Some applications of $l^\infty$-constraints in image processing
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Some applications of l ∞-constraints in image processing Pierre Weiss — Gilles Aubert — Laure Blanc-Féraud. N° 6115 November 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--6115--FR+ENG. Thème COG.
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(14) ÙÚÛÜ Ý{ÞiÞ~ßáàpâ{Ýã)àpÚäSåVÚtæ < ·, · >X. l∞. ç â{ÚIäSåã2ènÝàéälã2åàéä>àéÛÝ=ê0ÜQÞ~èdÚ!â{Üå{å{àéäaê. z_U!jSgIbnU!`QbnPSU'l`dlvI`ne=vivirylrdg_z_leb$gij < u, v >X :=. < ·, · >Y. ¥. | · |p p ∈ [1, ∞[. n X. k`bnPSU. lp. jSgirnT½g+j. X. kw`QbdPSU | · |∞. jlgirnT®gij l∞ . 0. X. 0 |u|∞ =. p ∈ [1, ∞[. uAjlz ¯ljlvI^0. kw`fvDjSgirnT½g+j. Y. n X 2 X. n X. g, h ∈ Y. 0. Ñ2_ Ò. gi,j hi,j. i=1. max. i∈{1,2,...,n}. ||g||p := (. Ñ2_ iÒ. n X i=1. Ñ2_ +Ò. |ui |. Ñ2_áùIÒ. |gi |p2 )1/p. max. i∈{1,2,...,n}. |gi |2. kw`fveg+ja iUª`cU=b ÑpvVlvi kj.ve=U=rdbnvIkjTDU=bdrnke,ÒØPSU=j ¡$kwzaKbnP 2α ¥le=U=j0bdU!rdU,zg+j f £ kjlvI^ bac ¥_kw`QbdPlU'kj0bdU!hiU!ryBvIrdbQgio a ∈ Y tvut. Ñ2_ +Ò. ui v i. |ui |p )1/p. z_U=¯ljSU!za^0. ||g||∞ :=. ¡#UPlvM +U0. u, v ∈ X. i=1 j=1. |u|p := (. ¥Sogir. £ +g r. z_U!jSgIbnU!`bdPlU'l`cBvI`ne=vIwvIrQySrng_z_lebQg+j i=1 +g r Y £ < g, h >Y :=. || · ||p. X. ù. ¥. Ñ2_ ü+Ò. Z=X K. k`stl`tbLvIj Pa^0yFU=r{elBU'gio. ez}| a@_nb|z a av ~. ´+õ ï ð!ð ¸ ï ò 2ì ^ q5rngiyFU=roSjlebnkg+j`ØuÔeg+ja iUª oSjlebdkgij. g+j kw`Aylrdg+yBU!rfkmovIjlz6g+jS^ ko wk z_U!j+bnke!vI^U!±0lvI~bng +∞ vIjlzbdPlvIbfkmbAzSg0U,`QjSgIb$b{vI²+UbnJPSU vIZSU −∞ gij Z ´+õ ï ð!ð ¸ ïØòQ wòQ ^áW#gaU=r{ek 0kbt^`«u oSjBebdkgij J : Z → Y kw`f`dvikzbdgVFU egaU=r{ek iUkoL0 ||u||2 → ∞ ⇒ J(u) → ∞. 76798-:<;=>=@?. J. k`jSgIb Ñ)_ +Ò.
(15) A Ü=àwånåCBEDF<G{Ü=èãHB6IfßÝä~â çJLK èdÝ-F<M. ü ´+õ ï ð!ð ¸ ï°ò wýQ ^ V jlz_kwe=vIbdg+r;oSjBebdkgij.`\U=b. FUvjSg+jËU=TVy_bt^Ëe=g0`cU,ze=gija iU=ªV`dSl`dUbQgIo Z Kkw`f∈z_UZ¯BjSU!z a^0 ¥ S χK ko x ∈ K Ñ)_!0Ò 0 gibdPSU!rd¡$kw`dU χK (x) = ∞ ko K kw`$vVeg+TDyBviebf`dUb,¥_bdPSU!j χK kw`fegaU!rne=k +Ui ´+õ ï ð!ð ¸ ïÎòQ 6 ^ \alFz_kú~U=rnU=j0bdkwvIvIjlz `dSShir{vizSkU!j+b `×OQPSU `clFz_kú~U=rnU=j0bdkwvIgIo J vIbfyBg+kj0b u ∈ Z ¥ kw`$z_U=¯ljSU!za ^ 0 Ñ2S+,Ò ∂J(u) = {η ∈ Z, J(u)+ < η, (x − u) >Z ≤ J(x), ∀x ∈ Z} kw`fe!vIU,z vV`dSShir{vizSkU!j+b, η ∈ ∂J(u) OQPSU `dS~z_kúFU!rdU!j+bnkvi e=vij FUbdPlgiSh+P+b$gio v+`bdPSU `dUb$gIo Pa^ayBU!rdySwvIjlU!`ylvi`n`dkjShDbnPSrdg+ShiPyFgikj+b ¡$Plke{PËljlz_U=rnU!`cbdkTvbnUfbdPlUAoSjBebdkgij J 5XjyBg+kj0bn`#¡$PSU!rdU J k`z_kúFU!rdU!j+bnkviSUi¥+bdPSU`clFz_kú~U=rnU=j0bdkwvIu rnU!zSleU,`bdgvV`ckjShiUbngi j 0 bdPSU e=v+`d`dkwe=vI~h+rnv+z_kU!j0b! ´+õ ï ð!ð ¸ ïØQò wQñ ^ \U!hiU=jBz_rdU ¤£ U=jle{PlU= W#gijstlh+vbnU ` \Ub G FUv e=gija iU=ªVylrdg+yBU!rviySySke!vbdkgijordg+T Z bng Y ∪ {∞} OQPSUeg+jstSh0vbdUoSjlebdkgij gIo G kw`$z_U=¯ljSU!z a ^ 0 Ñ2SMiÒ G∗ (u) = sup (< x, u >Z −G(x)) kw`fvVeg+ja iUªËylrdg+yBU!rQoSjlebnkg+j5¦6g+x∈Z rdU!g iU=r,¥0¡#UPlvM +U 0 G∗∗ = G G∗ OQPlULkjlz_kwe=vIbdg+rQoSjlebnkg+j gIo K ¥Sz_U!jSgIbnU!z. tvu. $~@ |.z.~ E.~ n¡.¢dzl£~¤3zQ.z6.~@ ny{zQ< 6n. Og`ckTVySkmo^bdPSU'jSgibnvIbdkgijl`!¥_¡#U'z_U!jSgIbnU u(i, j) bnPSU' vISUgIo u gijySkªaU! (i, j) Ogz_k`nernUbnk!ULbdPSU'h+rnv+z_kU!j0bQ¡Ul`dUbnPSU ewvi`n`dke!vIF¯lr{`tb$gir{z_U!rQ`ne{PSU!TDUFgirnrdg¡U!zËordg+T¥^áù`¤ £ ig r u∈X ∇u. 0. kw`$vIjU=U=TVU=j0bfgio Y . (∇u)(i, j) = ((∂1 u)(i, j), (∂2 u)(i, j)). ko i < nx kmo i = nx ko j < ny u(i, j + 1) − u(i, j) kmo j = ny (∂2 u)(i, j) = 0 OQPSkw`fz_U=¯ljSkbdkgij6vIg¡f`;bngz_U¯ljlUbdPSU z_k iU!rdh+U=jle=Uylrdg+yBU!rd^Ëa^z_lvikbt^i¥_kTVyBg0`ckjSh0 (∂1 u)(i, j) =. . u(i + 1, j) − u(i, j) 0. < ∇u, p >Y = − < u, div(p) >X. Ñ)_!+Ò Ñ)_ Ò Ñ)_,+Ò Ñ)_!0Ò NPOE7QNSR.
(16) ÙÚÛÜ Ý{ÞiÞ~ßáàpâ{Ýã)àpÚäSåVÚtæ. l∞. ç â{ÚIäSåã2ènÝàéälã2åàéä>àéÛÝ=ê0ÜQÞ~èdÚ!â{Üå{å{àéäaê . «U¡$k~`dU=UbdPBvbQPlvM akjSh `cle{Pv rnU=wvbnkg+jËk`#jSU,eU!`n`nvIrn^bngDe=giTVyS_bnUfbdPlUU=ª_v+eb$`clShir{viz_kU=j0b;gio bnPSU z_kw`de=rdU=bdULoSjlebdkgijlvi2;\_kTVySU vIhiU!Srnvhik iU,`X0 1 kmo p (i, j) − p1 (i − 1, j) ko 1 < i < nx 1 p (i, j) (div(p))(i, j) = kmo ii == n1 x −p1 (i − 1, j) Ñ)_MùiÒ 2 k o 2 p (i, j) − p (i, j − 1) ko 1 < j < ny p2 (i, j) + ko jj == n1 y −p2 (i, j − 1) ØkbdP bnPSk`Az_k`nernUbnk,vbnkg+j¥a¡#U'e=vij6z_U¯ljlUbnPSUbdgibnvi vIrnkvIbdkgij v+`X0 J1 (u) :=. n X. |(∇u)i |2. Ñ)_!ü0Ò. uAjSgIbnPSU=rA±0lvij0bdkbt^ËgIokj0bdU!rdU,`tb$kjbdPSkw`$ylviyBU!r$i=1k`QbnPSU'z_kw`de=rdU=bdk=U!zPa^ayBU!rn`dSrdopvie=Ugio u 0 J2 (u) :=. OQPSg0`cULbt¡gDoSjlebnkg+jl`fvirdUeg+j0 +Uª. n q X i=1. |(∇u)i |22 + 1. Ñ)_¦+Ò. ¨e© $ ¾ » ¼V»$ $ [6ª
(17) =»Ë= | {«
(18) f ¬ T3¦T® § vu@w. ¯°~@z}|z$n± ²@³.~. xfU=¡$rnkmbnkjShVySrngilU!T Ñt+!Òljlz_U=rfbdPSUog+rdT¥0 2Ñ S I+Ò ¡$PSU!rdU k`5bdPSUkjBz_ke!vbngiroSjBebdkgijgIo ¥+kmbk`#eU!vir5bdPBvb#ylrdg+SU!TÑ2S IiÒ5k`#egija +Uª¥iySrngiyFU=r vijlzegaU=r{ekχ iKULgij Z OQPal`!¥_¡U'U!vi`dk^h+Ub$bdPlKU'Uª_k`cbdU!jleUgiovVTVkjlkTVk=U=r, inf {J(u) + χK (u)}. u∈Z. vut. ´µ¶³d~@· ³z6z} ¸nf.¢zp@³~@d. ØPlU=j J kw``cbdrnkwebd^ e=gija iU=ª¥ ¡UPlvM iUVSjSkw±0SU=jlU!`n`'gIofv TVkjlkTVk=U=r,¥S_bkjÂbdPSUh+U=jSU!rnvi5eg+j0 +Uª e!vi`dUi¥0¡#UhiU=bvDegija +Uª`cU=b#gIo`cg+_bnkg+jl`!«Ukl`cbdr{vbnUAkb#bnPSrngiSh+PvIjËU=ªSvITVySUAkj 1 − D bnPlvb¡#U `cbdBz_^ËwvbdU!r$kj 2 − D Ub{vI²+U Z = X ¥ nx = n ¥ ny = 1 vIjlz`dUb30 J(u) =. n−1 X i=1. 76798-:<;=>=@?. |ui+1 − ui |. Ñ2S _MÒ.
(19) A Ü=àwånåCBEDF<G{Ü=èãHB6IfßÝä~â çJLK èdÝ-F<M. ,. OvI²+U ¥_¡$kbdP vIjlz QO PSUkfj+ibnU==rnySi/(n rnUbnvIbd−kgi1) j×gio$bdPSkw`Di TV∈ ga{0, zSU=Q2,k` ...,bnPSnU−og+1}g¡$kjShkα0Ë<¡0.5 U virdU gag+²0kjSh6og+rDbdPSU oSjlebdkgij¢gio TVkjSkTVvi z_kw`de=rdU=bdULbdgibnvi Mvirdkwvbnkg+j¥_eg+j+b{vIkjSU!z kj bnPSU'rnU!zbdSFU z_r{vM¡$j g+j £ khiSrnUÑt,Ò W#U!vird^i¥avIBTVg+jSgIbngijSg+l`kjlernU!v+`ckjSh`dU!±0SU!jleU,`;`nvbdkw`co^0kjSh'bnPSUegijB`tbnrnvikj0b!¥+hig+kjSh ordg+T vIjBz virdrnk 0kjSh'bng 1 − α PlvM iUAbdPSU'`nvITVUfbngIbnvi~ Mvirdkwvbnkg+j>Ñ 1 − 2αÒ¥SvIjBzvIrnU`dgi_bnkg+jl`;gioySrngilU!T α( + ) . j 0 zSvIbnv bdU!rdT x$U!z¹0'SySyFU=r vijlz«g¡U=r'e=gijl`cbdr{vIkj0bn` 2SUËvijlz Sv+e{²o0bt¡g £ khiSrnU j 0 YLrnU=U=¹ yFg+`n`dkSU'`cg+Sbdkgijl`. Ì6ª d¾$;$¾ »±«" ¾$ ¾$ $ «U h+gËlvie{²bdgbnPSU h+U=jSU!rnviySrngiSU=T Ñt+!Ò J kw`AvËhiU=jlU=r{vIySrngiyFU=regija +UªËoSjBebdkgij>gij Z ¥BvIjBz kw`Qhik iU!j a^>Ñci iÒ K V jbdPlULyBvi`cboU!¡Ö^+U!virn`!¥+Tvij0^Uú~girdbn`QPlvM +UFU=U!jz_U,z_kwe=vbnU!z bdgDbnPSU'z_U!`dkh+j gio opv+`tbfvIhig+rdkbdPlTV` bng`dgi iUQkTvihikjSh'ySrngiSU=T`!xfX £ ylrdg+SU!TÔog+r5kjl`tb{vIjle=Ufe=vijjSg¡ÌFUA`dgi iU,z kjËvITVg+`cb;rnU!vIabnkTVU ^ ü_¥+I¥M `) /Aj_ogirdbdSjBvbdU!^+¥a¡#ULe=vIjljSgIbPSgiyFUL`dgi akjlhÑt+MÒ¡$kbdP v PSkhiPËr{vbnUAgio egija +U=rnhiU=jBeUfvi`#¡UATviz_U jlg vi`n`dSTVy_bdkgijl`g+j J UªSeU!y_b$egija +Uª_kmbt^+6 V TvihikjSUfbnPlvb$¡UAPlvM +UvIjvIhig+rdkbdPST bnPlvbQhiU!jSU=r{vbnU!`;v `dU!±0SU!jleU {uk } bdPBvbfkw`$`cSylyBg0`cU,zËbngËvIySySrng+v+e{Pv`dgi_bdkgijgIoQÑti,Ò u ewvi`n`dke!vI;rnU!`dSbDgioeg+ja iUªÂgiySbdkTDk!vIbdkgiµ j ^ i+ `QbdU=w` B`bnPlvbVjSg«kmbnU=r{vbnk +UTVUbnPSg_z gijS^ B`ckjSh.bnPSU `cU,±+lU=jle=U!` J(uk ) vIjBz ∂J(uk ) e=vij¢vie{PSkU= +Uv.FUbdbdU!rDr{vbnUËbnPlvIj O(1) ¥ ogirDhiU!jSU=r{vI e=gija iU=ªËTVkjSkTVk,vbnkg+j ySrngilU!T Ñ k`QbnPSU z_U,`ckrdU,zylrdU,ekw`ckgijBÒ V¤o¡#UjSgIbnU J¯ bnPSUTVkjSkTST®gio#Ñt+!Ò¥SvIjlz J¯k = min i ¥_kbQTDU,vIjl`;bdPBvbbdgDhiUb i∈{1,2,...,k} (J(u )) v ySrnU!e=k`dkg+j k ¥vIja^6vIhigirnkbdPST ¡$krnU!±0SkrdU k bngFUDgIo#g+rnzSU=r b O(1) c Ñ2`cU!± U ^ I+ `og+rLv TVg+rdUySrnU!e=k`dUL|J¯`cbnvI−bdU!J|TD¯ U!≤j0b{Ò «UVySrdg+yBg0`cU kj>bdPSkw`ylviyBU!rbng B`cU bdPSUDySrngIstU,ebdU,z>`dSSh+rnv+z_kU!j0bTVUbnPSg_z6ogir¡$PSkwe{P.bnPSUeg+T ylU=ªakbt^¢kw`Ë`dPSg¡$jÌbdg×FU O(1) [\agÂbdPBvb!¥QjSg§TVU=bdPSg_zØe=vijÉzSg§BU=bcbdU!rËbnPlvIj°kbËogirËg+SrhiU!jSU=r{vI º. » ¼ ½. 2. 2. 2. NPOE7QNSR.
(20) ÙÚÛÜ Ý{ÞiÞ~ßáàpâ{Ýã)àpÚäSåVÚtæ. l∞. ç â{ÚIäSåã2ènÝàéälã2åàéä>àéÛÝ=ê0ÜQÞ~èdÚ!â{Üå{å{àéäaê. +. ylrdg+SU!T ¥+SyËbdg v TSbdkySke!vbdk iUAopviebdgir, O g l`cUAbdPSkw`;bdU!e{PljSk±0SU¡#UAjSU=U,zVbdg BULvISUfbng eg+TVyS_bdU vÂ`dSSh+rnv+z_kU!j0bVvijlzÌbdPSg U 5Bekz_U,vIj°ySrngIstU,ebdg+rVgij c V jÉbdPSU6jlUªab`cU,ebnkg+jl`V¡#U6`dPSg¡¿Plg¡½bdg e=giTVyS_bnU'`cSlhir{viz_kU=j0bn`!¦6girnU=g +U=r,¥IbnPS{U 5le=kwz_U!vij KySrdgistU!ebdgirQg+j K e=vij FU eg+TVyS_bdU,zU!vi`dk^+E V¤b e!vIjBU `dPSg¡$j bdPlvIb30 kmo |ui − fi |2 ≤ α ui Ñ i+Ò gIbdPlU=rn¡$k`dU (ΠK (u))i = u −f fi + α i. i. |ui −fi |2. OQPSkw`vIhig+rdkbdPSTøkw`¡U=²0jlg¡$j.kj.bdPSUDgiy_bnkTVk!vIbdkgij e=giTVTljSkmbt^§Ñ2`cU!U ogirLkjl`tb{vIjle=b U ^ i `Ò{ V¤b ¡Qvi`#ySrngiyFg+`dU!zrnU!e=U=j0bd^kjkTVvihiULySrngae=U!`n`ckjSh kj`cg+TDULylviyBU!rn`¿^ a¥Mù0¥~,+`)«U'z_U!`nernkBULkmbQSrnkU]Àl^ FU=g¡' Ádu@w u ¯. ¢zpy<~@|~y@zÃnf.¢dzl a@Änn<~¢Å. kw`$v`cg+SbdkgijgIoÑci!Ò¥SkmokbA`dvIbdkw`t¯BU!`#ogirfvij0^. t>0. 0. Ñ i+Ò. u ¯ = ΠK (¯ u − tη). _z U=jSgibdU,`bdPSU{5Bekz_U,vIj ySrngIstU,ebdg+rQgij K ¥SvIjlz η kw`fvij0^U=U=TVU=j0b$gIo ∂J(¯u) OgËeg+TDyl_bdU u¯ ¥_¡U'l`dULbdPSU'ySrngIstU,ebnU!z`cSlhir{viz_kU=j0b$vIhigirnkbdPST½z_U=¯ljSU,z a^0. ΠK (·). AU=rnUi¥ Ádut. tk > 0. og+rfvIja^. k. vIjlz. (. ηk. Ñ Ò. u0 ∈ K k uk+1 = ΠK (uk − tk ||ηηk ||2 ). k`fvija^U=U=TVU=j0bfgIo. Ƶ6ÇÈÉ|<¢zµ.¢zÊz}· ³z}|z. ∂J(uk ). . {tk } Ë. QO PlUQe=gija iU!rdh+U=jle=U5gIoBbdPSkw` kbdU=r{vbnk +U`ne{PSU!TDUQz_U!yBU!jlzS`g+j bdPlUFU=PBvM 0kgilrgiolbnPSU$`cU,±+lU=jle=U \U=b FUQbdPlUf`cU=b5gIo`cg+_bnkg+jl` gIoÑt+,ÒOQPSUQogig¡$kjShLbnPSU=g+rdU!T|ySrng iU!`bnPlvbog+r5v¡#U!le{Plg+`dU=jVt`ckU,±+lU=jle=UU ¥_¡#U'PlvM +ULe=gija iU!rdh+U=jle=ÌU ^ i `0 tk Ýä M P∞ t = ∞ B;Pã ÏSÜ=ä Í ´a¸ ´ Æ 6 pì *Î æ lim k→∞ tk = 0 k Ý ä M lim k=0 d(u k limk→∞ J(uk ) = inf u∈K J(u) , U) = 0 k→∞ ÏÐ lÜèdÜ d(·, ·) àwå'Sã Ïl¿ Ü Ñ FlâßáHà M+Ü{Ýh ä Màwåã¤ÝIäFâ{Ü#æ{èdÚÛ ÝÞlÚàéäBã#ã ÚËÝåÜ=ã Ò. ¦.vija^ `cU,±+lU=jle=U!` `dvIbdkw`to^bnPSUQPa^0yFgIbnPSU!`dkw`giolbnPSk`bnPSU=g+rdU!T £ g+r kjl`cbnvijleU tk = 1 ¡$kbdP p ∈ gir OQPSk`¡$kwz_UQe{PSgikweU#TVvi²iU,` bnPSU#bdPSU!girnU=T|z_kZËeSbbngl(k+1) `cU;ogir ySr{viebdkwe=vI 1 ]0, 1] tk = (k+1) log((k+1)) e=giTVyS_b{vbnkg+jl¡ ` 0vITVg+jSh.vI;yFg+`n`ckSU`dU!±0SU=jBeU!`!¥ TvIja^«e=giSwzÂU!viz«bng> +U=rn^ `dg¡¬eg+j0 +U=rnhiU!jleU+ \_giTVUfkwz_U!v+`5¡U=rnU$ySrdg+yBg0`cU,zDbdg'b{vie{²aU$bdPSkw`5ylrdg+SU!T vijlzDbdPlUfbdPlU=girn^Dk`;jSg¡°¡#U!BU!`cbnvIlkw`cPlU!z6Ñp`dU=U ^,_¥M¦ù `og+rf`cg+TVULrnUoU!rdU!jleU,`nÒ p. 76798-:<;=>=@?.
(21) A Ü=àwånåCBEDF<G{Ü=èãHB6IfßÝä~â çJLK èdÝ-F<M. M V¤o. J. kw`v\kyl`ne{PSkmbn eg+j+bnkjlgil`¡$kbdP\kyl`ne{PSkbd'eg+jl`tb{vIj0b. L. k2 Uiv0. |J(u) − J(v)| ≤ L||u − v||2. ¡ U e=vij¯ljBzvDylvirnviTVUbdU!rdU!`n`#giy_bnkTvI`cU,±+lU=jle=UVÑ2eotv^ +ù¦`Ò¿0 Í ´a¸ ´ Æ 6 wòQ Þ Õ Ó ÏSÜfÞ~èdÚ Ô=Ü{â=ã¤CÜ MËCå F<G¤êièdjÝ MIàpÜäB ã M+Üå=â{Ü=älã Ð àéPã Ïåã ܤo D tk = √ k. Ü=äSå>F_èdÜåLãPÏSÝIã Õ Ð ÏlÜèdÜ. LD k = J¯k − J¯ ≤ O(1) √ k. Ñ i+Ò. Ñ I0Ò Ñ +ùiÒ. wà å ãPÏSÜ¿ÑLFlâßmàHMiÜ{ÝähMàpÝIÛVÜ=ã¤Ü=è Úcæ'ãSÏlÜå=Ü=ã K Ò OQPSkw`kw` vijU!vi`d^e{PlgikweU bdgAU=jl`dSrnU5v$higag_zLrnvIbdUgIoSe=gija iU!rdh+U=jle=Ui¦6girnU5`dgiySPSkw`cbdkwe=vbnU!zbdU,e{PSjSkw±0SU!` e!vIj BU'z_U,`ckhijSU,zËlv+`cU,zËg+jkbdU=r{vbnk +UU,`tbnkTvIbdU!`gio J¯ Ñp`dU=ÖU ^!+ `og+rkjl`cbnvijleUMÒO gVg+Sr#²ajSg¡$U!zShiUi¥ bngrnU!vi^h0vIkjËU ZËekU=jBe^D¡$kmbnPËbdPlg+`dU$bdU,e{PSjSkw±+lU!`!¥+g+jSUAgioébdU=jjSU=U,zS`;U=ªabdr{v'kjSogirnTVvIbdkgijl`#vIFgi_b#bdPSU `dgi_bnkg+j¥Sk²+UbdPSUviySySrngMª_kTvbnU'z_k`cbnvijleUorngiT uk bngVbdPSU `cU=bAgIo`cg+Sbdkgijl`! V¤be!vIjbdPal`ABU,eg+TDU rnU!vi^Ëz_k ZËeSbfbdgviySyS^Ëkj ySr{viebdkweU+ P \kyl`ne{PSkmbn'hir{viz_kU=j0b,¥abdPSU!j6¡#U z_g £ kjlvI^+¥l¡U'rdU,e=vi~bnPlvbAko kw`Ae=gija iU=ª~¥SzSkmú~U=rnU=j0bdkwvIlU+¥S¡$kmbn jlgIb$jSU=U,z bdgVjSgirnTvIk!ULbdPSU'h+rnJv+z_kU!j0b!¥_vIjlz vVeg+jl`cbnvIj0b$`tbnU=y6`dk!ULU!jl`dSrdU,`$eg+j0 +U=rnhiU!jleU0 æ Õ Í ´a¸ ´ Æ 6 }ý *Î 3 Ñ Iü0Ò ||∇J(u) − ∇J(v)||2 ≤ L0 ||u − v||2 Pã ÏSÜ=ä>Pã ÏSÜ$ÞFèdÚ Ô=Ü{âã CÜ M êièdjÝ MIàpÜäB ã M+Üå=â{Ü=älLã Õ D. Ð àéãPÏ>â{ÚIäSåã¤ÝIälã;å{ã Ü Þ Ádu. . t=. 2 L0. u0 ∈ K uk+1 = ΠK (uk − t∇J(uk )). Ñ m0Ò. ÜälåCF_ènÜåãSÏlÝã Õ. d(uk , U ) → 0. Ñ i+Ò. ×.nÄn< ÅÉÅÉ~@dÄØ ay{z}|.o ad£¥|Ųy¿³v.~@dÄØ.~@ÅÙz. V¤o¡#UySrdg_e=U=U!zË¡$kmbnPe!vIrnUi¥+bdPSULTVU=TVgirn^VrdU,±+lkrnU=TVU=j0b{`5gIo bdPSkw`QvIhigirnkbdPST¬vIrnUAjSg TVgirnUfbdPBvIjËbt¡$kweU bnPSU`ck=UgIobdPlUkTvIhiULvIjlzËbdPSULU=jShibdPl`#gIog+Sr c e=gazSU!`z_gjSgibU=ªSeU=U,z 300 kjSU!`!;XjSUAkTVyBg+rcb{vIj0b v+z_ vIj0bnvihiUQgIoFbdPlU!`dUfvIhigirnkbdPST` kw` bdPal`5bdg vIg¡×opvi`cb;kTVySU!TVU=j0bnvIbdkgijVgIovTVgazSU=) OQPlU$Skh'kw`n`cSU kw`bdPBvbAeg+TVyS_bdkjShDbdkTVU!`fe!vIjBU'wvIrnhiU+. NPOE7QNSR.
(22) ÙÚÛÜ Ý{ÞiÞ~ßáàpâ{Ýã)àpÚäSåVÚtæ. l∞. ç â{ÚIäSåã2ènÝàéälã2åàéä>àéÛÝ=ê0ÜQÞ~èdÚ!â{Üå{å{àéäaê. ,. «UAe=vijD`dU=UQkb;¡$PSU!j bnrd^akjShLbdg¯ljBzDv`cbdg+ySySkjSh ernkmbnU=rnkg+j}V¤b5¡gilz BUAjSgijl`dU=jB`cUbdg'v+`d`dU!`n`bdPSU =e gijlzSkmbnkg+j 0 ∈ ∂J(uk ) nV jlzSU=U!z¥ J TVkhiP0b$BU'jSg+j `cTVgagIbnP`dgDbdPlvIb$bdPSU'jSg+rdT¿gIobdPSU `dSSh+rnv+z_kU!j0b TVkhiP0bfFU virdh+U'U= +U=j iU!rd^e=g0`cUbngbdPlU`cg+Sbdkgij§Ñpe=gijl`dkzSU=r J(x) = ||x||2 og+rfkjl`tb{vIjle=U,ÒOQPal`fv `dkTDylU'e{PSg+ke=ULkw`QbngV`dUb$bnPSU'jaSTFU=r$giokbdU!rnvIbdkgijl`! «U¡$k`cU!UkjbnPSUvIySySkwe=vbnkg+jl`#bdPlvIb#bnPSUrdU!vIbdkgij k ≤ O(1) LD vig¡f`;bdg ¯ljlzbdPSULjaSTFU=r √ k giokbdU!rnvIbdkgijl`QjSU!U!z_U,z bdgVhiU=b$vzSU!`dkrnU!z vie=e=Sr{vie^+ £ girQkjB`tb{vIjle=U¡UAh+Ub¨ yFU=r{eU!y_bdlvi^Ë`cbnvIlU+¨a`dgi_bnkg+jl`;ylrdU=bcbt^Vopvi`cbogirbdPSU BV − L∞ ySrngiSU=T Ñ `cU,eg+jlzS` ogi¡ r \ U=jlv0Ò¥;l_bogirDbdPlU6TDg+rdU eg+TVySU=ª¢ySrngiSU=T gioAÍD;¦6U=^+U=r,¥5¡UjlU=U!zÉvIrngiljlz V T k 0 j S d b ! U ' ` o i g r Ì Q 2 vIrnlvirnvkTvIh+UiË«UVbdPlkjS² bdPBvb'bdPlk`'h+U=jSU!rdkweVvIhig+rdkbdPST`cPSg+Swz>FUËl`dU!z«¡$PSU!j 5 ¡$kkjSh bdg>hiUbDv6±+lke{² g iU=rn akU!¡ÔgiobnPSUviSkkbdkU!`'gio$v TVg_z_U!26OQPSU!j§TDg+rdUrnU¯BjSU!z bdU,e{PSjSkw±0SU!` Bvi`dU!z gij vIj vIjlvi^_`dk`Qgio bdPlU `tbnrdlebdSrnUgIo J `dPSgiSwzBULbnPSgiSh+P0bfgIot uATDg+jSh'bnPSUAeg+jlelrdrnU=j0b;vIhig+rdkbdPlTV`!¥IUbl`5TVU=j0bnkg+jSSjlz_UfTVUbnPSg_zS` ^M `)¥0¡$PSkwe{PËvIrnU$²ajSg¡$j bng e=gija iU!rdh+U'opvi`cbdU!r!¥~S_b'vIrnU TVgirnU zS[k ZËelmbbdg kTVySU=TVU!j+b'kj>ySr{viebdkweU+¥BrnU!±0SkrdU PSkhiPSU!rTVU=TVg+rd^ `dylv+eUi¥QvIjBz¢PlvM +UbdPSU.`nvITVUbdPSU!girnUbnke!vIfeg+ja iU=rnhiU!jleUr{vbdU+Îx$U!e=U=j0bd^i¥#Í5SrnkkfU!`cbdU!rdg ¢kp j ^ _> ` ylrdg+yBg0`cU,zvDjSU=¡ TVUbnPSg_z¡$kbdP6vVeg+TDylU=ªakbt^Ëgio O(1) 5OQPSk`QTVU=bdPSg_z vIySylkU!`bngVvV`dTvIU!rf`cU=bQgio ylrdg+SU!T`=¥~S_bkb'`dU=U!T`fbnPlvbkmb'TVkhiP0bLFUVvIySySkU!z6 bngbnPSUDySrdg+SU=T`A¡UDySrdU,`cU!j0bkj.bdPlk`LylviyBU!r! V¤bf¡$kBUbnPSU'ySSrnyBg0`cUgioo_bnSrnU'kja iU,`tbnkh0vbnkg+jl`= Ú àvu@w. ½
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(32) f. Ųv£dz}. V jËbdPSkw`Q`cU,ebdkgij ¥+¡UAgagi²vbQv`dU!eg+jlzvIylySkwe=vIbdkgijËgiogiSrhiU!jSU=r{vISornviTDU!¡#g+rd²~ UvIrnUfkj+bnU=rnU!`cbdU,z kjbdPSUog+g¡$kjSh ylrdg+SU!T¥0 inf λ|u − f |pp + J1 (u). Ñ2S _,Ò. og+r L«UDTDkjSkTVk!UbnPSUbngIb{vI vIrnkvIbdkgij.`dS_stU!ebbngv p e=gijl`cbdr{vIkj0b!LØPSU=j p = 2 bnPSkw`$TVgapzSU=∈]1, kw`Q∞[ bdPSU xfX £ TVg_z_U!n^ i+`¤ UbnrdU,vbQbdPSU e!vi`dU p = 1 `cU!ylvIr{lvbnU=^i AgIbdUbdPBvbbdPSU`cg+SbdkgijËgio#Ñpl S!Ò5kw`SjSkw±0SULzSSUAbng `cbdrnkebQe=gija iU=ª_kmbt^VvIjlz egaU=r{ek akmbt^DgIo bdPSU lp jlgirnT «U¯lr{`tbfrnU!e!vI~`dgiTVULopviebn`$gioe=gija iU=ªvIjlvi^_`dk`Ñp`dU=fU ^!- `)¥_og+rfvVegiTVySUbnUrdU=oU=rnU=jle=U,Ò z_U!jSgIbnU!`LvkjSU,vIrLz_kmú~U=rnU=j0bnkvi giyFU=r{vbngir, Λ∗ : Y → X kw`kmb{`z_BvI g+yBU!rnvIbdg+r!v 2^ zSU¯lΛjSkbd:kgiXj ¥_→ogirfY vij0^ u ∈ X ¥ q ∈ Y ¥_¡U'PlvM iUÖ0 Ñ2S i+Ò < Λu, q >Y =< u, Λ∗ q >X u∈X. NPOE7QNSR.
(33) ÙÚÛÜ Ý{ÞiÞ~ßáàpâ{Ýã)àpÚäSåVÚtæ. l∞. ç â{ÚIäSåã2ènÝàéälã2åàéä>àéÛÝ=ê0ÜQÞ~èdÚ!â{Üå{å{àéäaê. . ûEüþýjÿ
(34) ¿û!"$#$&%(') *+,.-÷ÿ/ 0 ¿û12$#$3%4')56
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(37) A Ü=àwånåCBEDF<G{Ü=èãHB6IfßÝä~â çJLK èdÝ-F<M. i \ Ub vIjBz ylrdg+SU!T¥F0 : X → Y. OQPSU zSlvIySrngilU!T. G:Y →. Y BU bt¡g eg+ja iUª ySrngiyFU=roSjlebdkgijl`!{\Ub. P. ¥~BU bdPlU ySrnkTVvi Ñpl i+Ò. inf G(Λu) + F (u). kw`bdPSU!j zSU¯ljSU,z0^0 u∈X. P∗. 2Ñ S Ò rdU,`cyFU!ebdk iU!^+ OQPSg0`cU.`dgi_bnkg+jl`ËvIrnU6rnU=wvbdU,z. inf G∗ (−q) + F ∗ (Λ∗ q). \ Ub vijlz BU6bnPSU `dgi_bnkg+jl`gIo bnPSrngiSh+Pu¯bnPSU'Uªabdq¯rnU=Tvikbt^rdU!vIbdkgijl`¿0. q∈Y. P. vIjlz. P∗. Ñ2S i+Ò. F (¯ u) + F ∗ (Λ∗ q¯) =< Λ∗ q¯, u ¯ >X. V j¢bdPSUogig¡$kjSh. ØPSU=j p0 = ∞. G(Λ¯ u) + G∗ (−¯ q ) =< −¯ q, Λ¯ u >Y. ¥. kw` bnPSU egijIstShilvIbdUËgio p ØPSU=j . p0 p = ∞ p0 = 1. ê ¸< #¸ ,ð,ð ¸ ï ö wòQ. Ñ2S I0Ò ¥. p ∈]1, ∞[ p0 =. p p−1. ØPlU=j. Ó ÏSÜ1M-FlÝIßaÞFèdÚGßÜÛ Útæ1ì7RjÒSîS]òDàwåM+ÜUT;äFÜCM¡G3V±Õ 0. Ð ÏlÜèdÜ1Õ ê ¸¸ V jg+Srfe=v+`cU. inf. {q∈Y,||q||∞ ≤1}. ¥. < −div(q), f >X −β|div(q)|pp0. β = (λp)−1/(p−1) − λ(λp)−p. 0. ¥ ¥SvIjBz G(q) = ||q||1 £ krn`cb!¥S¡UrdU,e=vi~bnPlvbQorngiT JWiwz_U=r$kjSU,±+BvIkmbt^Ë¡UPlvM iUogir p ∈ [1, ∞[∪{∞} 0. Ñ2S +ùiÒ Ñ2S Iü0Ò. Λ = ∇ Λ∗ = −div F (u) = λ|u − f |pp sup. OQPal`!¥. p = 1. {r∈Y,||r||p=t}. G∗ (−q) :=. q · r = t||q||p0. sup < −q, r >Y −||r||1. r∈Y. =. sup sup < −q, r >Y −t t>0 ||r||1 =t. =. sup t||q||∞ − t. =. χK (q). t>0. Ñ2S m0Ò Ñ2S i0Ò Ñ2S S,Ò Ñ2S +iÒ Ñ2S i+Ò NPOE7QNSR. ¥.
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