An infeasible Predictor-Corrector Interior Point Method Applied to Image Denoising
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. An infeasible Predictor-Corrector Interior Point method applied to Image denoising Cecilia Pola and Claudia A. Sagastiz´abal. N ˚ 3205 Juillet 1997 ` THEME 4. ISSN 0249-6399. apport de recherche.
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(83) ! #" . [v ±b } kmy|s|kmw[:n[kmy ^ s ^H bb 1qty|b1y ¼© « ·D@ A?G ¹ 121¢ sq ² b 6 + ¨,¶!b1s R_¡Cb s ^ b v | by|xCv w } k¤wHkmw } by|bs¨ ¼© « o ·D@ A?G ¹ 1¢ sq ² b 6 + Z :¨»¶Rbts ¡Cbs ^ b µv | by|xCv w } kmw[okmw } by|bs¨ ¼ *Hv o s ^ b byusfv{«%kmw } krb1yWª @ · R t ¹k ªly|v:pm b t t · o D· @ A?G ¹|¹ 6 a k + Z · 9 ' e 1 9 6 P 6 o ·l@BAHG ¹ ¹ G a z qlkmw[ ªlx vbWs k¤s_v:wls|( v ^ Z F AW _ qtw } szq ² b : +¢cq! 9 Az# " qtw } : + c0q Z A$ "G * k¤wCqtpmpm~®ªt±b ^ vlv yub +J·u· ¹ : 9 · ¹ : ¹ >8·¾ ¹ ¨ w£v } b s|^ vob1w[y|n b xCv yu kms|kmlk¤s?~dkmw ª 7ª : £ } qtw } : [ª[±b bxHp§q:b ¡l~Es ^ b byuxUbWµs|km} bo} v cdxUv:w[b wls|y yu¥c0b vUqtpm¨ pmb s qtw¢ ª " q krqt¡[pmby/s|vt±_q y xUv:y|kms|km:b q{p¤nHbyWª y|nl´0µk¤b1w®supm~¡Cv n[w b q±_q~« v c *. . . . . .
(84)
(85). . . . . . . . . . . &. .
(86) . . . qtp¤s s ^ ^ b v nH ^ b s k°«<^ ~ b ±yus|^ v bxHwÀx[kmw[v:x[©s| kmc kms|qtb pm k°kms?v ~Ewkmv w w }#%km$'s|&%kmv (w[y¿ ·¾vt:«» ·¹ q ¸ |b£,y¹ qtkmsuy k¤yE=U1 b } ª n[3x¢ª[sukmvNwls|yub v ckmv b¿ xUs|v v:pmkmbw® s_qtqtwUpmµ:b v ¨ kms ]_^ ^ck¤yy yus|v xH*Hx[v km w[ok¤w8s|«=bWbyuqtsyukmkm¡[y_pmb»y|s x v:b w[} krbsuv s ^ q{v w|s b^ b s|v v wHbfckmbw s ^ #%v $'} &%yW( ª q{ x[ª[x y| b1vb 8km] cqtqt¡[s|pmbb yz:q{¨ s|kmy?«?q suk¤v:w v{«av x[sukmc0qtpmkms?~fµv w } k ¼ sulk¤qtv:xw[ykmw£c·¾bW q{¹ w[ªHy_y|b·=KbnHx0^ 7su_|v©¨ ]_q ^ bbw prqtª:su«=kmv b:suv k¤:pmbbwqtwUsuv pmb b ¹ qts|vwU^ b1qy b $ x kmc0Rtª q{pU$ qtw }}1¢nU&q{p[±»«<bWb qtyu^k¤¡HbWk¤ pm²kms?~ q{w } ¥b v } nUq{pmk¤s?~ k1 Z 9
(87) ø[é<åÝâÚì mï|Úß åÝètåÝì å°×rà $ R i o i 3 · Z ' e 1 9 o Z pA Z 9 : Z A 9 : Z ¹ i 3 $ R Õ ñ Úì ¤ïuÚßråÝètåÝì å°× à i ·? ¹ ·i pAb A ¹ i 3 9 : 9 : ô Ø ñ ÚìÝå°×rà ÚÖ ·V l1SASA ¹ V 9 $ k A ± ^ b b D· 1pA SA ¹ yuszqtw } y_«=v s ^ b v:¡ bWsuk¤:bk¤w¿·¾ ¹ ¨ ) _Q+ d ºE¡Cb b s ^q{pbf bWv:q ¡ } ~bWcsukm bbwls|«=kmnHv:wUw[b s|} kmv:s w^ qtkmwAs/·=·=ee ¹¹ ) kmyfqw[v wHy|cvlv:s ^ x v:¡[pmbc¨ %v b x bWkmy|b1p¤~lª!pmbs ) g l· 1 ¹ + b e ' 1 1 9 ' e % 1 9 i7o 1 i RpA . 9. A(. . . . . . . õ õ äþHúû. 9. . 9. . 9. .
(88) . : ?"b D
(89) ;. (. ±cd^ bbs ^ b v }o ªlkmv{y!« s ^} kmbcc0bwHq{sy|kmv k°w « - v c k «%·¾ s ¹ ^ ¨ b]_kmc0^ bqtwo:·=b e ^ ¹ qty q{w- ¡U.Kby|- v:p¤:x[b k }[bn[pmy|yWkm¨ w[ºEq*bw[v:v n[w s|pm} kmw[k ¯ b b ^ bb wl b&s|krqtq ¡H p¤bQbv x[b suyzk¤cdq{pakm¥W¦ qtnCsuqtk¤y|v:k w ¼ k ± ^ kr ^ kmy_b[x[prqtkmw[b } kmwo«=nHp¤p } bszq{k¤pmykmw ^Ýlª'f#_|¨ *b±_s|v:w nHw } pmb qtpm v kms ^ c" ^ «=v qtwly|~oyun[ ckm:b,bsw 9[qts,¨q{*w «=s|b - kmsub v cdq{s|x[kmv n[w[s|yb1ªly s ^ ·b 9 ¹ qt w l} q_" yu n[¡[ q c} kmvbwl} sb1pU[v{·69« ¹ kmw k¤y_s ¡H^ n[bkmy|p¤n[s?)¡ } k ¯ b bwls|krq{p ·9 ¹ ª 6· 9 ¹ ) + c0q! 6 9 ·69 Z 9 6 ¹ 6 ):@p+ bAjA?GHG?GDA - "A ·? ¹ 1 ± ^ b ]_ bo^ b,6· 9 ¡[6 n[ª w } 6 p¤bQ+ c bW· 9 ^ 6 ¹q{ª w[kmy|6 cJ+ Hv ·w[9 yu6 km¹y|s| ykmw :6· b9 w[6 ¹b A @qts|+kmw[F AHs?G?±»GHv_GA8yu- bW¹ ¦:n[«=v b wUcµbsyH^)b/= ¡[yun[sWwª }q*pmy|b bW¨ ¦:n[bwCb_vt«[yzq{cx[pmkmw[ xC;1v Ukmwl" s|y.kmy 9y|b6 pm" bW6 ª{sus|b v } ¨v ]_wH^y|s b1 ~nUqts,kmc s ^ b_q{sc v nC} q b1 p[q{·¾wl:s| b¹ b¨&kmw[* v:qocJyunls ^´0byukmb_bwlxUs v:km} w®bsuyW ª bWq q{y|yubdn[¡[k¤y|wb¦ s ^nHb bwUv b km vtkm« wU q{7pC «= n[ wU µ s|xCkmv v w kmwl s|y ) 9 1&k¤y } bWprq b } q&y|b kmv n[y_yus|bx³l· 1 UH\ R + 921 ¹ ± ^ bw 6· 9 1 ¹ D· 1 U ¹ Z R A ·? ¹ 1 ±cd^ v b } bb p?¨ R _=jAb^{k¤y_qtw ck v ¼ pmk ² b»suv pmb q{wUb qtw } 1 kme y_s ^ b ; I Qx b } krs|b } ¡l~os ^ b ^^ e bwEq} yqtcd x[pmk¤wH o xUv:k¤wl} s } vlb y } wHv s^ yzq{ s|kmy?«=~³·¾ ¹ ª s ^ bo cdv } bp 1 kmy } v:w[y|k } b b } w[v:s q 1n q{s|b º bkmy_wHv n[b1w[ b q{q{s|w b } ¨ q kmy bWprq b ),s b b kmy_w[vow[b±(y|b kmv n[y_xCv kmwlsqtw q&w[b±(yzq{cx[pmkmw[xUv:kmw®s ^ } ^ ^ ¼ ^ ]_^ ^ ·km?y_¸_s|i vo¹ sQ:k¤k¤wbyQw[xU^Ýb ?v: _|qty|¨ y|sukmb +4¡[pmyzk¤b_:q{bcsuw v x[c1 pmkmU w[b q{ w xCb_} v s kmqowlkms|y,xCyv ¡[¡lyun[~kmws|kmy| v:p¤b kmpmw[®jkmHw[su bW8 !w[ kr ¦: n[ b±_jkms dq/n E:b y|xUbWq q qt cd¨ bs|b k¤'yQk¤U yQ1y|n[¡UHq{ª[pms ^ v b kmks } bWc q ca kmw c 1 ·9 ¹ 9 ' U ·9 Z 1 U ¹ »·69 Z 1 U ¹ A ·? ¹ «=v s k§q{p!y|subx[y|km¥b1y 1 H¨ yuy|v[krqtsub } s|vq/y|v:pmn[s|kmv w 9l· ¹ vt«_·? ¹ s ^ b b&kmy»s ^ b ¦:nUqtwlsuk¤s?~ :· ¹ + l· 1 U ¹ Z 1 · 98· ¹|¹ Z ' U · 9l· ¹ Z 1 U ¹ · 9l· ¹ Z 1 U ¹ G ]_^ bxCv ¼ yuy|km¡[pmbb[kms|y&vt« ·?^¸_i ¹ q Fb )yub kmv nHyoy|s|} b1xaª,wln[pm pDy| s|b1x q{w } qt w¬ bcb b1wU~E^ y|suv ^ xaªQ± ^ bw¬^ s ^ b ·? nH s|s|¹*kmw[^ v p xH} p§yq{ªCw[¡[bn[«=s nHwUyus|szkmq v: w s|y:qtv ykmw[¡Cobbs|w¿v ckmw[ªUkm¡CcbWk¥q{b n[y|¨ b 98· ¹ ckmw["km cokm ¥by q{w } v xC¡CbbWyv:±_ckms bys v b&w[yuszqtq{yuw®b s±¨ *Hbv w s ^ b yzq w[² bb±vtv« y|} kmcqtxH¡Cp¤v krnHkms»s?~ls ^ªt±bbfyus|v:v cxHx[kms,kmw[b wlsusubby|kmsw[¨ o] kmv wl:s|bv&s ^cb v ±_bkmsus bW^ ^ bWw[q:kr ^ q{p yqt} cb1 sxHqtp¤kmkmpmw[y ^ xCb v b km¨wls 9l· ¹ ªH±»b b1w[b q{s|b q&xUq suk§µn[prq y|nH¡[ q } kmbwls ¹ ' U U ¹|¹ ¹|¹ · + l· 1 Z 98· 1 · 9l· yq qts|kmpmy½km«=w[~lbWk¤q wH km¥Ws q{^ s|b kmv w boprqtb s||kmv: v w qt·w }.¹
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