Questions variationnelles autour d'un problème de restauration d'images
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Universite Pierre et Marie Curie (Paris VI) Laboratoire Jacques-Louis Lions Memoire de synthese en vue d'une Habilitation a Diriger des Recherches Specialite : Mathematiques Questions variationnelles autour d'un probleme de restauration d'images Simon Masnou soutenu le 1er decembre 2008 devant le jury compose de MM. Fabrice Bethuel Universite Paris 6 Antonin Chambolle Ecole Polytechnique (Rapporteur) Albert Cohen Universite Paris 6 Gianni Dal Maso SISSA (Rapporteur) Guy David Universite Paris-Sud (President) Yves Meyer ENS Cachan Joachim Weickert Univ. des Saarlandes (Rapporteur)
- journees agreablement studieuses
- fac¸on fondamentale aux travaux
- diversifiee d'outils et de modeles
- magnifique lieu de travail et d'echange
- source constante d'inspiration et de motivation
- traitement de l'image
- fruit de collaborations techniques
20Habilitationvierunivjane2010rsitaire22.3.2T.able.des.mati?res.1.A.p.erturbationtedexpansionehametho.dP7.1.0.1.I.n.tro.duction....3.1.1...........b...ork.Random.30.....lab.........3.eakly.....results...24.........267.1.0.2.P.erturbation.Expansionthe:.Outlines.of.the.Metho.d..........path.....18.maximal..8.1.0.3generalSome.denitions....23.sums.random.ti.................sc...................tro.........metho9.1.0.4.Main.resultscomp.....A.................tro...................eigh.the...........W.to.el10.1.0.5.A.w.ork2.3.4ed-out.example............Limit.Asymptotic.fo.some.endan.ariables.tro.................3.1.2........11.2.Some.limit.resultsDononsrandom.trees.13.2.1.In.tr.o.duction25.a...................I.................3.2.2.di...............3.2.3.o.urn.......813ject2.2.Distances.in.T.rees......30.tation...............I.................3...................17.W13ted2.2.1toIminimalneltro.duction................2.3.3.eigh.path.the.lab...................19.Some.questions..............13.2.2.2.D.i.gital.trees21.Some.Theorems.3.1.b.vior.r.of.w.dep.t.v.23.in.duc.on...............................2313Main2.2.3.Notation.and.metho.dolo.gy......................3.1.3.i.ussi....................15.2.2.4.D.i.stanc.es.in3.2DSToly.urns.....................................3.2.1.n.duction............15.2.2.5.T.ries............26.Em.ed.ng.d...........................27.Asymptotic.osition.f.discrete..............1622.33.2.4Binaryprotreew...............................3.3.fragmen...............................3.3.1.n.duction17.2.3.1.I.n.tro.duction....................30...3.3.2ofHomogeneous.random.fragmen.tation.pro3.4.4cess39.......................Some..31.3.3.3.Exp3.4.3onenmotia.lorksfragmen.tation.probabilit.yduction...............36................32enerating3.3.4generalizedA.pro.ject.wfutureork..............tro.................................3.4.2.examples........36.3.4.A.pattern.matc.hing.probl.e.m....37.G.function.the.Moran.del.........37.Some.w........................36.3.4.1.I4nη = (I−μF )η +ξ ,n+1 n n n+1
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θ =θ +wt+1 t t+1
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μ Lt
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˜ ˆ ˜θ = θ −θ θt t t t+1
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u T u u˜ ˜ ˜ ˜θ = (I−μLφ )θ , θ =θ =−θ ,t 0 0t+1 t t 0
v T v v˜ ˜ ˜θ = (I−μLφ )θ +Lv, θ = 0,t t tt+1 t t 0
w T w w˜ ˜ ˜θ = (I−μLφ )θ −w , θ = 0.t t+1t+1 t t 0
u˜{θ }t
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w˜{v} {θ } {w}t tt+1
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δ = (I−μF )δ +ξ, δ = 0t+1 t t t 0
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= Φ(t,s)ξs
s=0
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¯I−μF = (I−μF )+μZ.t t t
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(0) (0) (0) (0)¯H = (I−μF )H +μZJ , H = 0t tt+1 t t 0
(0) (0)
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(r) (n)
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(r) (r) (r−1) (r)¯J = (I−μF )J +μZJ ; J = 0, 0≤t<rt tt+1 t t t
(n) (n) (n) (n)
H = (I−μF )H +μZJ ; J = 0, 0≤t<nt tt+1 t t t
q≥ 1 X ={X } (l×1)n n≥0
δ = (δ(r)) Xr∈N
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t <t +r≤t ≤···≤tm m m+1 s
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i ,···,i1 s
X i Xn,i n
p ≥ 1 n ∈ N G = {G} (d×d)t t≥0
(δ,p(n+2))
X
p(n+2)/2−1(r +1) δ(r)<∞.
r
D (G) j ∈ {1,···,n} 0 ≤p,n
s≤t<∞
X j/2G ···G ≤D (G)(t−s) . i i p,n1 j
pn/j
s≤i <···<i ≤t1 j
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