap sau. (i) Hiˆe ˙’ n thi . mˆo . t gi´a tri . cao cho tˆa ´ tca ˙’ c´ac m´u . c x´am trong pha . m vi quan tˆam v`a mˆo . t gi´a tri . thˆa ´ p cho tˆa ´ tca ˙’ c´ac m´u . c x´am kh´ac. Ph´ep biˆe ´ nd¯ˆo ˙’ i n`ay ta . omˆo . ta ˙’ nh nhi . phˆan (xem H`ınh 4.8(a)). V´ıdu . T (r):=    α nˆe ´ u r ∈ [0,A) ∪ (B,L − 1], β nˆe ´ u r ∈ [A, B](A<B). (ii) C´ach tiˆe ´ pcˆa . nth´u . hai du . . a trˆen ph´ep biˆe ´ nd¯ˆo ˙’ i (xem H`ınh 4.8(b)): T (r):=    r nˆe ´ u r ∈ [0,A) ∪(B,L− 1], β nˆe ´ u r ∈ [A, B](A<B). Su . ˙’ du . ng bit d¯ˇa . c tru . ng Thay v`ı l`am mo ˙’ ng c´ac m´u . c x´am ta l`am s´ang lˆen cho to`an a ˙’ nh bˇa ` ng c´ac bit d¯ ˇa . c tru . ng. Gia ˙’ su . ˙’ mˆo ˜ i pixel trong a ˙’ nh chiˆe ´ m B (chˇa ˙’ ng ha . n, bˇa ` ng 8) bit. Mˆo ˜ i pixel tu . o . ng ´u . ng v´o . i B bit trˆen B mˇa . t phˇa ˙’ ng 1 bit d¯u . o . . c d¯´anh sˆo ´ t`u . 0 (mˇa . t phˇa ˙’ ng ´ıt ´y ngh˜ıa nhˆa ´ t, k ´yhiˆe . u LSB) d¯ˆe ´ nmˇa . t phˇa ˙’ ng B −1 (mˇa . t phˇa ˙’ ng c´o ´y ngh˜ıa nhˆa ´ t, k´yhiˆe . u MSB). C´ac bit bˆa . c cao ch´u . a nhiˆe ` u thˆong tin (nh`ın thˆa ´ yd¯u . o . . c). C´ac mˇa . t phˇa ˙’ ng bit kh´ac tham gia nhiˆe ` u o . ˙’ nh˜u . ng chi tiˆe ´ ta ˙’ nh. Gia ˙’ su . ˙’ ta muˆo ´ nlˆa ´ y ra bit quan tˆam th´u . n v`a hiˆe ˙’ n thi . n´o. X´et 77 H`ınh 4.7: A ˙’ nh gˆo ´ cv`aa ˙’ nh sau khi d˜an d¯ˆo . tu . o . ng pha ˙’ nsu . ˙’ du . ng h`am logarithm. biˆe ˙’ udiˆe ˜ n nhi . phˆan r = k 1 2 B−1 + k 2 2 B−2 + ···+ k n 2 B−n + ···+ k B−1 2+k B , v´o . i k 1 = 0 hoˇa . c1. Ta muˆo ´ n xuˆa ´ tra s =    L nˆe ´ u k n =1, 0nˆe ´ u ngu . o . . cla . i. Dˆe ˜ d`ang ch´u . ng minh rˇa ` ng k n = i n − 2i n−1 , trong d¯´o i n l`a phˆa ` n nguyˆen cu ˙’ a r 2 B−n . 4.2.2 Phu . o . ng ph´ap biˆe ˙’ ud¯ˆo ` cˆo . t Biˆe ˙’ ud¯ˆo ` cˆo . tcu ˙’ amˆo . ta ˙’ nh sˆo ´ v´o . i c´ac m´u . c x´am thay d¯ˆo ˙’ i trong pha . m vi [0,L−1] l`a mˆo . t h`am r`o . ira . c p(r k ):=n k /n, trong d¯´o r k l`a m´u . c x´am th´u . k,n k l`a sˆo ´ c´ac pixel trong a ˙’ nh c´o m´u . c x´am n`ay, n l`a sˆo ´ c´ac pixel trong a ˙’ nh, v`a k =0, 1, ,L− 1. N´oi c´ach kh´ac, p(r k ) l`a x´ac suˆa ´ t xuˆa ´ thiˆe . ncu ˙’ am´u . c x´am r k . D - ˆo ` thi . biˆe ˙’ udiˆe ˜ n h`am n`ay cho ta mˆo . t mˆo ta ˙’ to`an cu . ccu ˙’ aa ˙’ nh. Chˇa ˙’ ng ha . n, d¯ˆo ` thi . H`ınh 4.9(a) chı ˙’ ra c´ac m´u . c x´am tˆa . p trung o . ˙’ v`ung d¯en; ngu . o . . cla . il`aH`ınh 4.9(b). D - ˆo ` thi . trong H`ınh 4.9(c) c´o 78 . pha ˙’ nsu . ˙’ du . ng h`am logarithm. biˆe ˙’ udiˆe ˜ n nhi . phˆan r = k 1 2 B−1 + k 2 2 B 2 + ···+ k n 2 B−n + ···+ k B−1 2+ k B , v´o . i k 1 = 0 hoˇa . c1. Ta muˆo ´ n xuˆa ´ tra s =    L nˆe ´ u. v)c´ok´ıch thu . ´o . c 5 12 ×5 12 th`ı phu . o . ng tr`ınh (4 .2) cho kˆe ´ t qua ˙’ tu . o . ng d¯u . o . ng chı ˙’ nˆe ´ u h(x, y)c˜ung c´o k´ıch thu . ´o . c 5 12 ×5 12. Nhu . d¯ u . o . . cchı ˙’ ra. trong d¯´o (r 1 ,s 1 )v`a(r 2 ,s 2 ) l`a hai d¯iˆe ˙’ md¯iˆe ` u khiˆe ˙’ ncu ˙’ a ph´ep biˆe ´ nd¯ˆo ˙’ i tuyˆe ´ nt´ınht`u . ng kh´uc T. V´o . i r 1 = r 2 ,s 1 =0,s 2 = L − 1, ta c´o h`am ngu . ˜o . ng
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