R 2 (orR 3). Letus onsideraPartial Dierential Equation (PDE) problem dened in Ω, written in its variational formu-

presented to obtain the

Habilitation à Diriger des Reherhes

Université Paul Sabatier Toulouse 3

Mention: Applied Mathematis

by

Didier Auroux

Fast algorithms for image proessing and data

assimilation

Defended on 26 November 2008

After reviews by:

G.Aubert, Professor Université de NieSophia-Antipolis

P. Rouhon , Professor Mines ParisTeh

F. Santosa ,Professor University of Minnesota

Commitee members:

G.Aubert, Professor Université de NieSophia-Antipolis (Reviewer)

J. Blum , Professor Université de NieSophia-Antipolis (Examiner)

L. Cohen , Researh diretor CNRS & Université ParisDauphine (Examiner)

P. Degond , Researh diretor CNRS & Université PaulSabatier (Examiner)

M. Masmoudi , Professor Université PaulSabatier, Toulouse (Advisor)

J.-P. Puel , Professor Université de VersaillesSaint-Quentin (Examiner)

1 Introdution 7

2 Imageproessing 11

2.1 Introdution . . . 11

2.2 Topologial asymptotianalysis . . . 12

2.2.1 Presentation ofthemethod . . . 12

2.2.2 Mainresult . . . 12

2.3 Inpainting [9,13℄ . . . 13

2.3.1 Crakloalization problem . . . 14

2.3.2 Dirihlet and Neumannformulations forthe inpainting problem 14 2.3.3 Asymptoti expansion . . . 15

2.3.4 Algorithm . . . 16

2.3.5 Remarks . . . 17

2.4 Restoration [16, 22℄. . . 17

2.4.1 Variational formulation . . . 17

2.4.2 Topologial gradient . . . 18

2.4.3 Algorithm . . . 19

2.4.4 Remarks . . . 19

2.4.5 Extension to olorimages . . . 19

2.5 Classiation [10,16℄ . . . 21

2.5.1 Introdutionto the lassiation problem . . . 21

2.5.2 Restoration and lassiation oupling . . . 21

2.5.3 Extension to unsupervised lassiation . . . 22

2.6 Segmentation [12,13℄. . . 23

2.6.1 From restorationto segmentation . . . 23

2.6.2 Power seriesexpansion . . . 24

2.6.3 Algorithm . . . 25

2.7 Complexityand speedingup [13, 16℄ . . . 26

2.7.1 Disreteosine transform . . . 26

2.7.2 Preonditi onedonjugate gradient . . . 27

2.8 Topologial gradientand minimalpaths [26℄ . . . 27

2.8.1 Minimal paths . . . 28

2.8.2 Fast marhing. . . 28

2.8.3 Coupled algorithm . . . 29

2.9 Conlusions and perspetives . . . 30

3 Bak and forth nudging 31 3.1 Introdution . . . 31

3.2 BakandForthNudging (BFN)algorithm [8, 11,18℄ . . . 33

3.2.1 Forward nudging . . . 33

3.2.2 Bakward nudging . . . 34

3.2.3 BFN algorithm . . . 34

3.2.4 Choie ofthe nudgingmatriesand interpretation . . . 35

3.3 Numerial experiments [11,14,21℄ . . . 37

3.3.1 Numerial hoie of the nudgingmatries . . . 37

3.3.2 Experimentalapproah . . . 37

3.3.3 Physial models. . . 38

Lorenz equations . . . 38

Visous Burgers' equation . . . 38

Shallow watermodel . . . 39

Layered quasi-geostrophi oean model. . . 39

3.3.4 Conlusions emerging fromthe numerial experiments . . . 40

3.4 Theoretial onvergene results [18, 24℄ . . . 41

3.4.1 Linear ase . . . 41

3.4.2 Transportequations . . . 42

Linear visous transport . . . 42

Visous Burgers. . . 44

Invisid lineartransport . . . 45

Invisid Burgers . . . 46

3.5 Nudging and observers[25℄ . . . 47

3.5.1 Observers for ashallow watermodel . . . 48

3.5.2 Invariant orretion terms . . . 48

3.5.3 Convergene study ona linearized simpliedsystem . . . 49

3.5.4 Numerial experiments . . . 51

3.6 Conlusion. . . 52

4 Image data assimilation 53 4.1 Introdution . . . 53

4.2 Desription of the algorithm [23℄ . . . 54

4.2.1 Constant brightness assumption . . . 54

4.2.2 Cost funtion . . . 55

4.2.3 Regularization . . . 55

4.2.4 Muti-grid approah and optimizatio n . . . 55

4.2.5 Qualityestimate . . . 57

4.3 Numerial experiments [23℄ . . . 57

4.3.1 Simulated data . . . 57

4.4 Conlusions . . . 59

5 General onlusions and perspetives 61

List of publiations 63

Publiations linked to the PhD thesis. . . 63

Publiations afterthe PhD thesis . . . 64

Referenes 67

Introdution

In this work, several problems have been studied with the ommon goal of pro-

viding robust, partiularly easy to implement, fast and powerful algorithms. The

eieny of the algorithms is required by the operational ontext of the methods,

and by a need to proess more and more data in an inreasingly short time. The

other onstraint thatwe partiularly tookinto aount isthe easeof use and imple-

mentation of the methods we have developed.

In hapter 2, we will takle various problems in image proessing by an original

approah intheeld: topologialasymptotianalysis, ormoresimplythe topologial

gradient.

There has reently been a renewed interest in image proessing thanks to new

appliations in teleommuniations and mediine: on one hand, new tehnologie s

in teleommun iat ions and diusion of information, whih now involve sending and

reeiving massive ows of numerial data (e.g. images), and on the other hand the

medial world, in whih huge progress has been made, in partiular for the early

detetion oftumors, thanksto morepowerfulimaging tehniques.

Our study is motivated by several observations. First, the topologial gradient

is generallyusedfor strutural mehanis, design,and shape optimization problems.

Also,ithasbeen suessfullyapplied in eletromagnetism for the detetionofraks

or hiddenobjets. However,manyimage proessing problemsrelyon the good iden-

tiation of a subset of the image, for instane edges or harateristi objets. This

ommonfeatureseemedinterestingtous,andallowedustoadaptthetopologialgra-

dient method, initiallyusedfor rakdetetion, toseveralimage proessingproblems

(restoration, lassiation, segmentation, inpainting).

Theseondinteresting aspet isthe speedof the method. In various elds,topo-

logial asymptoti analysis hasmade it possible to obtain good results very quikly.

However,medial imagingand audiovisual diusion(e.g. satellite television orinter-

net broadasting) both requirethe proessingtime to be negligible. Ifthe proessing

time is too large, it will delay the medial diagnosis, or the ow of data. It is thus

important to buildextremely fastshemes for solving thesevarious problems,in real

time formoviesand anegligibletime (e.g. smaller than oneseond) for images.

Aswewillseehereafter,thetopologialgradientmethodatuallyadaptsperfetly

toimageproessingproblems,allowingustoobtainveryinterestingresultsforapar-

tiularlysmall omputation ost.

In hapter 3, we will study data assimilation for environmental and geophysial

problems, and more partiularly within the framework of atmospheri and oeani

observations. For several years, one of the major onerns has been to appreiably

improve our knowledge of these turbulent systems,one ofthe major goals being the

abilityto predit their evolution with ahighreliability.

Several dierent hallenges appear in data assimilation: short-range (e.g. a few

days)weatherforeasting,thestudyofglobalwarmingandlimatehange,detetion

of extreme limati phenomena several weeks in advane, ...For all these problems,

the goals are almost similar. They onsist of estimating quikly and with a very

high degree of auray the state of a turbulent system, from the ombined knowl-

edgeofmodels anddata: on onehandmathemati al equations modelingthe oupled

atmosphere-oeansystem,andontheotherhandobservationsofdierentnature(e.g.

in situ,or satelliteobservations), orresponding tovarious physial quantities.

Beyondthe extreme size ofthe problem tobesolved(several billionsofvalues to

be identied from hundreds of millions of observations)and the omputation al time

neededtosolve it,another fatorappears: the ostof development anduseofa data

assimilationmethod. Presently, itisextremelydiult toimplement suhamethod,

even on a relatively simple problem. This motivated us to study the possibility of

improving one ofthe simplestmethods of data assimilation, nudging (also known as

Newtonian relaxation), in order to obtain muh better results without ompliating

the method.

By applying the nudging method to the bakward (in time) problem, we noted

thatitis possible to stabilizethe bakward system, whihis unstablebeause of the

irreversibility of the physial problem. Thus, as detailed in hapter 3, we an go

bak in time, and obtain a more reliable estimate of the system at a previous time,

fromwhih foreasts maybe dedued. By applyingalternatively and repeatedly the

standard nudgingmethod to the forward and bakward models, we obtain an itera-

tive algorithm that is very easy to implement and provides denitely better results

than the standard nudging. Indeed, the results are of similar quality, and are often

obtainedmuhmorequiklythan byusingthestandardvariational dataassimilation

method.

Chapter4presentsastudyat theinterfae ofthesetwo elds: the assimilationof

images. Presently, ahuge quantityofobservations omingfromsatelliteimagesises-

sentiallynotusedto improve the knowledgeofthe systemstate. However,sequenes

ofimagesobtained bysatellitesdenitely showvarious harateristi strutures(hur-

rianes,swirls, urrentsof hot water, pollution, ...) moving and evolvingin time.

Several approahesan be onsideredto solvethis kindof problem,and wemade

Thatappearedtoustobethemostadaptedhoieforrapidlyextratingonventional

data (i.e. diretly related to the model variables), and then being able to use them

in a standard assimilationsystem.

Theideathatwedevelopinhapter4isbasedontheonstantbrightnessassump-

tion, whih onsists of looking for a displaement eld that transports an image to

another one. The originalityof our approah liesin the nonlinearization of the ost

funtiontobeminimized, ombinedwith afastmethodtoassembletheJaobianma-

trix. Finally, a multi-grid approah makes itpossible to guarantee the qualityofthe

minimum. Thanks to all these tehniques, we are able to extrat omplete veloity

elds in avery shorttime, anditis alsopossibleto provide aqualityestimateofthe

identied elds, whih an be viewed aserror statistisof these pseudo-observations

within the framework ofdata assimilation.

Finally some general onlusions and researh perspetives are given in hapter

5.

Image proessing by topologial

asymptoti analysis

Thishapter summarizes the work presented in [9, 10, 12, 13, 16, 22, 26℄.

2.1 Introdution

Theideaoftopologial asymptoti analysisisto measurethe impatofa pertur-

bationofthedomainonaostfuntion. Weonlyonsiderheretheapproahthathas

beenintrodued for topologialoptimizatio n purpose, in whihthe goalistoidentify

an optimalshape and itsomplement ar yin a givendomain [133, 98, 104℄.

Topologial shape optimizatio n seems partiularly well adapted to solve image

proessingproblems(likelassiation,segmentation, enhanement ,inpainting,...),

asthey mainly onsistof identifyinga partiular subdomain ofthe image: itsedges.

Atrstsight,themainissueoftopologialshapeanalysisisthenon-dierentiability

of the problem. To nd the optimal domain is indeed equivalent to identify its

harateristi funtion. Several lassial approahes have been developed to make

this problem dierentiable. We an ite here the relaxation tehnique, whih al-

lows the harateristi funtion to take all possible values in the interval

[0; 1]

^{, and}

the level set approah where the harateristi funtion is replaed by a regular

level set funtion whih is positive inside the optimal domain and negative outside

[133, 34, 33, 36, 51, 157℄.

Theideaoftopologialasymptotianalysisistoswiththe harateristi funtion

fromonetozero(orfromzerotoone)ina(innitely)smallarea. Thus,thevariation

of the ost funtion is small when we swith a very small part from the subdomain

to its omplement ary. The topologial asymptoti expansionprovides this variation,

andallowsoneto deriveatopologial gradient oftheostfuntion [133,98, 158,157℄.

Inthis hapter, werstpresent thebasitoolsoftopologialasymptoti analysis,

and we then study several appliations to image proessing problems: inpainting

(where the goal is to ll a hidden part of an image), restoration and enhanement ,

lassiation, and segmentation. Then, we present a very eient way to speed up

allthealgorithmsintrodued inthishapter,basedondisreteosine transformsand

an appropriate preonditioning. Finally, we present a oupled approah ombining

thetopologialgradientandtheminimalpath tehniqueinordertoimprovethe edge

detetion, andto avoidnon-onnexontours.

2.2 Topologial asymptoti analysis

2.2.1 Presentation of the method

Let

Ω

^{bearegular openboundeddomain of}

R ²

(or

R ³

). Letus onsideraPartial Dierential Equation (PDE) problem dened in

Ω

^{, written in its variational formu-}

lation:

nd

u ∈ V

^suhthat

a(u, w) = l(w), ∀ w ∈ V ,

^(2.1)

where

V

^{isaHilbertspaeon}

Ω

^,usually

H ¹ (Ω)

^,

a

^{isabilinearontinuousandoerive}

formdened on

V

^,and

l

^{isalinearontinuousformon}

V

^{. Wenally onsideraost}

funtion

J (Ω, u)

^{tobe minimized, where}

u

^{isthe solution ofequation(2.1).}

We now onsider a small perturbation of the domain, e.g. by the insertion of a

rak

σ _ρ = x ₀ + ρσ(n)

^,where

x ₀ ∈ Ω

^{representsthepointwhere therakisinserted,}

σ(n)

^{is a straight rak ontaining the origin of the domain, and}

n

^{is a unit vetor}

normal to the rak. Finally,

ρ > 0

^{representsthe size of the} perturbation ,assumed to be small. Let

Ω ρ = Ω \ σ _ρ

^{be the perturbed domain. We an onsider the same}

PDE problemasbefore, but on the perturbed domain:

nd

u _ρ ∈ V ρ

^suhthat

a _ρ (u _ρ , w) = l _ρ (w), ∀ w ∈ V ρ ,

^(2.2)

where

V ρ

^,

a _ρ

^and

l _ρ

^{represent the restrition of the Hilbert spae}

V

^to

Ω _ρ

^{, and the}

perturbedbilinear and linearforms respetively.

We anrewrite theostfuntion

J

^asafuntionof

ρ

^{byonsideringthe following}

map:

j : ρ 7→ Ω _ρ 7→ u _ρ

^{solution of(2.2)}

7→ j(ρ) := J(Ω _ρ , u _ρ ).

^(2.3)

The topologial sensitivitytheory providesan asymptoti expansion of

j

^when

ρ

tendsto zero. Ittakesthe general form:

j(ρ) − j(0) = f(ρ)G(x 0 ) + o(f (ρ)),

^(2.4)

where

f (ρ)

^{is an expliit positive funtion going to zero with}

ρ

^{, and}

G(x ₀ )

^{is alled}

the topologial gradient at point

x ₀

^[133℄.

Then to minimize the riterion

j

^{, one has to insert small holes (or raks) at}

points where the topologial gradient

G

^{is the most negative, in order to make the}

ostfuntion

j

^{dereasequikly(see the asymptoti expansion(2.4)).}

2.2.2 Main result

Theorem 2.1 Ifthereexistsalinearform

L _ρ

^denedon

V ρ

^,afuntion

f : R ⁺ → R ⁺

, and fourreal numbers

δJ ₁

^,

δJ ₂

^,

δa

^and

δl

^{suh that}

• lim

ρ→0 f (ρ) = 0

^,

• J (Ω _ρ , u _ρ ) − J (Ω _ρ , u ₀ ) = L _ρ (u _ρ − u ₀ ) + f(ρ)δJ ₁ + o(f (ρ)),

• J (Ω _ρ , u ₀ ) − J (Ω, u ₀ ) = f (ρ)δJ ₂ + o(f (ρ)),

• (a _ρ − a ₀ )(u ₀ , p _ρ ) = f (ρ)δa + o(f (ρ)),

• (l _ρ − l ₀ )(p _ρ ) = f (ρ)δl + o(f (ρ))

^,

where the adjoint state

p _ρ

^{is the solutionof the adjoint equation}

a _ρ (w, p _ρ ) = − L _ρ (w), ∀ w ∈ V ρ ,

^(2.5)

and

u _ρ

^{is the solution of the diret equation (2.2), then the ost funtion}

j

^{has the}

asymptoti expansion (2.4),where the topologial gradient

G(x)

^{isgiven by}

G(x) = δJ ₁ + δJ ₂ + δa − δl.

^(2.6)

2.3 Inpainting [9, 13℄

In this setion, we present an appliation of the topologial asymptoti analysis

to theinpaintingproblem. Thegoalofinpaintingistolla hiddenpartof animage.

In other words, if we denote by

Ω

^{the original image and}

ω

^{the hidden part of the}

image,thegoalistoreoverthehiddenpart

ω

^{fromtheknownpartoftheimage}

Ω \ ω

^.

There aremanyappliations, for instaneremoving somespotson abadly preserved

movie or image, ordeleting enrustedlogos and imageson television programs, ...

This problem has been widely studied. Several methods have been onsidered:

learningapproahes(neuralnetworks,radialbasisfuntions,supportvetormahine,

...),in whih the learning data istaken in

Ω \ ω

^{, and then the} approximate funtion is evaluated in

ω

^{[177, 178℄;} minimization of an energy ost funtion in

ω

^{based on}

a total variation norm[67, 68℄; morphologial analysisfor the reonstrutionof both

artoonand texture[87℄; ...

In order to study the inpainting problem, we rst onsider a rak loalizatio n

method. Crak detetion allows us to identify the edges of the hidden part of the

image, and the inpainting problem an then be easily solved. We will onsider the

lassial thermal diusiontehnique [142, 66, 174, 175, 150℄ and improve itbymod-

eling the edges byraks. Theseraks aresupposed to be highly insulating and to

allow the temperature to jump aross edges. As both the Dirihlet and Neumann

onditionsareknownonthe boundaryofthe hiddensubset,wean deneariterion

measuringthe disrepanybetween thesolutionsofaDirihletandaNeumannprob-

lem respetively [118℄. This problem is similar to the inverse ondutivity problem,

ofapartialdierentialequationfromtheknowledgeoftheDirihlettoNeumannoper-

ator. Onlytwomeasurementsareneededtoreoverseveralsimpleraks[30,31,48℄.

Fromthe numerialpoint ofview,severalmethods [40, 49, 50, 64, 96, 152, 151℄ have

been proposed, but the topologial gradient approah seemsto be the most eient

methodfor rakloalizatio n. Theminimization ofthe riterionallowsus to identify

themainedgesinsidethehiddenpartoftheimage. Theimageisnallylledbetween

the edgesthanksto the Laplae operator.

This setion summarizes the work introdued in [9, 13℄. We also refer to these

referenesfor the results ofmanynumerial experiments.

2.3.1 Crak loalization problem

Let

Ω

^{be abounded open setof}

R ²

. We assumein this setionthat

Ω

^{ontains a}

perfetly insulating rak

σ ^∗

^{. We impose a ux}

φ ∈ H ^−1/2 (Γ)

^{on the boundary}

Γ

^of

Ω

^{, and we want to nd}

σ ⊂ Ω

^{suh thatthe solution}

u ∈ H ¹ (Ω \ σ)

^of

 



∆u = 0 in Ω \ σ,

∂ _n u = φ on Γ,

∂ _n u = 0 on σ,

(2.7)

satises

u | Γ = T

^{, where}

T ∈ H ^1/2 (Γ)

^{is a given funtion. We also assume some}

ompatibilityonditions in orderto have awell-poseddiret problem.

Atopologialgradientapproahhasbeenintrodued in[37℄,andonsistsofden-

ing a Dirihlet and a Neumann problem, as we have an over-determina tion in the

boundary onditions:

u _D ∈ H ¹ (Ω \ σ)

^suhthat

 



∆u _D = 0 in Ω \ σ, u _D = T on Γ,

∂ _n u _D = 0 on σ,

(2.8)

u _N ∈ H ¹ (Ω \ σ)

^suhthat

 



∆u N = 0 in Ω \ σ,

∂ _n u _N = φ on Γ,

∂ _n u _N = 0 on σ.

(2.9)

Itislear thatfortheatualrak

σ ^∗

^{,the twosolution}

u _D

^and

u _N

^{areequal. The}

ideaisthen to onsider andminimize the following ostfuntion

J (σ) = 1

2 k u _D − u _N k ² _L ² _(Ω) .

^(2.10)

Thetopologial asymptoti expansionof this ostfuntion isdetailed in [37℄.

2.3.2 Dirihlet and Neumann formulations for the inpainting prob-

lem

In our approah, we now denote by

Ω

^{the image and}

Γ

^{its boundary,}

ω ⊂ Ω

^the

missingpart of the image and

γ

^{its boundary. Let}

v

^{be the image that we want to}

restore. Weassume that

v

^{isknown in}

Ω \ ω

^{, and unknown in}

ω

^.

Theideaisto adapt the rakloalization method to inpainting: rakdetetion

rstallowsustoidentifytheraks(oredges)

σ

^{ofthehiddenpart}

ω

^{oftheimage,and}

then we willimposethattheLaplaianof therestored imageisequal to zeroin

ω \ σ

^.

For agivenrak

σ ⊂ ω

^,as

v

^{(Dirihletondition) and}

∂ _n v

^{(Neumann ondition)are}

known onthe boundary

γ

^of

ω

^{, we ansolvetwo dierent problems inside}

ω

^.

For a given rak

σ

^{, we denote by}

u _D ∈ H ¹ (Ω \ σ)

^{the solution of the following}

Dirihletproblem:

 

 

 



∆u _D = 0 in ω \ σ, u _D = v on γ,

∂ _n u _D = 0 on σ, u _D = v in Ω \ ω.

(2.11)

Outside

ω

^{, the solution isequal to the original image, andinside}

ω

^{, we useequation}

(2.8).

Inthesameway,ifweassume

v

^{tobeenoughregular,weanonsiderthesolution}

u _N ∈ H ¹ (Ω \ σ)

^{ofthe following Neumann problem:}

 

 

 



∆u _N = 0 in ω \ σ,

∂ _n u _N = ∂ _n v on γ,

∂ _n u _N = 0 on σ, u _N = v in Ω \ ω.

(2.12)

Notethatfromthe numerial point ofview,itismuhmoreeasyto solvean approx-

imatedNeumann problem:

 

 

 



∆u _N = 0 in ω \ σ,

∂ _n u _N = ∂ _n v on γ,

∂ _n u _N = 0 on σ,

− α∆u _N + u _N = v in Ω \ ω,

(2.13)

where

α

^{isa small positive number.}

2.3.3 Asymptoti expansion

The ost funtion remains unhanged, and is still dened by (2.10), as the idea

is to nd some raks

σ ⊂ ω

^{thatminimize the dierene between the two solutions}

u _N

^and

u _D

^{. We assume thatthe rak}

σ

^{is equal to}

x + ρσ

^{, where}

x

^{is the point of}

insertion oftherak,

ρ

^{isthe sizeof theinsertedrak(assumedto be small),and}

σ

is areferene rak, of unitnormal vetor

n

^{. Then, we anrewrite the ost funtion}

J

^{dened by equation (2.10) as a funtion}

j(ρ)

^of

ρ

^{. The asymptoti expansion is}

then the following:

j(ρ) − j(0) = f (ρ)g(x, n) + o(f (ρ)),

^(2.14)

where the topologial gradient

g

^{is dened by}

g(x, n) = − [( ∇ u _D (x).n)( ∇ p _D (x).n) + ( ∇ u _N (x).n)( ∇ p _N (x).n)] ,

^(2.15)

where

u _D

^and

u _N

^{are the solutions of (2.11) and (2.12)} respetively, but without any inserted rak (

σ = ∅

^{). Also,}

p _D

^and

p _N

^{are the} orresponding adjoint states, respetively solutions in

H ¹ (Ω)

^{of the following equations:}

 



p _D = 0 in Ω \ ω, p _D = 0 on γ,

− ∆p _D = − (u _D − u _N ) in ω,

(2.16)

 



p _N = 0 in Ω \ ω,

∂ _n p _N = 0 on γ,

− ∆p N = +(u D − u _N ) in ω.

(2.17)

The topologial gradient dened by equation (2.15) an be rewritten in the fol-

lowing way:

g(x, n) = n ^T M (x)n,

^(2.18)

where

M (x)

^{is the}

2 × 2

^(resp.

3 × 3

^{in the aseof3Dimages, or movies) symmetri}

matrixdened by

M(x) = − sym( ∇ u _D (x) ⊗ ∇ p _D (x) + ∇ u _N (x) ⊗ ∇ p _N (x)).

^(2.19)

From this equation, we an dedue thatthe minimum of

g(x, n)

^{is reahed when}

n

^{isthe eigenvetor assoiated to the lowesteigenvalue}

λ _min (M(x))

^of

M(x)

^.

2.3.4 Algorithm

Theinpainting algorithm isthen the following:

•

^{Calulation of}

u _D

^and

u _N

^{, solutions of the diret problems (2.11) and (2.12)}

respetively, without anyinsertedrak(unperturbed problem:

σ = ∅

^).

•

^{Calulation of}

p _D

^and

p _N

^{the two} orresponding adjoint states, respetively solutions of equations (2.16) and (2.17).

•

Computation ofthe matrix

M (x)

^{dened byequation (2.19).}

•

Loalizati onof the raks: dene

σ = { x ∈ ω; λ _min (M (x)) < δ < 0 } ,

^(2.20)

where

δ

^{isa negative threshold.}

•

^{Calulation of the solution of the Neumann problem (2.12) perturbed by the}

insertion of

σ

^.

Thisimage isthen equal to theoriginal image in

Ω \ ω

^{, andithasbeen}reonstruted in

ω

^.

2.3.5 Remarks

From the numerial point of view, raks are modeled by a very small ondu-

tivityinstead of onsidering real holes in the domain. The previous algorithm has a

omplexityof

O (n. log(n))

^,where

n

^{isthesizeoftheimage,i.e. thenumberofpixels,}

asexplained in setion2.7.

The main advantage of this algorithm is that the reonstrution is done in only

one iteration of the topologial gradient algorithm, whih onsistsof

5

resolutions of a PDE (the two diret and two adjoint unperturbed problems, and then one diret

perturbedproblem)inthedomain

Ω

representingtheimage. Severalnumerialresults arepresented in [9℄ andshow the qualityand eieny of the reonstrution.

Theonlyontrolparameterofthismethodisthenegativethreshold: belowagiven

value,thepixelsareonsideredasbeingpartoftheedgeset,whereasitisnotthease

beyond the threshold. The reonstruted image isprovided by the resolution of the

diret perturbed problem(2.12),andthe qualityof theimage relies onthe onnexity

of the identied edges. Ifa givenidentiededge isnot onnex,the Laplaian indeed

produes a blurred zone. Then, the threshold is usually set suh that the main

identied edges are onnex. Of ourse, it may lead to the wrong identiation of

edges. But the various numerial experiments have shown that the threshold an

be xed to an a priori value, as the optimal threshold is almost independent of the

images.

Anothersolution to thisproblem ispresented in setion2.8.

2.4 Restoration [16, 22℄

In this setion, we onsider the restoration problem, with the aim of restoring

noisyimages. Themainideaisto usethetopologialgradientfor detetingthe edges

of the noisyimage in orderto preserve themduring the restorationproess.

Thismethod is based onthermal diusion, like manyother variational methods.

Inordertoavoidblurringeets, severalnonlinearisotropi andanisotropimethods

havebeenintrodued,someofthemrelyingontheminimizationofthetotalvariation

[142,66,124, 174,175,43℄. Weshouldmentionthatsomenonvariational approahes

also exist,mainly statistialmethods [86℄.

This setion summarizes the work presented in [16, 22℄. We also refer to these

referenes for the resultsof numerial experiments.

2.4.1 Variational formulati on

Let

Ω ⊂ R ²

bean openboundeddomain,and

v ∈ L ² (Ω)

^{bethe noisyimage. The}

enhanement of

v

^{isbasedon the resolution of the following problem:}

nd

u ∈ H ¹ (Ω)

^suhthat

− div(c ∇ u) + u = v in Ω,

∂ _n u = 0 on ∂Ω,

^(2.21)

where

n

^{is the outward unit normal to}

∂Ω

^{, and}

c

^{is the} ondutivity, to be de- ned in the following. Several hoies an be made for the ondutivity, mainly

c

equal to a onstant value (linear diusion method: it is fast, but it blurs important

strutures), or

c

^{dened bya nonlinear funtion of}

∇ u

^{(nonlinear diusion method,}

edge-preserving [175, 43℄). In the topologial gradient approah,

c

^{takes only two}

values: a onstant value

c ₀

^(loseto

1

^{) in the smooth part of the image, and a very}

smallvalues

ε

^(loseto

0

^{) onthe edges or raks in orderto preservethem.}

Setting

c = 0

^{on a part of the image is equivalent to perturbing the domain by}

the insertion of raks. For a given point

x ₀ ∈ Ω

^{and for a given small parameter}

ρ > 0

^{, we onsider}

Ω _ρ = Ω \ σ _ρ

^{the perturbed domain by the insertion of a rak}

σ _ρ = x ₀ + ρσ(n)

^{, where}

σ(n)

^{isa straight rakand}

n

^{is aunit vetor normal to the}

rak. The variational formulation of the perturbed problemis the following:

nd

u _ρ ∈ H ¹ (Ω _ρ )

^{suh that}

a _ρ (u _ρ , w) = l _ρ (w), ∀ w ∈ H ¹ (Ω _ρ ),

^(2.22)

where

a _ρ

^(resp.

l _ρ

^{) is the following bilinear (resp. linear) form dened on}

H ¹ (Ω ρ )

(resp.

L ² (Ω _ρ )

^)by

a _ρ (u, w) =

Z

Ω ρ

(c ∇ u ∇ w + uw) dx, l _ρ (w) = Z

Ω ρ

vw dx.

^(2.23)

Edge detetion ifequivalent to looking for asubdomain of

Ω

^{where the energy is}

small. So our goalis to minimizethe energy norm outsideedges:

j(ρ) = J (Ω _ρ , u _ρ ) = Z

Ω ρ

k∇ u _ρ k ² .

^(2.24)

2.4.2 Topologial gradient

From theorem 2.1, we an derive the following asymptoti expansion of the ost

funtion(2.24):

j(ρ) − j(0) = ρ ² G(x ₀ , n) + o(ρ ² ),

^(2.25)

where

G(x ₀ , n) = − πc( ∇ u ₀ (x ₀ ).n)( ∇ p ₀ (x ₀ ).n) − π |∇ u ₀ (x ₀ ).n | ² ,

^(2.26)

where

p ₀

^{is the solutionof the} unperturbed adjoint problem:

− div(c ∇ p ₀ ) + p ₀ = − ∂ _u J (Ω, u ₀ ) in Ω,

∂ _n p ₀ = 0 on ∂Ω.

^(2.27)

Aspreviouslyseen,thetopologialgradientanberewritten:

G(x, n) = h M (x)n, n i

^,

where

M(x)

^{isthe following}

2 × 2

^{symmetrimatrix:}

M(x) = − πc ∇ u ₀ (x) ∇ p ₀ (x) ^T + ∇ p ₀ (x) ∇ u ₀ (x) ^T

2 − π ∇ u ₀ (x) ∇ u ₀ (x) ^T .

^(2.28)

2.4.3 Algorithm

Our algorithm onsists of inserting small heterogenei ti es (or raks) in regions

where the topologial gradient is smaller than a given threshold. There regions are

the edges ofthe image. Thealgorithm isasfollows:

•

Initialization:

c = c ₀

^{(onstant value}everywhere).

•

^{Calulation of}

u ₀

^and

p ₀

^, respetivelysolutions of the diret (2.21) andadjoint (2.27) unperturbed problems.

•

Computation of the

2 × 2

^matrix

M (x)

^{dened by (2.28), and of its lowest}

eigenvalue

λ _min (M(x))

^{at eah point of the domain.}

•

^{Set the new} ondutivity:

c ₁ =

ε

^if

x ∈ Ω

^{is suhthat}

λ _min (M(x)) < α < 0,

c ₀

^{elsewhere, (2.29)}

where

ε > 0

^{isassumed to be small,and}

α

^{is anegative threshold.}

•

^{Calulation of}

u ₁

^{, the solution of the perturbed diret problem (2.21) using}

c = c ₁

^.

Theimage

u ₁

^{isthe restored image.}

2.4.4 Remarks

Fromthenumerialpointofview,itismoreonvenienttosimulatetheraksbya

smallvalueof

c

^{insteadof onsideringtopologial} perturbation sof

Ω

^{. Theresolution}

of problem(2.21) with

c = c ₁

^{is an}approximation ofthe resolution ofthe perturbed problem(2.22),beoming morepreise as

ε

^goesto

0

^.

Asin the previous setion (inpainting problems), our algorithm is extremely ef-

ient as it requires only

3

resolutions of a partial dierential equation in

Ω

^{: the}

diret and adjoint original problems, and then the diret perturbed problem. And

the omplexityofthis algorithm is still

O (n. log(n))

^{(seesetion 2.7).}

Asshown in [16℄, the quality of the numerial results is very good. One again,

thealgorithmreliesonathresholdingofthetopologialgradientinordertodenethe

edge set. Contrary to inpainting problems, the onnexity of the edges isnot ruial

sine it does not hange signiantly the quality of the restored image. However,

setion2.8presentsawaytoidentifyonnexedges, withfewer badlyidentiededges.

2.4.5 Extension to olor images

Inthissetion, weadaptthe topologialgradient approahtoolorimages. Color

images an be represented or modeled in various ways, for instane the RGB (Red-

Green-Blue) spae in whih images are viewed asfuntions from

Ω

^to

R ³

instead of

R

. A rst approah onsists of deoupling the three hannels, and in solving diret

andadjoint problemsforeahhannel. Butitisalsopossible toonsiderdiretlythe

vetorial minimization problem, involving the resolution of vetorial problems. The

topologial asymptoti expansion is still given by equations (2.25-2.26) and (2.28),

whereallfuntionsarevetorial,i.e. thetopologialgradientisthesumonallhannels

ofthe orresponding expressionsfor eah hannel[22℄.

Another approah hasalsobeenstudied in [22℄,in whih we useadierent norm

for oupling the dierent hannels. In order to identify the loal variations of the

olor image, Di Zenzo denes a multi-spetraltensor assoiated to the image vetor

eld[83℄:

T =

t ₁₁ t ₁₂ t ₂₁ t ₂₂

, t _ij = X 3

k=1

∂u ^k

∂x _i

∂u ^k

∂x _j , 1 ≤ i, j ≤ 2,

^(2.30)

in the ase of bidimensional images. This tensor desribes the rst orderdierential

struture ofthe image, andthe DiZenzo gradient isgiven bythe square root of the

largesteigenvalue ofthe struture tensor:

k∇ u k DZ = 1

√ 2

t ₁₁ + t ₂₂ + q

(t ₁₁ − t ₂₂ ) ² + 4t ² ₁₂ ¹ ₂

.

^(2.31)

Itis possible to rewrite this gradient in a dierent waywith the following funtion:

H( ∇ u) = 1 + p

1 − 4f ( ∇ u)

2 ,

^(2.32)

where

f ( ∇ u) = det ² ( ∇ u ¹ , ∇ u ² ) + det ² ( ∇ u ¹ , ∇ u ³ ) + det ² ( ∇ u ² , ∇ u ³ )

( |∇ u ¹ | ² + |∇ u ² | ² + |∇ u ³ | ² ) ² ,

^(2.33)

det ² ( ∇ u ^s , ∇ u ^t ) = ∂u ^s

∂x ₁

∂u ^t

∂x ₂ − ∂u ^t

∂x ₁

∂u ^s

∂x ₂ 2

.

^(2.34)

Then,weanderivetheasymptotiexpansionoftheostfuntiondenedbyequation

(2.24) in whihthe norm isthe DiZenzo norm (2.31):

G(x ₀ , n) = X 3

k=1

h − πc( ∇ u ^k ₀ (x ₀ ).n)( ∇ v ^k ₀ (x ₀ ).n) − πH( ∇ u ₀ (x ₀ )) |∇ u ^k ₀ (x ₀ ).n | ² i

(2.35)

with our standardnotations.

In [22℄, we showthat this approah has the same omputation al ost asthe ve-

torialapproah(inwhihthe dierent hannels aredeoupled), whileitimprovesthe

edge detetion, and hene it produes a better restored image, more preise on the

2.5 Classiation [10, 16℄

Inthis setion, we now fouson the regularized and unsupervised image lassi-

ation problem.

Inspiredbytheworkpresented in [150, 43℄,in whihtheauthors proposealassi-

ation modeloupledwith a restorationproess,we adapthereour approahbased

on the topologialasymptoti analysis.

Thissetionsummarizestheworkpresentedin[16,10℄. Werefertothesereferenes

for the numerial results.

2.5.1 Introdution to the lassiati on problem

Let

v

^{betheoriginalimagedenedonanopenset}

Ω

^of

R ²

,andlet

C _i , 1 ≤ i ≤ n

^,be

n

^{lasses(i.e. greyorolorlevels). Werstassumethatthesselassesarepredened.}

Thegoal ofimage lassiation isto nda partitionof

Ω

^{in subsets}

{ Ω i } i=1...n

^{, suh}

that

v

^{is lose to}

C _i

ⁱⁿ

Ω _i

^.

A variational approah an be dened: it onsists of a ost funtion measuring

the dierene between the original image andthe lassied image:

J((Ω _i ) _i=1...n ) = X n

i=1

Z

Ω i

(v(x) − C _i ) ² dx + α X

i6=j

| Γ _ij | ,

^(2.36)

where

Γ _ij

^{representsthe interfae}

Ω _i ∩ Ω _j

^{between two subsets.}

Themain diultyof this approah isthat the unknowns aresets, and not vari-

ables. This is why the topologial asymptoti analysis seems to be appropriate for

solving this problem. The topologial gradient and the orresponding numerial re-

sultsarepresented in [10℄.

2.5.2 Restoration and lassiation oupling

Anothersolutiononsistsofouplinglassiation with restoration,and toadapt

theapproahintrodued insetion2.4. Theideaistorstonsideraniterationofthe

topologialasymptoti analysisfor theimage restorationprobleminorderto smooth

the image, and then to lassify this smooth image without anyregularizati on. Ifwe

removetheregularizati ontermfromequation(2.36),whihleadstotheunregularized

lassiation problem, then the optimal subset

Ω i

^{is the set of pixels that areloser}

to

C _i

^{than to any other}

C _j

^{. In other words, eah pixel is assigned to the subset}

orresponding to itslosest lass.

In the perturbed problem (2.29), instead of setting

c = 0

^(or

c = ε

^{from the}

numerial point of view) onthe edge set and

c = c ₀

^{elsewhere, we set}

c ₁ =

( ε

^{on the edge set,}

c ₀

ε

^elsewhere.

(2.37)

•

^{Appliation of the} restorationalgorithm dened in setion 2.4, with

c ₁

^dened

by(2.37) insteadof (2.29).

•

Unregularized lassiation ofthe image

u ₁

^{, usingfor examplethe losest lass}

algorithm (in whih eah pixel is assigned to the subset orresponding to its

losest lass).

Aspreviouslyseen,the omplexityofthisalgorithm is

O (n. log(n))

^{, andthevari-}

ousnumerialresultspresentedin[10℄showtherelativeeienyoftheseapproahes.

Moreover,itispossible toregularize moreorlessthe imagebyhoosingdierent val-

uesof

c ₁

^{, and itallows us toalso obtain good resultson noisyimages.}

2.5.3 Extension to unsupervised lassiati on

If the number

n

^{of lasses is given, but not their values}

C _i

^{, it is possible to}

determinethem in an optimalway. Thislassiation probleman be dened as

(Ω min i ),(C i ) j((Ω _i ), (C _i )) = X n

i=1

Z

Ω i

| v(x) − C _i | ² dx + α X

i6=j

| Γ _ij | .

^(2.38)

Theideaistominimizetheostfuntion

j((Ω i ), (C i ))

alternativelywithrespetto

Ω _i

^{andwithrespetto}

C _i

^{. The}minimizationwith respetto

Ω _i

^{onsistsoflassifying}

the image, while the minimization with respet to

C _i

^isobtained straightforward by the meanof the image in eahlass:

C _i = 1

| Ω _i | Z

Ω i

v(x) dx.

^(2.39)

Theunsupervised lassiation algorithm is then asfollows:

•

Initialization: dene an initial guess

C ₁ , . . . , C _n

^(e.g. equi-distributed lasses).

•

^{Repeatuntil onvergene:}

Calulate the lassied image using the lasses

C ₁ , . . . , C _n

^{(see previous}

algorithm).

Updatethe values ofthe lasses using (2.39).

If the number

n

^{of lasses is not given, we an add a} penalizatio n term

+βn

in the ostfuntion (2.38), measuringthe number oflasses. Theminimization with

respetto

n

^{providesthe optimalnumber oflasses. The number oflasses islearly}

relatedto the hoie of the weighting oeient

β

^.

2.6 Segmentation [12, 13℄

Thissetionisonernedwithimagesegmentation,whihaimistondapartition

ofanimageintoitsonstituentparts. Theideaisstilltoapplyourtopologialgradient

basedalgorithm forthe detetion ofedges in the image.

Several approahes have been studied in the literature. One an ite variational

methods, for example based on the minimization of the Mumford-Shah funtional

[136℄,theativeontoursandsnakemethods [55,156℄,stohastiapproahes[54, 61℄,

wavelets, ...[43, 45, 42, 140, 149, 150, 176℄.

This setion summarizes the main results presented in [12, 13℄. Several numer-

ial experiments are also detailed in these referenes and show the eieny of our

approah.

2.6.1 From restoration to segmentation

Westill onsiderthe restorationalgorithm,in whihthe following ondutivityis

usedfor the perturbed problem:

c(ε) =

( ε

ⁱⁿ

ω, 1

ε

^outside

ω,

^(2.40)

where

ω ⊂ Ω

^{represents the edge set. We rst assume that}

ω

^{is thikened (i.e. of}

odimension

0

ⁱⁿ

Ω

^{). Fromequation(2.40),the algorithmnowonsistsofsolvingthe}

following problem:

( P ε )

 

 

 



− div(ε ∇ u _ε ) + u _ε = v in ω,

− div 1

ε ∇ u _ε

+ u _ε = v in Ω \ ω,

∂ _n u _ε = 0 on ∂Ω,

(2.41)

where

u _ε ∈ H ¹ (Ω)

^{, i.e. with the impliit boundary ondition that}

c(ε)∂ _n u _ε

^hasthe

same valueon both sidesof

∂ω

^.

Thenwe havethe following asymptoti result[12℄:

Theorem 2.2 If wedenoteby

u _ε

^{the uniquesolutionofproblem}

( P ε )

ⁱⁿ

H ¹ (Ω)

^{, then}

ε→0 lim ( k∇ u _ε − ∇ u ₀ k L ² (Ω\ω) + k u _ε − u ₀ k L ² (ω) ) = 0,

^(2.42)

where

u ₀ ∈ H ¹ (Ω \ ω) ∩ L ² (Ω)

^{isthe solution tothe following problem}

( P 0 )

 

 

 



u ₀ = v in ω,

− div ( ∇ u ₀ ) = 0 in Ω \ ω,

∂ _n u ₀ = 0 on ∂ω,

∂ _n u ₀ = 0 on ∂Ω.

(2.43)

This result proves that the segmented image

u ₀

^{an be} approximated by

u _ε

^if

ε

is small. We now assume that the edge set

ω

^{is of odimension}

1

ⁱⁿ

Ω

^{. From the}

point of view of appliations, it is ompletely natural to assume that the edges are

atintheimage. Inordertohaveoherentnotations, wewillfurtherdenoteby

σ

^the

edgeset. Weassumethat

σ

^{isknown, e.g. provided bytherakdetetionalgorithm}

previouslyseen.

We an rewrite the approximated segmentation problem

( P ε )

^asfollows:

( ˜ P ε )

 

 

 



− div 1

ε ∇ u _ε

+ u _ε = v in Ω \ σ,

∂ _n u _ε = 0 on σ,

∂ _n u _ε = 0 on ∂ Ω,

(2.44)

where

u _ε ∈ H ¹ (Ω \ σ)

^{. If}

v ∈ L ² (Ω)

^{, then problem}

( ˜ P ε )

^{has a unique solution in}

H ¹ (Ω \ σ)

^{. Asa orollaryof the previousresult, we have the following one [12℄:}

Theorem 2.3 If we denote by

u _ε

^{the unique solution of problem}

( ˜ P ε )

ⁱⁿ

H ¹ (Ω \ σ)

^,

then

k u _ε k L ² (Ω) ≤ k v k L ² (Ω) , k∇ u _ε k L ² (Ω\σ) ≤ √

ε k v k L ² (Ω) ,

^(2.45)

and

ε→0 lim k∇ u _ε − ∇ u ₀ k L ² (Ω\σ) = 0,

^(2.46)

where

u ₀ ∈ H ¹ (Ω \ σ)

^{is the unique solutionto the following problem:}

( ˜ P 0 )

 

 

 

 

 



− div ( ∇ u ₀ ) = 0 in Ω \ σ, Z

Ω i

u ₀ = Z

Ω i

v ∀ Ω i

^{onnex omponent of}

Ω \ σ,

∂ _n u ₀ = 0 on σ,

∂ _n u ₀ = 0 on ∂ Ω.

(2.47)

For numerial reasons, itan be verydiult tosolvediretlyproblem

( ˜ P 0 )

^{, and}

evenproblem

( ˜ P ε )

^{fortoosmallvaluesof}

ε > 0

^{. Indeedthe}onditioningofthesystem to be solved goes to innity when

ε → 0

^{. In order to overome this issue, we will}

expandthe solution

u _ε

^{of problem}

( ˜ P ε )

^{into a power series of}

ε

^.

2.6.2 Power series expansion

From the knowledge of the power series expansion of

u _ε

^{and the omputation of}

several solutions

u _ε

^{for not toosmall oeients}

ε > 0

^{, itispossible to} approximate the asymptotisolution

u ₀

^[12℄:

Theorem 2.4 There exist a onstant

ε _R > 0

^{and a family of funtions}

(u _n ) _n∈ N

^of

H ¹ (Ω \ σ)

^{suh that for all}

0 ≤ ε ≤ ε _R

^,

u _ε = X ∞

n=0

u _n ε ⁿ .

^(2.48)

Moreover,

u ₀

^{isthe unique solutionin}

H ¹ (Ω \ σ)

^{of problem}

( ˜ P 0 )

^{,andthe other fun-}

tions

(u _n )

^{are the unique solutions in}

H ¹ (Ω \ σ)

^{of the following problems:}

( ˜ P 1 )

 

 

 

 

 



− div ( ∇ u ₁ ) = − u ₀ + v in Ω \ σ, Z

Ω i

u ₁ = 0 ∀ Ω _i

^{onnex omponent of}

Ω \ σ,

∂ _n u ₁ = 0 on σ,

∂ _n u ₁ = 0 on ∂Ω,

(2.49)

n ≥ 2, ( ˜ P n )

 

 

 

 

 



− div ( ∇ u _n ) = − u _n−1 in Ω \ σ, Z

Ω i

u _n = 0 ∀ Ω _i

^{onnex omponent of}

Ω \ σ,

∂ _n u _n = 0 on σ,

∂ _n u _n = 0 on ∂Ω.

(2.50)

We andene a funtion of

ε ∈ R ⁺

asfollows

f (ε) := u _ε ∈ H ¹ (Ω \ σ).

^(2.51)

From the previous theorem, we know that

f

^{has a power series expansion at the}

origin given by (2.48). We onsider a family of

N

^points

(ε i )

ⁱⁿ

[ε c , ε _R ]

^{, where}

ε _c

^is

the smallest value of

ε

^{for whih it is easy to numerially ompute}

f (ε)

^{, and}

ε _R

^is

smaller than the onvergene radius of the power series. We an then ompute an

interpolatio n polynomial

g _N

^{of degree}

N − 1

^{dened by:}

g _N (ε) = X N

i=1



 Y N

j=1,j6=i

ε − ε _j ε _i − ε _j



 u _{ε i} ,

^(2.52)

where

N

^{isthe number of points}

ε _i

^.

Theanalyityof

f

^{allowsus to estimatethe} approximation error:

k u ₀ − g _N (0) k H ¹ (Ω\σ) = O (ε ^N _c ).

^(2.53)

2.6.3 Algorithm

Wean thendene asegmentation algorithm,basedon therestorationalgorithm

previously dened in setion2.4:

•

^{Solve the diret (2.21) and adjoint (2.27)} unperturbed problems with

c = c ₀

everywhere.

•

^{Compute the}

2 × 2

^matrix

M (x)

^{dened by equation (2.28) and its lowest}

eigenvalue

λ _min (M(x))

^{at eah point of the domain}

Ω

^.

•

^Dene

σ = { x ∈ Ω; λ _min < α < 0 }

^{the edge set, where}

α

^{is a small negative}

•

^Set

ε _c > 0

^{the minimal value of}

ε

^{for whih it is easy to ompute numerially}

the solution

u _ε

^{of problem}

( ˜ P ε )

^.

•

^Choose

N ∈ N ^∗

in order to have an approximationerror in

O (ε ^N _c )

^{, and hoose}

N

^{dierent values}

(ε _i )

^.

•

^{Compute the solutions}

(u _{ε i} )

ⁱⁿ

H ¹ (Ω \ σ)

^{of problems}

( ˜ P ε i )

^.

•

^{Compute the}interpolatio n polynomial

g _N

^ofdegree

N − 1

^{,dened byequation}

(2.52),for

ε = 0

^.

Thisalgorithmhasaomplexityin

O (N.n. log(n))

^,where

n

^{isthenumberofpixels}

in the image, and

N

^{is the degree of the} interpolatio n approximation. In numerial experiments,

N

^{is typiallyof the orderof}

2

^to

5

^.

Several numerial testsaredetailedin [12℄.

2.7 Complexity and speeding up [13, 16℄

Inthis setion, we present the tehniques thatwe have usedfor solving the PDE

problems previously seen, and that lead to a theoretial omplexity in

O (n. log(n))

[13℄. Several numerial experimentshave onrmed thisomplexity[16, 13℄.

2.7.1 Disrete osine transform

Inall the algorithmswepresented in the previous setions,we only have to solve

the following PDE

− div(c ∇ u) + u = v in Ω,

∂ _n u = 0 on ∂Ω,

^(2.54)

forvarious oeients

c

^{. Therst}resolutions aredonewith a onstant valueof

c

^{. It}

isthen possibletolargely speeduptheomputationtime byusingthe disreteosine

transform(DCT) method. Problem(2.54) isthen equivalent to

X

m,n

1 + c(mπ) ² + c(nπ) ²

u _m,n φ _m,n = X

m,n

v _m,n φ _m,n ,

^(2.55)

where we denote by

φ _m,n = δ _m,n cos(mπx) cos(nπy)

^{a osine basisof}

R ²

, and where

(v _m,n )

^{representtheDCToeientsoftheoriginalimage}

v

^{. Itisthen}straightforward toidentify

(u _m,n )

^{, the DCToeients of}

u

^{in equation(2.55):}

u _m,n = v _m,n

1 + c(mπ) ² + c(nπ) ² .

^(2.56)

Theomplexityofsuharesolutionis

O (n. log(n))

^,where

n

^{isthenumberofpixelsof}

the image. The resolution of all unperturbed problems is then done in the following

way:

•

Computation of

v _m,n

^{, the DCToeientsof the original image}

v

^.

•

Computationof

u _m,n

^{, the DCToeientsof}

u

^{fromequation(2.56).}

•

Computationof

u

^{usingan inverse DCT.}

2.7.2 Preonditioned onjugate gradient

Then,thesolutionofallpreviouslydetailedproblemsomesfromtheresolutionof

aperturbed problem. Forthe lastresolution ofa diretproblemwith anon onstant

oeient

c

^{, we an rewrite the problemin the following way:}

A(c)u = B,

^(2.57)

where

u

^{is the unknown image. If}

c

^{is onstant, equation (2.57) is easy to solve.}

The idea is to preondition equation (2.57) with the DCT solver used in the rst

resolution. Problem(2.57) isequivalent to

A(c ₀ ) ⁻¹ A(c) u =

A(c ₀ ) ⁻¹ B

.

^(2.58)

As

c

^{islose to}

c ₀

⁽

c

^{isindeedequal to}

c ₀

^{,exeptin anegligible partofthe domain),}

the systemmatrix

[A(c ₀ ) ⁻¹ A(c)]

^{is lose to the identityoperator,and the resolution}

of (2.58) isthen easy: weusea preonditioned onjugate gradient (PCG)methodto

solve this problem. As the oeient

c

^{is lose to}

c ₀

^{, we an expet a}

O (n. log(n))

omplexityfor the resolution of the perturbed problem. The numerial experiments

learly onrm this omplexity, both for smalland large problems.

The main advantage is that it allows us to proess images in a very short time

(e.g.

1600 × 1200

^{images in lessthan one seond) and movies in real time (provided}

the movie is splitinto shortsequenes of afew seonds)with a ++ode.

2.8 Coupling between the topologial gradient and the

minimal path tehnique [26℄

As previously seen, e.g. in setion 2.3, it is ruial to identify onneted (or

ontinuous) ontours. Up to now, we had to threshold the topologial gradient with

a nottoo smallvalue,in order toidentifyonneted ontours,but thisleads to thik

identied edges, and alsoto onsider morenoisypointsaspotential edges.

We notied thatthe edges orrespond to valleylines of the topologial gradient.

It isofourse possibleto identify them byadapting the thresholdoeient, butwe

propose here to use the minimal path and fast marhing tehniques for identifying

the valleylinesof the topologial gradient [71, 73, 82, 84, 179, 145, 164℄.

Inthe following,we onsideranyofthe previousimage proessing problems. We

only assume that the topologial gradient

g

^{has been dened and omputed every-}

where. The goal is to identify the valley lines orresponding to the most negative

parts ofthe topologial gradient.

Thissetion summarizes the study presented in [26℄, in whih several numerial