HAL Id: inria-00114051https://hal.inria.fr/inria-00114051v2

Submitted on 9 Feb 2007

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Some applications of l∞-constraints in image processingPierre Weiss, Gilles Aubert, Laure Blanc-Féraud

To cite this version:Pierre Weiss, Gilles Aubert, Laure Blanc-Féraud. Some applications of l∞-constraints in image pro-cessing. [Research Report] RR-6115, INRIA. 2006, pp.33. <inria-00114051v2>

appor t de r ech er ch e

ISSN

0249

-639

9IS

RNIN

RIA/

RR--6

115-

-FR+

ENG

Thème COG

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° 6115November 2006

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

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X £ g+r u, v ∈ X

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Y £ g+r g, h ∈ Y

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|| · ||p¥Sogir

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Y

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Ñ)_ Ò

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+

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fi + α ui−fi

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||ηk||2 )

Ñ ÒAU=rnUi¥

tk > 0og+rfvIja^

kvIjlz

ηk k`fvija^U=U=TVU=j0bfgIo ∂J(uk)

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UFUQbdPlUf`cU=b5gIo`cg+_bnkg+jl`gIo­Ñt+,Ò­OQPSUQogig¡$kjShLbnPSU=g+rdU!T|ySrngiU!`bnPlvb­og+r5v¡#U!lePlg+`dU=jV`cU,±+lU=jle=Utk¥_¡#U'PlvM+ULe=gijaiU!rdh+U=jle=UÌ^ i`0Í ´a¸ ´ Æ 6 pì *Î æ

limk→∞ tk = 0ÝäM ∑∞

k=0 tk = ∞ B;ãPÏSÜ=älimk→∞ J(uk) = infu∈K J(u)

ÝäMlimk→∞ d(uk, U) = 0

Ð ÏlÜèdÜ d(·, ·) àwå'ãSÏlÜ¿ÑFlâßáàHM+ÜÝähMàwåã¤ÝIäFâÜ#æèdÚÛ ÝÞlÚàéäBã#ã ÚËÝåÜ=ãÒ¦.vija^`cU,±+lU=jle=U!``dvIbdkw`to^bnPSUQPa^0yFgIbnPSU!`dkw`giolbnPSk`bnPSU=g+rdU!T £ g+rkjl`cbnvijleU tk = 1

(k+1)p

¡$kbdPp ∈

]0, 1]gir

tk = 1(k+1) log((k+1))

OQPSk`¡$kwz_UQePSgikweU#TVvi²iU,`bnPSU#bdPSU!girnU=T|z_kZËeSbbngl`cU;ogirySrviebdkwe=vIe=giTVyS_bvbnkg+jl`¡0vITVg+jSh.vI;yFg+`n`ckSU`dU!±0SU=jBeU!`!¥TvIja^«e=giSwzÂU!viz«bng>+U=rn^ `dg¡¬eg+j0+U=rnhiU!jleU+\_giTVUfkwz_U!v+`5¡U=rnU$ySrdg+yBg0`cU,zDbdg'bvie²aU$bdPSkw`5ylrdg+SU!T vijlzDbdPlUfbdPlU=girn^Dk`;jSg¡°¡#U!BU!`cbnvIlkw`cPlU!z6Ñp`dU=U^,_¥Mù¦`og+rf`cg+TVULrnUoU!rdU!jleU,`nÒ

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Lk2 Uiv0

|J(u) − J(v)| ≤ L||u − v||2Ñ i+Ò

¡Ue=vij¯ljBzvDylvirnviTVUbdU!rdU!`n`#giy_bnkTvI`cU,±+lU=jle=UVÑ2eotv^ +ù¦`Ò¿0Í ´a¸ ´ Æ 6 wòQ Ó ÏSÜfÞ~èdÚÔ=Üâ=ã¤ÜCMËåCF<G¤êièdÝjMIàpÜäBãM+Üå=âÜ=älã Ð àéãPÏåã ܤÞoÕ

tk =D√k

Ñ I0ÒÜ=äSå>F_èdÜåLãPÏSÝIãÕ

εk = Jk − J ≤ O(1)LD√

k

Ñ +ùiÒÐ ÏlÜèdÜ D àwåãPÏSÜ¿ÑLFlâßmàHMiÜÝähMàpÝIÛVÜ=ã¤Ü=èÚcæ'ãSÏlÜå=Ü=ã K ÒOQPSkw`kw`vijU!vi`d^ePlgikweUbdgAU=jl`dSrnU5v$higag_zLrnvIbdU­gIoSe=gijaiU!rdh+U=jle=Ui¦6girnU5`dgiySPSkw`cbdkwe=vbnU!zbdU,ePSjSkw±0SU!`

e!vIjBU'z_U,`ckhijSU,zËlv+`cU,zËg+jkbdU=rvbnk+UU,`tbnkTvIbdU!`gioJÑp`dU=UÖ^!+`og+rkjl`cbnvijleUMÒ­OgVg+Sr#²ajSg¡$U!zShiUi¥

bngrnU!vi^h0vIkjËUZËekU=jBe^D¡$kmbnPËbdPlg+`dU$bdU,ePSjSkw±+lU!`!¥+g+jSUAgioébdU=jjSU=U,zS`;U=ªabdrv'kjSogirnTVvIbdkgijl`#vIFgi_b#bdPSU`dgi_bnkg+j¥Sk²+UbdPSUviySySrngMª_kTvbnU'z_k`cbnvijleUorngiT

uk bngVbdPSU`cU=bAgIo­`cg+Sbdkgijl`!V¤be!vIjbdPal`ABU,eg+TDUrnU!vi^Ëz_kZËeSbfbdgviySyS^Ëkj ySrviebdkweU+£ kjlvI^+¥l¡U'rdU,e=vi~bnPlvbAko J kw`Ae=gijaiU=ª~¥SzSkmú~U=rnU=j0bdkwvIlU+¥S¡$kmbnP\kyl`nePSkmbn'hirviz_kU=j0b,¥abdPSU!j6¡#Uz_gjlgIb$jSU=U,zbdgVjSgirnTvIk!ULbdPSU'h+rnv+z_kU!j0b!¥_vIjlz vVeg+jl`cbnvIj0b$`tbnU=y6`dk!ULU!jl`dSrdU,`$eg+j0+U=rnhiU!jleU0

Í ´a¸ ´ Æ 6 ý *Î æ3Õ||∇J(u) −∇J(v)||2 ≤ L′||u − v||2

Ñ Iü0ÒãPÏSÜ=ä>ãPÏSÜ$ÞFèdÚÔ=Üâã ÜCMêièdÝjMIàpÜäBãM+Üå=âÜ=älãLÕ

u0 ∈ Kuk+1 = ΠK(uk − t∇J(uk))

Ñ m0ÒÐ àéãPÏ>âÚIäSåã¤ÝIälã;åã Ü Þ t = 2

L′

ÜälåCF_ènÜåãSÏlÝãÕ

d(uk, U) → 0Ñ i+Ò

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c e=gazSU!`z_gjSgibU=ªSeU=U,z 300kjSU!`!;XjSUAkTVyBg+rcbvIj0b

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0 ∈ ∂J(uk)nV jlzSU=U!z¥

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J(x) = ||x||2og+rfkjl`tbvIjle=U,ÒOQPal`fv

`dkTDylU'ePSg+ke=ULkw`QbngV`dUb$bnPSU'jaSTFU=r$giokbdU!rnvIbdkgijl`!«U¡$k`cU!UkjbnPSUvIySySkwe=vbnkg+jl`#bdPlvIb#bnPSUrdU!vIbdkgij

εk ≤ O(1)LD√k

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J`dPSgiSwzBULbnPSgiSh+P0bfgIot

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ε

5OQPSk`QTVU=bdPSg_z vIySylkU!`bngVvV`dTvIU!rf`cU=bQgioylrdg+SU!T`=¥~S_bkb'`dU=U!T`fbnPlvbkmb'TVkhiP0bLFUVvIySySkU!z6bngbnPSUDySrdg+SU=T`A¡UDySrdU,`cU!j0bkj.bdPlk`LylviyBU!r!V¤bf¡$kBUbnPSU'ySSrnyBg0`cUgioo_bnSrnU'kjaiU,`tbnkh0vbnkg+jl`=

Ú Û ½ =»Ë= ÝÜ÷ÞQQÁc # ß © »¾$ « f

àvu@w á*âd<~@zQ¡y.zz6.~@9nf.¢zÊÅÙv£zOQPlk`#ylvIrvIh+rnviySPvIjlzËbdPSkw`ylvIrvIh+rnviySPVgijS^Vk`#ySrnU!`dU=j0bdU,zkjËbdPlULe=gij0bdkjaSgil`#`cU=bcbnkjShB

Ωz_U=jlgIbdU,`#v

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BV (Ω) = u ∈ L1(Ω), supξ∈C1

c (Ω; ã 2),‖ξ‖L∞(Ω)≤1

∫

Ω

u(x)div(ξ(x))dx < ∞ Ñ)_ SMÒ

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||v||G = infg||g||∞, div(g) = v, g = (g1, g2), |g| =

√

g21 + g2

2Ñ)_ ++Ò

vV`dylvie=UÖ0G = v, ||v||G < ∞ ¥

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uvIjBzÉv§bdUªabnSrdU

vl`dkjSh×bdPSU

og+g¡$kjShTVg_z_U!n0

infu∈BV (Ω),v∈G,f=u+v

∫

Ω

|Du| + λ||v||G Ñ)_ i0Ò

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og+g¡$kjSh±0ä ||sin(nx)||L2([0,2π]) = π ∀n ∈ åä ||sin(nx)||G([0,2π]) = 1/n ∀n ∈ åàvut æ_³..z6¹L³ÅÙzQ<~@|aÌEyyd.|¢zÍD#¦6U!^iU!rËz_kz°jlgIbySrngiyFg+`dU.vIja^×jaSTVU!rdkwe=vifTVUbdPlgazÉbng¢`dgiiU PSkw`ËTVg_z_U=) OQPlU6¯lr`tbvi_bdPSg+rn`¡$Plg bnrdkU!zÌbng×egiTVySSbdU.v§`dgi_bdkgijÉ¡#U!rdUg\;nç­U!`dU.vIjlzØ\~QXL`cPSU!rkjÊ^ `) OQPlU=^ÌviySySrngMª_kTvbnUbnPSU

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pvirdg+Sjlz

2

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f = u + v0^yBU!jlvIk,vbnkg+j

OQPlU=krfz_k`nernUbnU'vIySySrngMª_kTvIbdkgij¡$rnkbdU!`X0

infu,v

TV (u) + µ∑

i,j

|f − u − v|2i,j + TV ∗(v

γ)

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f = u + vÉOQPlU vi_bdPlgir`D`dgiiUkbËl`ckjSh§vÂ\aU!e=gijlz

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infu∈X

J1(u) + λ infg∈Y,div(g)=f−u

||g||∞) Ñ)_ ÒØPSU!rdU

J1k`$z_U=¯ljSU,zkj§Ñ2S,üiÒ­«U'PlvMiUbdPlUogig¡$kjShVrdU,`clmb0

ê ¸<#¸ ,ð,ð ¸ ï°ñQ pì ë ènÚjG=ßÜÛíìîjÒðï¦ñòâÝä_GÜèdÜ2æÚèÛÖF_ßÝIã¤ÜCMÝå5æÚßéßÚ Ð åÖÕ

infg∈Y,||g||∞≤α

J1(f − div(g)) Ñ2_ ++Ò

ê ¸¸ OQPSU¯lr`tb$kwz_U!vDkw`QbdPlvIbf¡#U'e!vIjl`cULbnPSUePlvIjlhiUgIoMvirdkwvIlU10

u = f − div(g)Ñ2S ++Ò

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Ybdg

X

infu∈X

J1(u) + λ infg∈Y,div(g)=f−u

||g||∞ = infg∈Y

J1(f − div(g)) + λ||g||∞ Ñ)_ 0ùiÒ

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og+rQbdPSU'kjS²ËFUbt¡U=U=jbnPSUbt¡#gVTVg_z_U=w`Òd0

infg∈Y,||g||∞≤α

J1(f − div(g)) Ñ2_ iü0Ò

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infg∈Y,||g||∞≤α

J(g) Ñ2S +ÒØkbdPh0

J(g) :=n

∑

i=1

|(∇(f − div(g)))i|2Ñ)_ 0Ò

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ãPÏSÜ=äÕ

∂(φ A)(u) = A∗0(∂φ(Au))

Ñ2S ,ÒW#PSgag0`ckjSh0

φ(x) = ||x||1Ñ)_ +Ò

«Ue=vij¡$rnkmbnUf0

J(g) = φ(∇(f − div(g)))Ñ)_ 0Ò

¦6g+rdU!giU!r!¥ ∇(f − div(g)) = A0g + b¥S¡$kbdP

b = ∇fvijlz0

A0(·) = A∗0(·) = −∇div(·) Ñ2S Ò

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gi

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∇(div(∇(f − div(gn))

|∇(f − div(gn))|2))

Ñ)_ ùiÒ

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nÑ)_ 0Ò

V jlzSU=U!z¥∂J(g)

e!vIj¢FU Uª_ySrnU!`n`cU,z×vi` ∇(div(p))¥#ogirË`dgiTVU

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16√

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α

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α = 0.1

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infu∈X

λ|u − f |pp + J1(u)Ñ2S _,Ò

og+rp ∈]1,∞[

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infu∈X

G(Λu) + F (u)Ñpl i+Ò

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infq∈Y

G∗(−q) + F ∗(Λ∗q)Ñ2S Ò

\Ubuvijlz

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V j¢bdPSUogig¡$kjShp′kw`bnPSU egijIstShilvIbdUËgio

p ØPSU=j

p ∈]1,∞[¥p′ = p

p−1

ØPlU=jp = 1¥

p′ = ∞ ØPSU=j p = ∞ ¥ p′ = 1

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infq∈Y,||q||∞≤1

< −div(q), f >X −β|div(q)|p′

p′

Ñ2S +ùiÒ

Ð ÏlÜèdÜ1Õ

β = (λp)−1/(p−1) − λ(λp)−p′ Ñ2S Iü0Ò

ê ¸¸ V jg+Srfe=v+`cUΛ = ∇ ¥ Λ∗ = −div

¥F (u) = λ|u − f |pp

¥SvIjBzG(q) = ||q||1

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supr∈Y,||r||p=t

q · r = t||q||p′

Ñ2S m0ÒOQPal`!¥

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< −q, r >Y −||r||1Ñ2S i0Ò

= supt>0

sup||r||1=t

< −q, r >Y −tÑ2S S,Ò

= supt>0

t||q||∞ − tÑ2S +iÒ

= χK(q)Ñ2S i+Ò

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£ gir 1 < p < ∞ 0

F ∗(−div(q)) := supu∈X

< −div(q), u >X −λ|u − f |ppÑ2S Ò

= supv∈X

< −div(q), (v + f) >X −λ|v|ppÑ2S ++Ò

= < −div(q), f >X + supt>0

sup|v|p=t

− < div(q), v >X −λtpÑ2S i0Ò

= < −div(q), f >X + supt>0

t|div(q)|p′ − λtpÑ2S 0ùiÒ

= < −div(q), f >X +β|div(q)|p′

p′

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t → t|div(q)|p′ −λtp Ç g+kjlhV`dgl¥_¡UL¯ljlz

β = (λp)−1/(p−1) − λ(λp)−p′

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u = f − βp′|div(q)|p′−2div(q)Ñpl +Ò

ê ¸¸ £ kr`tbf¡UrdU,e=vIFbdPlvIb F ∗∗(u) = F (u)

\agVbnPlvbF (u) = supv∈X < u, v >X −F ∗(v)

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v →< u, v >X −F ∗(v)

\agVbnPlvb0

(u − f) + βp′|div(q)|p′−2div(q) = 0Ñpl ùI+Ò

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infu∈X,|u−f |1≤λ

J1(u)Ñ2Sáùa,Ò

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infq∈Y,||q||∞≤1

< −div(q), f >X +λ|div(q)|∞Ñ2SáùI+Ò

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ui = fi + λγi(−div(q))i

|(div(q))i|Ñ2Sáù0Ò

A àéãSÏγ = (γ1, γ2, ..., γn) ∈ Y n å>Flâ÷Ï6ãSÏlÝãvÕ

γi ≥ 0 ∀i ∈ 1, 2, ..., n|γ|1 = 1γi = 0

à æ |(div(q))i| < |div(q)|∞

Ñ2Sáù Ò

ê ¸¸ /A`dkjShDbnPSU¯lr`tbfU=ª0bnrdU!TvIkmbt^rnU=wvbnkg+j¥SvIjlzbdPlUopvieb$bnPlvbF ∗∗ = F

¥_¡#Uh+Ub30

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= supu∈X

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v 6= 0¥abdPSU!j_0

(∂F ∗(v))i = fi + λγivi

|vi|Ñ2Sáùü0Ò

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ØkbdPγi`dvIbdkw`to^akjSh0

γi ≥ 0 ∀i ∈ 1, 2, ..., n|γ|1 = 1γi = 0

ko |vi|2 < |v|∞

Ñ2Sáù-0Ò

OQPSkw`rdU!vIbdkgijËz_gaU,`;jSgibQvig¡ÉbdgrdU=bdrnkU!iUuordg+T

q5AU=+U=rdbdPSU!U,`d`;kmbQhikiU,`;`dgiTVUAkj_ogirnTvbnkg+j

g+ju«Ue=vij jlgIbdkweUVbnPlvbgij«ySkmª_U!`'¡$PlU=rnU |(div(q))i| 6= |div(q)|∞

¥ui = fi

OQPlk`'TVU,vIjl`bnPlvbTvij0^ ySkª_U=w`L¡$krdU!TVvikj SjlePlvijShiU,zOQPSkw`Lk`Lkj vIh+rdU!U=TVU=j0bL¡$kmbnP.bdPlUopv+ebLbdPlvIbbnPSkw`TDg_z_U!k`v+zSvIySbdU!zbdgVrnU=TVgiULkTDylS`dkiU'jSg+k`dU10BstB`tbfbnPSU'jSg+k`d^ySkªaU!`A`cPSg+SwzFUePlvIjlhiU!z«U'PlvMiUjSgib$l`cU,zbdPSU`dU!e=gijlzUªabdrnU=Tvikbt^rdU!vIbdkgij^iU=b!V¤bfU,vizS`bngVbdPSUog+g¡$kjShrnU!`dSb0

ê ¸<#¸ ,ð,ð ¸ ï ö ö

(∇u)i = |(∇u)i|2qiÑ2S üi0Ò

ê ¸¸ G(∇u) = G∗∗(∇u)

Ñ2S üS,Ò= sup

q∈Y< ∇u, q >Y −G∗(q)

Ñ2S ü+iÒ

= < −q,∇u >Y −G∗(−q)Ñpl ü++Ò

OQPal` −q`cg++U!`;ySrdg+SU=T Ñpl ü0Ò5OQPSkw`#^0kU=wzS`#bdPlUAUª_kw`tbnU=jle=UDÑ2eot^ i+`Ò5gioTlmbnkySkU=r`

µi`cBeP

bnPlvb30

(∇u)i − µiqi = 0Ñ2S ü Ò

ØkbdPµi = 0

ko |qi|2 < 1¥_gir

µi > 0kmo |qi|2 = 1

aV j BgibdP6e=v+`cU,`¡#UhiU=bµi = |(∇u)i|2

Ñ2S üiiÒ#bnU=`$B`bdPlvIbqrnU=ySrnU!`dU=j0bn`#bnPSU'girnkU!j0bnvbnkg+jgiobnPSUU!iU=kjSU!`Qgio

u

OQPSUjaSTVU=rnke!vIkj0bdU=rnU!`cbAgio;z_lvIkbt^og+rfbdPlUBV − l1

ySrngilU!T¿kw`AjSgibeU!vir!$q;rdg+SU!T Ñ2SáùIIÒe!vIjÌBU6`dgiiU!zס$kmbnPÌg+SrvIhigirnkbdPST ¥­S_bËz_gaU,`jlgIbË`cU!U=T U,`cyFU!e=kvi^×`ckTVySU=rDbdPlvij¢bnPSUySrdkTvIylrdg+SU!T

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infu∈X,|u−f |∞≤α

J(u)Ѥùa ü+iÒ

ØkbdPX = Y n «UL¡$k`tbnlz_^VbdPlk`TVg_z_U=F¡$kmbnPËbt¡gz_kú~U=rnU=j0bylrdkgir` J ­OQPSU¯lrn`cb#g+jSUAkw`#bdPSUbngIbvIvIrnkvIbdkgij0

TV (u) = J1(u) :=

n∑

i=1

|∇u|2Ѥùa üi0Ò

OQPSU`dU!e=gijlzkw`bdPSUz_kw`nernUbdk=U,zPa^ayFU=r`cSrdopvie=Ugiou0

J2(u) :=

n∑

i=1

√

|∇u|22 + 1Ѥùa ü0ùIÒ

OQPSU'ylrdg+SU!T¿¡$kbdPJ = J1

kw`fz_U!jSgIbnU!z P1 ¥BvIjlzbdPSUgijSU¡$kbdP J = J2kw`$z_U!jSgIbnU!z P2 ;OQPlU=^PBvMiULrdU,`cyFU!ebdkiU'`cU=bn`Qgio`dgi_bdkgijB`

U1 vIjlz U2

_ u@w `$d~@·³dzzkn¡.¢zp@³.~u®zSk[ZËelmbt^ bdPlvIb¡#UU!jleg+Sj0bdU!r'l`ckjSh

TVv+`vySrdkgir'kw`bnPSUËjSg+j«ljSk±0SU!jSU!`n`'gio#bnPSUË`cg+Sbdkgij

u$ebdlvi^i¥MbnPSU=rnUU=iU!jkw`vAPaSh+U`dUbgIo~`cg+_bnkg+jl`kj1−D¥Ivi``cPSg¡$jkjbdPlU#kj0bdrngazSlebngirn^LU=ªSvITVySUi

U1k`QbnPal`$veg+j0+Uª`dUb,¥_hiU=jlU=rvI^jSgibfrdU,z_le=U!zbdgv`ckjSh+U=bdgijXjbdPlUegij0bnrnvird^+¥ibnPSU`cg+Sbdkgijl`Qgio P2 vIrnULhiU=jlU=rvI^Sjlk±0SU'U=ª_e=U=y_bfkj jSgijrnU=U=vIj0bfe=v+`cU,`=

ê ¸<#¸ ,ð,ð ¸ ï ( ( *Î æ

amplitude(f) := maxi

(fi) − mini

(fi) ≥ 2αѤùa üiü+Ò

ãPÏSÜ=ä>ãPÏSÜå=Úß F_ã)àpÚäÂÚtæ P2àwåF_äBàUaFlÜÒ

ê ¸¸ Og.ySrngiUDbnPSk`ySrngiyFg+`dkmbnkg+j¥¯Brn`cbrnU=TvIrn²6bdPBvbkoamplitude(f) < 2α

bdPSU!j§bdPSU`cg _bnkg+jl`VvIrnUËbdPlU e=gijl`cbnvij0bn`gIoAbnPSUkj0bdU!rdvi[mini(fi) + α, maxi(fi) − α]

oV jÌbdPlUgibdPSU!rVe!vi`dU!`ljSk±0SU!jSU!`n`rnU!`dSmb`Lordg+T bnPSUVopvieb'bdPlvIb

J2k`'`cbdrnkwe^>eg+j0+Uª>kj§vI5z_krdU,ebdkgijB`UªSeU!y_b'bdPSUËe=gij `cbnvij0bn`!OQPal`;bdPSU`dgi_bdkgijËkw`#SjSkw±+lUASybdgveg+jl`tbvIj0b!ØPSU!j

amplitude(f) ≥ 2αkmbQkw`;U,vi`d^bng

ylrdg+UAbnPlvbfbnPSUTVkjSkTSTøgIobdPSU`dgi_bnkg+jTl`cbfBUU,±0lvIbngmini fi + α

¥F`cgDbnPlvbfbnPSUe=gijl`cbnvij+bkw`$viebdlvi^¯Sª_U,z5OQPSkw`QU=jlzS`$bdPSU'ySrngagIot

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f = u + b¥~¡$PSU!rdU

bkw`vËSjSkogirnT ¡$PSkbdUDjSgikw`dUgij

[−α, α] £ SrdbdPSU!rL`dSySyFg+`dUbnPlvb#¡#U$PlvM+U$vySrdg+lvIlkkmbt^

P (u)giU=rbnPSUfkTvIhiU,`bdPBvb5kw`5ySrngiyFgirdbdkgijlviabdg

exp(−J(u)) f ­OQPSU!jkbfkw`QrdU,vi`dgijlviSUbdgVrnU!e=giU=ruorngiT

f`cg+akjShbnPSkw`$ySrngiSU=T¥0

supuP (u|f) = sup

uP (f |u)

P (u)

P (f) Ѥùa üj0Ò

ØPSkweP k`fU!±0SkMviU!j0bbng0

infu−log(P (f |u))− log(P (u)) Ѥùa i+Ò

AgIbdUDbnPlvbP (fi|ui) = 1

2α g [ui−α,ui+α]

uA`LbnPSUjSg+k`dUkw`ySkª_U=¡$k`dUVkjlzSU=yFU=jlzSvij0b!¥P (f |u)

kw`h+k+U=j0^0

P (f |u) =

0kmo |f − u|∞ > α

1(2α)n

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infu,|u−f |∞≤α

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vircbnkmopv+eb`fvIySyFU!virdkjShkj6bdPlUÌiiq#Y'I+iVkTVvihiUe=giTVySrnU!`n`ckgijOQPSg0`cUvircbnkmopv+eb`fvIrnUFgiSjlzSU!z6vIjBzTVkhiP0b$BUbnPSU'ySrnkjBekylvIviySySke!vbdkgijgIobdPlk`QTVg_z_U=)

_ u hk³d.~E.~d~@zl.zÅÙauAQz_khikbnvi#kTvIhiU,`virdU±+BvIj0bdk=U,z¥bt^aySke!vI^«gij

256h+rdU!^ U=+U=w`=V¤bDkw`jSgIbDkTVyFU!z_kjSh ogirbdPSU

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Q : Y → 2α åui → 2αb u

2αc + α

Ѥùa i0ÒYLkiU!jÉv ySrnkg+r

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Q−1(f) = u, f = Q(u) = u, |u− f |∞ ≤ α Ѥùa ÒikjmlonqpmnqpmrsPpZpZnqtBuqv n w$xyv^pZzBrorsPpZv XlA|nq~0|PvkpmlA|k.v(|tsPz 8o v |~rouqn Xz vP

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v+`$vDySrdkgirQkw`fvDlvizkwz_U!vogirQbnPSU!`dU'²akjBzS`$gIoz_kw`tbngirdbdkgijl`!OQPSUTvIkj6stl`cbdk¯Be=vIbdkgij§og+rbnPSUl`dUgio

TVv+`v6ylrdkgir,¥k`bdPlvIbkbDrnU=h+SwvIrnk!U!`bnPSUkTvIhiU,`=¥

¡$kbdPlgi_bSSrnrdkjSh«bdPSU U,z_hiU,`=¹V jÉbdPSU6e=v+`cU gioLvIjl∞ eg+jl`tbnrnvikj0b,¥5bdPlk`v+z_vIj0bnvihiUk`VjSgibËeU!vIrvija^0TVg+rdU+

V jlzSU=U!z¥BkTvIhikjSUbnPlvbf(i + 1, j) − f(i, j) > C > 2α

bdPSU!j bdPSUD`cU=b u ∈ X, |u − f |∞ ≤ αg+jS^>eg+j+bvIkjl`LkTvihiU!`L¡$PSkwePÂ`dvIbdkw`to^u(i + 1, j) − u(i, j) > C − 2α

VOQPal`LbdPSUl∞ e=gijl`cbdrvIkj0bU!jl`dSrdU,`#bdPlvIb$bdPSUQstSTVyl`fhirnU!vbnU=rbdPBvIj

2αvIrnUySrnU!`dU=rniU!z¥0¡$kbdP6vDTvª_kTVviz_kw`tbngirdbdkgijgIo

2α

uAjSgIbnPSU=rfrnU=Tvird²giokj+bnU=rnU!`cb$k`QbnPlvb30ê ¸<#¸ ,ð,ð ¸ ï ( þ' *Î ä«S çU¬ ì¤àHÒéÜÒ Ð ÏSÜ=ä ny = 1

ò+BU2 ⊂ U1

OQPSkw`DrdU,`clmbDk`D±0SkmbnUkj0bdU!rdU,`tbnkjShB¥­kmb`dPSg¡f`'bnPlvbvIj×U,vi`d^¡vM^ bdg«ySkwe²Âv `cg+Sbdkgij×gioU1k`

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OQPSU2 − D

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1 − De=v+`cU+¥SvIjlzySrngiyFg+`dkbdkgij Ѥùa üiÒQz_gjlgIb$PSgiwz

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bngmax(f) − α

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Ψ =

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0£¥&¡ á ºEÂ â­Eã

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Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

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ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)!#""%$'& ()(+**,*-/.10324. 5'- 62

ISSN 0249-6399