A formal $\Gamma$-convergence approach for the detection of points in $2$-D images

(1)

HAL Id: inria-00418526

https://hal.inria.fr/inria-00418526

Submitted on 19 Sep 2009

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

points in 2-D images

Daniele Graziani, Laure Blanc-Féraud, Gilles Aubert

To cite this version:

Daniele Graziani, Laure Blanc-Féraud, Gilles Aubert. A formal Γ-convergence approach for the

de-tection of points in 2-D images. [Research Report] RR-7038, INRIA. 2009, pp.25. �inria-00418526�

(2)

a p p o r t

d e r e c h e r c h e

9 -6

3

9

9 IS

R

N

IN

R

IA

/R

R

--7

0

3

8 --F

R

+

E

N

G

Thème COG

A formal Γ-convergence approach for the detection

of points in 2-D images.

Daniele Graziani, Laure BLANC-FÉRAUD, Gilles Aubert

N° 7038

2009

2

(3)

(4)

2

Daniele Graziani

∗

, Laure BLANC-FÉRAUD

∗

,Gilles Aubert

†

5

ThèmeCOG Systèmes ognitifs 6

ProjetARIANA 7

Rapportdere her he n° 70382009xxivpages 8

Abstra t: Weproposeanewvariationalmodeltoisolatepointsina

2

-dimensionalimage. 9

Tothispurposeweintrodu easuitablefun tionalwhoseminimizersaregivenbythepoints 10

wewanttodete t. Inordertoprovidenumeri alexperimentswerepla ethisenergywitha 11

sequen eofamoretreatablefun tionalsbymeansofthenotionof

Γ

- onvergen e. 12

Key-words: pointsdete tion, urvature-dependingfun tionals,divergen e-measureelds, 13

Γ

- onvergen e,biologi al

2

-Dimages. 14

∗

ARIANAProje t-team, CNRS/INRIA/UNSA,2004Routedeslu ioles-BP93,06902Sophia-Antipolis Cedex,Fran e

†

LABORATOIREJ.A.DIEUDONNÉUniversitédeNi eSOPHIAANTIPOLIS,par valrose06108Ni e CEDEX2,FRANCE.

(5)

2

-D images. 16 Résumé: 17 Mots- lés : 18

(6)

Contents 19 1 Introdu tion ii 20 2 Preliminaries v 21

2.1 Convergen eforasetofpoints . . . v 22

2.2 Distributionaldivergen e . . . v 23

2.3 TheDiri hletproblemwithmeasuredata . . . vi 24

3 Existen eresult vii

25

4

Γ

- onvergen e x

26

4.1 Modi aMortola'sapproa h . . . x 27

4.2 DeGiorgi's onje ture . . . x 28

5 The approximating fun tionals xii

29

6 Dete tion xv

30

6.1 Dis retization . . . xv 31

6.2 Dis retizationintime . . . xvi 32

7 Computer examples xvii

33

7.1 Parametersettings . . . xvii 34 7.2 Commentaries. . . xvii 35 8 Con lusion xxii 36 Referen es xxiii 37

(7)

1 Introdu tion 38

Dete tingne stru tures, like pointsor urvesin twoorthree dimensional images respe -39

tively,is animportantissue in image analysis. In biologi alimagesapoint may represent 40

aviralparti lewhosevisibilityis ompromisedbythepresen eof otherstru tureslike ell 41

membranesor somenoise. 42

Fromavariationalpointofviewtheproblemofisolatingpointsisadi ulttask,sin eit 43

isnot learhowthesesingularitiesmustbe lassiedintermsofsomedierentialoperator. 44

Indeed, sin e these are usually dened asdis ontinuity without jump, we annot use the 45

gradientoperator as in the lassi alproblem of ontours dete tion. As a onsequen ethe 46

fun tionalframework wehavetodealwithmaybenot lear. 47

Onepossiblestrategyto over omethis obsta leis onsidering thesekindsof pathology 48

asa

k

- odimensionobje t,meaningthattheyshouldberegardedasasingularityofamap 49

U

: R

k+m

_{→ R}

k

, with

k

≥ 2

and

m

≥ 0

(see[7℄fora ompletesurveyonthis subje t). So 50

thatthedete tingpoint ase orrespondsto the ase

k

= 2

and

m

= 0

. 51

Inthisdire tion,in[5℄,theauthorshavesuggestedavariationalapproa hbasedonthe 52

theory of Ginzburg-Landau systems. In their work the isolated points in

2

-D images are 53

regardedasthetopologi alsingularitiesofamap

U

: R

2 _{→ S}

1

,where

S

1

isaunit sphereof 54

R

2

. Sothatitis ru ialto onstru t, startingfrom theinitialimage

I

: R

2 _{→ R}

,aninitial 55

ve toreld

U

0 : R

2 _{→ S}

1

with atopologi alsingularityofdegree

1

at pointsin theinitial 56

image

I

where theintensity is high. Howto dothis in arigorousway,it isasubje tof a 57

urrentinvestigation. 58

Hereourpurposeistoprovidealightervariationalformulation,inwhi hthesingularity 59

in the image is dire tlygivenin terms of a properdierentialoperator dened on ve tor 60

elds. Another importantdieren e is that in [5℄ points and urves are dete tedboth as 61

singularities,whilein the presentpaperouraim is toisolatefrom the initialimage points 62

andatsametimeremoveanyothersingularitieslike urvespre isely. Inourmodel

Ω ⊂ R

2

63

isanopensetandrepresentstheimagedomain. 64

In order to dete t the singularities of the image, we have to nd a fun tional spa e 65

whose elements generate, in a suitable sense, a measure on entrated on points. Su h a 66

spa eis

DM

p

_(Ω)

introdu ed in[4℄where

1 < p < 2

,thespa e ofve torelds

U

: Ω → R

2

67

whose distributional divergen eis a Radonmeasure(see Subse tion2.2 for denitions and 68

examples). Therestri tion

1 < p < 2

isdue totheinizialization(seesubse tion

2.3

). 69

Unfortunatelyevenifweare apableof onstru tinganinitialve toreld

U

0

(seebelow 70

for su h a onstru tion) belonging to the spa e

DM

p

_(Ω)

, its singular set ould ontains 71

severalstru tureswewanttoremovefromtheoriginalimagelikeforinstan e urveorsome 72

noise. Hen e, after the initialization we haveto lear away all the stru tures we are not 73

interestedin bybuilding up,startingfrom theinitial data

U

0

, anewve toreld

U

whose 74

singularitiesaregivenbythepointsoftheimage

I

wewantto isolate. 75

Thus, fromonehand wehavetofor e the on entrationset ofthedistributional diver-76

gen eof

U

0

to ontainonly the pointswewant to at h, and on theother hand we have 77

toregularizetheinitialdata

U

0

outsidethepointsof singularities. Tothisend wepropose 78

tominimizeanenergyinvolvinga ompetitionbetweenadivergen etermandthe ounting 79

(8)

Hausdormeasure

H

0

. Morepre iselytheenergyisthefollowing 80

Z

Ω\P

|divU |

2 + λ

Z

Ω

|U − U

0 |

p

+ H

0 (P ).

(1) where

U

∈ L

p,2

_{(div; Ω \ P )}

is thespa eof

L

p

-ve torelds whosedistributional divergen e 81

belongsto

L

2 _{(Ω \ P )}

,

P

istheatomi setwewanttotargetand

λ

isapositiveweight. The 82

rstintegralfor es

U

toberegularoutside

P

,whiletheterm

H

0 _{(P )}

penalizesthepresen e 83

ofsingular urvesin theimage. 84

Fromapra ti alpointofview,this hoi eallowsustoworkwitharstorderdierential 85

operator and permits us to formulate the minimization problem in a ommon fun tional 86

framework. 87

Clearly for inizialiting the minimization pro ess weneed to onstru t from the initial 88

image,ave toreld

U

0

belongingto

DM

p

_(Ω)

. Su have toreld anbeprovidedbythe 89

gradientofweaksolutionofthe lassi alDiri hletproblemwithmeasure data. 90

(

∆f = I

on

Ω

f

= 0

on

∂Ω.

(2)

Inordertoprovide omputerexamples,wemustapproximatefun tional(1)bymeansofa 91

sequen eofmore onvenientfun tionals. 92

The approximation, wesuggest in this paper, is basedon the so alled

Γ

- onvergen e, 93

thenotion of variational onvergen e introdu ed byDe Giorgi (see [13, 14℄). This theory 94

is designed to approximate a variational problem by a sequen e of dierent variational 95

problems. Themostimportantfeatureofthe

Γ

- onvergen ereliesonthefa tthatitimplies 96

the onvergen e of minimizers of the approximating fun tionals to those of the limiting 97

fun tional. Inthisworkwesuggestapossible

Γ

- onvergen eapproa h forthedete tionof 98

points. Bythewaywe stressoutthat the

Γ

onvergen eresultis only onje tured in this 99

paper, whose purpose is to test anew variational method from an experimental point of 100

view. Forarigourousvariationalapproximationin aparti ular ase,wereferthereaderto 101

[6℄. 102

Classi allyvariationalapproximationte hniques su h as

Γ

- onvergen eor ontinuation 103

method(see[2,3℄and[20℄respe tively)havebeensu esfullyemployedinimageandsignal 104

pro essing. 105

Forinstan ein ([2,3℄)AmbrosioandTortorellihaveproventhatthe lassi al Mumford-106

Shah's fun tionalfor dete ting

1

-dimensional smoothboundaries, anbeapproximatedby 107

asequen eofellipti fun tionalsthat arenumeri allymoretreatable. 108

Themaindi ultyhereisrelatedtothepresen eofa odimension

2

obje t,whi hisnot a ontour:theset

P

. Inordertoobtainavariationalapproximation losetotheoneprovided in ([2,3℄), the ru ial stepis then to repla ethe term

H

0 _{(P )}

of fun tional (1) bya more handy, from a variationalpoint of view, fun tional involvinga smooth boundary and his perimetergivenbythe

1

-dimensionalHausdormeasure

H

1

. Followingsomesuggestionfrom thepaperofBraidesand Mal hiodi andBraides and Mar h (see[9,10℄) su h afun tional

(9)

isgivenby:

G

β

ε

(D) =

1 4π

Z

∂D

1 β

ε

+ β

ε

κ

2 (x)dH

1 (x),

where

D

is a properregular set ontainingthe atomi set

P

,

κ

is the urvature of its 109

boundary,the onstant

1 4π

isanormalizationfa tor,and

β

ε

innitesimalas

ε

→ 0

. Roughly 110

speakingtheminimaofthisfun tionalarea hievedontheunionofballsofsmallradius,so 111

thatwhen

β

ε

→ 0

thefun tionalshrinks totheatomi measure

H

0 _{(P )}

. Ontheotherhand 112

theintrodu tion ofa urvaturetermrequiresanontrivialand onvenient,foranumeri al 113

pointofview,approximationofthe urvature-dependentfun tional. Su hanapproximation 114

isbasedona elebrated onje tureduetoDeGiorgi(see[12℄). Bymeansofthisargument 115

it is possibleto substitute the urvature-depending fun tional with an integralfun tional 116

involving the Lapla ian operator of smooth fun tions. Then it remains to approximate 117

the

H

1

-measure and this an be done by retrieving a lassi al gradient approa h used in 118

[15, 16℄. This strategy allows to deal with a fun tional whose Euler-Lagrange equations 119

anbedis retized. A simpleand intuitiveexplanation ofthe onstru tionof the omplete 120

approximatingfun tionals willbegiveninse tion

3

. 121

Thepaperis organizedasfollows: se tion

2

is intendedto remind thereaderof math-122

emati altoolsuseful in the following. In se tion

3

weaddressthe existen eresult forthe 123

fun tional

F

(U, P )

denedin (1). In se tion

4

westatethetwowell-known

Γ

- onvergen e 124

results we need in the sequel. In se tion

5

we build in a formal way the approximating 125

sequen e. Inse tion

6

wepresentthedis retemodelandthedete tionstrategy. Finallythe 126

lastse tionisdevotedtosome omputerexamples. 127

(10)

2 Preliminaries 128

2.1 Convergen e for a set of points 129

Forourpurposeitwillbe ru ialdealingwithanotionof onvergen efornitesetsofpoints 130

introdu edin[10℄. 131

Denition 2.1. We say that a sequen e of a nite set of points

{P

h

} ⊂ Ω

onverges to 132

a set

P

⊂ Ω

if ea h of the sets

P

h

ontains a number

N

of points

{x

1 h

, ..., x

N

h

}

, with

N

133

independent of

h

,su hthat

x

i

h

→ x

i

for any

i

= 1, ..., n

and

S

N

i=1

{x

i

} = P.

134

Lemma 2.1. Let

{P

h

}

be asequen e of a nite set of pointssu h that

H

0 _(P

h

) ≤ N

0

for 135

every

h

with

N

0 ∈ N

. Then there exists a subsequen e

{P

h

k

} ⊂ {P

h

}

and a set of points 136

P

⊂ Ω

su hthat

P

h

k

onvergeswithrespe ttothe onvergen e 2.1tothe set

P

. 137

Proof. Sin e

H

0 _(P

h

) ≤ N

0

,wemaynd

N

1 ≤ N

0

su hthateveryset

P

h

ontains

N

1

points. 138

Forevery

i

= 1, ..., N

1

andthereexists asubsequen e

x

i

h

k

⊂ x

i

h

onvergingto

x

i

_{∈ Ω}

. 139 Thenbysetting

P

h

k

=

S

N

1 i=1

x

i

h

k

and

P

=

S

N

1 i=1

x

i

,thethesisis a hieved. 140

Lemma2.2. Let

{P

h

} ⊂ Ω

beasequen eofnitesetof points onvergingtoanitesetof 141 points

P

. Then 142

H

0 _{(P ) ≤ lim inf}

h

→+∞

H

0 _(P

h

)

(3)

Proof. Fromdenition2.1itfollowsthat

lim inf

h→+∞

H

0 _(P

h

) ≥ lim inf

h→+∞

H

0 _({x

1 h

, ..., x

N

h

}) = N = H

0 _{(P ).}

2.2 Distributional divergen e 143

Inthissubse tionwere allthedenitionofthespa e

L

p,q

_{(div; Ω)}

and

DM

p

_(Ω)

,introdu ed 144 in[4℄. 145 Let

Ω ⊂ R

2

beanopenset andlet

U

: Ω ⊂ R

2 _{→ R}

2

beave toreld. 146

Denition 2.2. We say that

U

∈ L

p,q

_{(div; Ω)}

if

U

∈ L

p

_{(Ω; R}

2 ₎

and if its distributional 147 divergen e

divU ∈ L

q

_(Ω)

. If

p

= q

the spa e

L

p,q

_{(div; Ω)}

willbe denotedby

L

p

_{(div; Ω)}

. 148 Wesay that

U

∈ L

p,q

loc

(div; Ω)

if

U

∈ L

p,q

_{(div; A)}

for everyopen set

A

⊂⊂ Ω

. 149 Denition2.3. For

U

∈ L

p

_{(Ω; R}

2 ₎

,

1 ≤ p ≤ +∞

,set

|divU|(Ω) := sup{

Z

Ω

U

· ∇ϕdxdy : ϕ ∈ C

1

0 (Ω), |ϕ| ≤ 1}.

Wesaythat

U

isan

L

p

-divergen e measureeld, i.e.

U

∈ DM

p

_(Ω)

if

kU k

_DM

p

_(Ω)

:= kU k

_L

p

_(Ω;R

2 ₎

+ |divU|(Ω) < +∞.

Wesaythat

U

∈ DM

p

loc

(Ω)

if

U

∈ DM

p

_(A)

for everyopenset

A

⊂⊂ Ω

. 150

(11)

Remark2.1. If

U

∈ DM

p

_(Ω)

thenviaRieszTheorem itispossibletorepresentthe distri-151

butionaldivergen e of

U

by aRadonmeasure. Morepre isely thereexistsaRadon measure 152

µ

su hthat for every

ϕ

∈ C

1

0 (Ω)

the following equalityholds: 153

Z

Ω

U

· ∇ϕdxdy = −

Z

Ω

ϕdµ.

Forinstan ethe eld

U

(x, y) = (

x

2 _+y

2 ,

y

x

2 _+y

2 )

belongsto

DM

1 loc

(R

2 )

anditsdivergen e 154

measureisgiven by

−2πδ

0

,where

δ

0

isthe Dira mass. 155

Su h aresult an beprovenby approximation. Letus denethe following map: 156

U

ε

(x, y) :=

(

U

(x, y)

if

|x| ≥ ε

(

x

ε

2 ,

y

ε

2 )

if

|x| < ε.

Itisnot di ultto he kthat

u

ε

isLips hitz-map withdivergen egiven by 157

2 ε

2 χ

B(0,ε)

. 158

Then for every testfun tion

ϕ

∈ C

1

0 (R

2 )

wehave 159

Z

U

ε

· ∇ϕdxdy = −

Z

₂

ε

2 χ

B(0,ε)

ϕdxdy.

Byapplying the hange of variables

x

=

x

1 ε

,

y

=

y

1 ε

weobtain

Z

U

ε

· ∇ϕdxdy = −2

Z

χ

B(0,1)

ϕ(

x

1 ε

,

y

1 ε

)dx

1 dy

1 ,

sothat,letting

ε

→ 0

,by the dominated onvergen etheoremweobtain

Z

Ω

U

· ∇ϕdxdy = −2πϕ(0, 0) = −2π

Z

Ω

ϕdδ

0

2.3 The Diri hlet problem with measure data 160

Fortheinitializationofouralgorithmwemustbuildave toreld

U

0

whi hshouldbesu h 161

thatitsdivergen e issingularonpointsoftheimage

I

. Thereforewewilluse thegradient 162

ofthesolutionofthefollowingDiri hletproblem(appliedwith

µ

= I

) 163

(

∆f = µ

on

Ω

f

= 0

on

∂Ω

(4)

where

µ

isaRadonmeasure. 164

Classi alresults (see[19℄) guaranteethe existen eof auniquesolutionof problem (4). 165

Con erningtheregularityitisknownthen

f

∈ W

1,p

_(Ω)

with

p <

2

. 166

(12)

3 Existen e result 167

Inthisse tionweshowtheexisten eofaminimizingpair

(U, P )

forthefun tional

F

dened 168

in(1.) 169

Our argument needs two steps (see also [17℄ for a similar approa h to minimize the 170

lassi al Mumford-Shah's fun tional). The rst one onsists in proving the existen e a 171

minimizerofthefun tional(1)when theset

P

isxed. 172

Tothisaimweadoptethefollowingnotation: 173 174

F

(U ) = F (·, P ) =

Z

Ω\P

|divU |

2 _dxdy

_{+ λ}

Z

Ω

|U − U

0 |

p

dxdy

+ H

0 (P ).

(5)

Theorem 3.1. For every set

P

there existsa unique minimizer

U

P

∈ L

p,2

_{(Ω \ P ; div)}

of 175

the fun tional(5). 176

Proof. Let

U

n

beaminimizingsequen e. Thenwehavethefollowingbound 177

F

(U

n

) ≤ M.

(6)

Fromthebound (6)andthe lassi alinequality:

kU

n

k

p

_L

p

_{(Ω\P )}

≤ 2

p

_kU

n

− U

0 k

p

_L

p

_{(Ω\P )}

+ kU

0 k

p

L

p

_{(Ω\P )}

itfollowsthat

kU

n

k

p

_L

p

_{(Ω\P )}

≤ M + kU

0 k

p

_L

p

_{(Ω\P )}

:= C.

Moreoveorwealsohave:

kdivU

n

k

2 L

2 _{(Ω\P )}

≤ F (U

n

) ≤ M ;

sothat,upto subsequen es,weobtain 178

(

U

n

⇀ U

P

in

L

p

_{(Ω \ P )}

divU

n

⇀

divU

P

in

L

2 _{(Ω \ P ).}

(7)

Thereforewe an on ludethat

U

n

weakly onvergesin

L

p,2

_{(Ω \ P ; div)}

toave toreld 179

U

P

∈ L

p,2

(Ω \ P ; div)

. 180

Then wehave thanksto semi ontinuity properties of the

L

p

-norm with respe t to the weak onvergen e:

inf

U

F

(U ) ≤ F (U

P

) ≤ lim inf

n

→+∞

F

(U

n

) = inf

U

F

(U ).

Finallythestrong onvexityoffun tional(5)givestheuniquenessoftheminimizer

U

181

On e weobtainedtheexisten eoftheminimizer

U

P

foreveryset

P

xed,wefo uson 182

thefollowingfun tional. 183 184

E(P ) := F (U

P

, P

) =

Z

Ω\P

|divU

P

|

2 dxdy

+ λ

Z

Ω

|U

P

− U

0 |

p

dxdy

+ H

0 (P ).

(8)

(13)

Weextend

U

and

divU

byzeroon

P

. Howeverwekeeptheintegrationdomainof

divU

to 185

be

Ω \ P

. Wedothatinordertomake learthat

divU

isthedistributionaldivergen eof

U

186

on

Ω \ P

andnoton

Ω

. 187

Thefollowingsemi ontinuitylemmaplaysakeyrole. 188

Lemma3.1. Assumethatasequen eofnite setsofpoints

{P

n

} ⊂ Ω

onvergestoanite setof point

P

⊂ Ω

. Then

E(P ) ≤ lim inf

h

→∞

E(P

n

)

Proof. Letusset

U

n

= U

P

n

,we anassumethat

U

n

and

divU

n

arebothdened onallof

Ω

in thesense explainedabove. Thesequen e

U

n

is bounded in

L

p

_(Ω)

. Indeed,bytaking intoa ountthat

U

n

isaminimizerofthefun tional(5),

kU

n

k

p

_L

p

_(Ω)

≤ 2

p

_kU

n

− U

0 k

p

_L

p

_(Ω)

+ kU

0 k

p

L

p

_(Ω)

≤ 2

p

_F

_{(0) + kU}

0 k

p

_L

p

_(Ω)

= kU

0 k

p

L

p

_(Ω)

(2

p

_{+ 1).}

Inthesamewayone anshowthatthesequen e

divU

n

isboundedin

L

2 _(Ω)

Sothat,upto 189

subequen es,wemayassume 190

(

U

n

⇀ U

in

L

p

_(Ω)

divU

n

⇀ V

in

L

2 _(Ω).

(9)

We laimthat

divU = V

in

Ω \ P

. Infa t,takeanytestfun tion

ϕ

withsupportin

Ω \ P

, thensin e

P

n

→ P

,wehavefor

n

largeenough

supp(ϕ) ⊂ Ω \ P

n

and, onsequently,

Z

supp(ϕ)

U

n

∇ϕdx = −

Z

supp(ϕ)

divU

n

ϕdx.

Therefore,bytakingtheweaklimitby(9) weget

Z

supp(ϕ)

U

∇ϕdx = −

Z

supp(ϕ)

V ϕdx.

Thensin ethetestfun tion

ϕ

isarbitrary,we an on ludethat

divU = V

on

Ω \ P

. 191

The thesisfollowsbe ause, from the lower semi ontinuityof the

L

p

-norm and Lemma 2.2

E(P ) ≤ E(U, P ) ≤ lim inf

n

E(U

n

, P

n

)

Wearenowin positionofprovingthemain resultofthisse tion. 192

Theorem3.2. There exist aminimizer

(U, P )

of the fun tional

F

,with

U

∈ L

p,2

_{(div; Ω)}

193

and

P

⊂ Ω

anite setof points. 194

(14)

Proof. Forevery

P

let

U

P

aminimizerforthefun tional

F

,whoseexisten eisguaranteed 195

byTheorem

3.1

196

Thenwefo usonthethefun tional

E(P ) = F (U

P

, P

)

andwetakeaminimizingsequen e

{P

n

}

. ThenbyLemma2.1wehave(uptoasubsequen es)that

P

n

→ P ⊂ Ω

and

U

P

n

→ U

P

. ByLemma 3.1weget

E(P ) ≤ lim inf

n→+∞

E(P

n

).

Therefore 197

inf

(U,P )

F

(U, P ) ≤ F (U

P

, P

) ≤ lim inf

n

→+∞

E(P

n

) ≤ lim inf

h→+∞

F(U

n

, P

n

) = inf

P

F

(U

P

, P

) ≤ F (U, P ),

(10) Nowset

P

˜

:= P

\ ∂Ω

. Sin eforevery

P

,

U

P

isaminimizerwegetfrom (10)

F

(U

_P

, ˜

P

) ≤ F (U

_P

, P

) ≤ F (U

P

, P

) ≤ F (U, P ),

forevery

(U, P )

. Hen e we on ludethat

F

(U

_P

, ˜

P

) ≤ inf

(15)

4

Γ

- onvergen e 198

The key point of our strategy is to repla e the fun tional (1) by means of more regular 199

fun tionalsbyfollowinaformal

Γ

- onvergen eapproa h. 200

Thereforethisse tionisdevotedtoaverysimplepresentationofthetworesultsweneed: 201

Modi a-Mortola'stheorem(see[15,16℄) on erningtheapproximationoftheperimeterand 202

DeGiorgi's onje ture(see[12℄)abouttheapproximationof urvaturedepending fun tion-203

als. Forthedenition of the

Γ

- onvergen e andits main properties wereferthe readerto 204

[8, 11℄ andreferen estherein. 205

4.1 Modi a Mortola's approa h 206

Modi a-Mortola theorem states that it is possible to approximate, in the

Γ

- onvergen e 207

sense,aperimeterbymeansofthefollowingsequen eoffun tionals 208

F

_ε

1 (u) :=

(

_R

Ω

ε|∇u|

2 ₊

V

(u)

ε

dx

if

u

∈ W

1,2

_(Ω),

+∞

otherwise

,

where

V

(u) = u

2 _{(1 − u)}

2

is adouble well potential. Besides, sin e the minimizersof the 209

fun tional

F

1 ε

maybetrivial,some onstraintonthefun tions

u

ε

must beadded. Usually 210

avolume onstraintofthetype

R

Ω

udxdy

= m

,isassumed. 211

Letusgiveanintuitiveexplanationofsu haresult. Sin e

V

hastwoabsoluteminimizers 212

at

u

= 0, 1

,when

ε

issmall,alo al minimizer

u

ε

is losedto

1

onapartof

Ω

and loseto 213

0

ontheotherpart,makingarapidtransitionoforder

ε

between

0

and

1

. When

ε

→ 0

the 214

transitionset shrinkstoasetof dimension

1

,sothat

u

ε

goestoafun tion takingvalues

u

215

into

{0, 1}

andthefamilyoffun tionals

Γ

- onvergesto themeasureoftheperimeterofthe 216

dis ontinuitysetof

u

. Modi a-Mortola'sTheorem isthefollowing. 217

Theorem 4.1. The fun tionals

F

1 ε

: L

1 (Ω) → [0, +∞] Γ−

onverge with respe t to the 218

L

1

- onvergen etothe following fun tional 219

F

1 (u) =

(

C

V

H

1 (S

u

)

if

u

∈ {0, 1}

+∞

otherwise

where, as usual,

S

u

denotes the set of dis ontinuities of

u

and

C

V

is a suitable onstant 220

dependingon thepotential

V

. 221

4.2 De Giorgi's onje ture 222

The aim of De Giorgi was nding a variational approximation of a urvature depending fun tionalofthetype:

F

2 (D) =

Z

∂D

(1 + κ

2 )dH

1 ;

where

D

isaregularsetand

k

isa urvatureofitsboundary

∂D

. 223

(16)

Sin e

∂D

an be represented as the dis ontinuity set of the fun tion

u

0 = 1 − χ

D

, 224

by Modi a-Mortola's Theorem it follows that there is a sequen e of non onstant lo al 225

minimizerssu hthat

u

ε

→ u

0

withrespe tto the

L

1

- onvergen esu hthat 226

lim

ε

→0

F

1 ε

(u

ε

) := C

V

H

1 (∂D).

FurthermorelookingattheEuler-Lagrangeequationasso iatedtoa ontourlengthterm, 227

yields a ontour urvature term

κ

, while the Euler-Lagrange equations for the fun tional 228

F

1 ε

(u)

ontainsaterm

2ε∆u −

V

′

(u)

ε

. 229

Then De Giorgi suggested to approximate the fun tional

F

2

by adding to Modi a-230

Mortola'sapproximatingfun tionalstheterm 231

F

_ε

2 (u) =

Z

Ω

(2ε∆u −

V

′

(u)

ε

)

2 _(ε|∇u|

2 ₊

V

(u)

ε

)dx.

In[18℄theauthorshaveprovenasimpliedversionoftheDeGiorgi's onje ture,where theintegralaboveisrepla edbythefun tional

F

_ε

2 (u) =

Z

Ω

(2ε∆u −

V

′

(u)

ε

)

2 _dx.

(17)

5 The approximating fun tionals 232

Inthisse tionwepresenttheenergywedealwithandthe onstru tionoftheapproximating 233

sequen e. 234

Theenergyweareinterestedin isgivenby

Z

Ω\P

|divU |

2 _{+ λ}

Z

Ω

|U − U

0 |

p

+ H

0 (P ).

where

U

∈ L

p,2

_{(div; Ω \ P )}

,

U

0 ∈ DM

p

loc

(R

2 )

andnally

P

is anatomi set onsistingof a 235

nitenumber

N

ofpoints,i.e.

P

= {x

1 , ..., x

N

}

. 236

As pointedoutin theintrodu tion,therst stepisto substitutethe ountingmeasure

H

0 _{(P )}

withamoretreatable termgivenby:

G

β

ε

(D) =

1 4π

Z

∂D

1 β

ε

+ β

ε

κ

2 (x)dH

1 (x);

where

D

is an union of regular simply onne ted sets

{D

i

}

with

i

= 1, .., N

, su h that 237

x

i

∈ D

i

,

D

i

T D

j

= ∅

for

i

6= j

.

κ

is the urvature of the boundary of the set

D

, the 238

onstant

1 4π

isanormalizationfa torand

β

ε

isinnitesimalas

ε

→ 0

. 239

Tounderstand why we an approximate

H

0 _{(P )}

with

G

β

ε

(D)

one shouldnote that the 240

solutionofthefollowingminimumproblem 241

min

D

⊃P

G

β

ε

(D)

(11) isgiven by

D

=

S

N

i

B(x

i

, β

ε

),

where

x

i

arethe pointsof

P

. Wegiveanideaofapossible 242

proofinthe aseofasinglepoint. 243

BytheYoung'sinequalitywehave

G

β

ε

(D) ≥

1 4π

Z

∂D

2κdH

1 _(x)

andbyapplyingtheGauss-BonnetTheorem 244

G

β

ε

(D) ≥

1 4π

(2)(2π) = 1 = H

0 _{(P ).}

Finallyasimple al ulationshowsthat, ifweevaluatethefun tional

G

β

ε

on

B

(x

1 , β

ε

)

, 245

weobtainthe value

1

, i.e. thenumberof pointsin

P

, i.e.

H

0 _{(P )}

. The

N

points ase an 246

bere overedwithminor hangesbythesameargument. 247

Forwhatfollowsitis onvenientto splitthefun tional

G

β

ε

intwoterms:

G

β

ε

(D) = G

1 βε

(D) + G

2 βε

(D)

where 248

(18)

G

1 _β

_ε

(D) =

1 4π

Z

∂D

1 βε

dH

1 _(x);

and 249

G

2 _β

_ε

(D) :=

1 4π

Z

∂D

β

ε

κ

2 (x)dH

1 (x).

We anwrite anintermediateapproximationofenergy(1): 250

E

ε

(U, D) = G

1 β

ε

(D) + G

2 β

ε

(D) +

Z

Ω

(1 − χ

D

)|div(U )|

2 + λ

Z

Ω

|U − U

0 |

p

.

(12)

Theadvantageofsu h aformulationis that weknowhow to provideavariational ap-251

proximationoftheperimetermeasure

H

1 _⌊∂D

. FollowingModi a-Mortola'sapproa hsu h 252

anapproximation anbeobtainedbyusingthefollowingmeasure: 253

µ

ε

(w, ∇w)dx = ε|∇w|

2 +

V

(w)

ε

dx,

where

V

(w) = w

2 _{(1 − w)}

2

isadoublewellfun tional. 254

Next step is expressing the urvature term by means of the fun tion

w

. Thanks to 255

thesimplied versionoftheDe Giorgi's onje turewe anrepla etheterm

κ

bytheterm 256

2ε∆w −

V

′

_ε

(w)

. 257

Sothatwe an formallywritethe ompleteapproximatingfun tional:

Φ

ε

(U, w) : =

Z

Ω

w

2 |div(U )|

2 dx

+

1 4π

Z

Ω

β

ε

(2ε∆w −

V

′

_(w)

ε

)

2 dx

+

1 β

ε

Z

Ω

µ

ε

(w, ∇w)dx

+

λ

Z

Ω

|U − U

0 |

p

dx

+

2 µ

ε

Z

Ω

(1 − w)

2 dx,

(13) where

U

∈ L

p,2

_{(div; Ω)}

is equal to

0

on the

∂Ω

and

w

is smoothfun tion equal to

1

258

on the boundary, i.e.

1 − w ∈ C

∞

0 (Ω)

,

µ

ε

→ 0

when

ε

goes to

0

. The last integralis a 259

penalizationtermwhi hprevents

w

ε

from onvergingto thefun tion onstantlyequalto

0

260

as

ε

→ 0

. 261

Thenif

(U

ε

, w

ε

)

isaminimizingsequen eof

Φ

ε

,then

w

ε

mustbevery losetothevalues 262

1

when

ε

goesto

0

, sin ethe double well potential is positive ex eptfor

w

ε

= 0, 1

and

w

263

mustbeequalto

1

on

∂Ω

. Ontheotherhand,nearthepointswherethedivergen eisvery 264

big

w

ε

mustbe loseto

0

. Besideswhen

ε

→ 0

,

β

ε

→ 0

goesto

0

aswell,sothatthesingular 265

set

D

isgivenbyanunionofballsofasmallradius

β

ε

. 266

Therefore,whilethefun tions

U

ε

approximateaminimizer

U

oftheoriginalfun tional, 267

thelevelset

{w

ε

= 0}

approximatetheoriginalsingularset

P

. 268

Remark 5.1. We point outthat the

Γ

- onvergen e result is not proved in this paper, but 269

only onje tured. A ompleteproofofthe

Γ

- onvergen eresultandtheequi oer ivenessofthe 270

sequen e

Φ

ε

,inthe parti ular asewherethe ve toreld

U

isagradient, has beenprovided 271

bythe rstandthirdauthorin [6 ℄. 272

(19)

Therstvariationofthis fun tionalleadstothefollowinggradientowsystem

∂U

∂t

= 2∇(w

2 _{divU ) + λp|U − U}

0 |

p−2

(U − U

0 )

∂w

∂t

= −4

∆h

β

ε

+ β

ε

h

+

2 ε

2

1 β

ε

V

′′

(w)h − 2w|divU |

2 + 2

1 µ

ε

(1 − w),

(14)

where

h

isgivenbytheequation 273

h

= 2ε∆w −

1 ε

V

(20)

6 Dete tion 274

Inourmodeltheimage ontainsanatomi Radonmeasure. Thus,inordertondaninitial 275

ve tor eld whi h opies the singularities of the initial image, we use the gradient of the 276

solutionofthefollowingDiri hletproblem: 277

(

∆f = I

on

Ω

f

= 0

on

∂Ω.

(15)

In this way we obtain a ve tor eld whose divergen e is singular on a proper set whi h 278

ontainsthe points we wantto dete t. In generalthis set ould ontain other stru tures. 279

For instan e if the initial image is a Radon measure on entrated both on points and on 280

urves,thedivergen eof

∇f

issingularonpointsandon urvesat thesametime. Besides 281

ifthere is somenoisein theimage, it ould be not lear how to dierentiate thesingular 282

pointsduetothenoise,fromthosewewantto at h. Asa onsequen e,bysolvingproblem 283

(15),wegetjust apredete tion,whi hhastoberened. 284

Tothispurposewesear hfor aminimizerof theenergy

Φ

ε

(U, w)

viasolvingequations 285

(14)withinitialdata

U

0

givenby

∇f

. Sothat weobtainave toreld

U

whosedivergen e 286

isrelevantonlyontheset

P

and afun tion

w

whosezerosaregivenbytheset

P

. 287

6.1 Dis retization 288

Theimage is anarray ofsize

N

2

. Weendowedthe spa e

R

N

×N

withthe standards alar produ tandstandardnorm. Thegradient

∇I ∈ (R

N

×N

_{) × (R}

N

×N

₎

isgivenby:

(∇I)

i,j

= ((∇I)

1 i,j

,

(∇I)

2 i,j

)

where

(∇I)

1 i,j

=

(

I

i+1,j

− I

i,j

if

i < N

0

if

i

= N,

(∇I)

2 i,j

=

(

I

i,j+1

− I

i,j

if

j < N

0

if

j

= 0.

We also introdu e the dis rete version of the divergen e operator simply dened as the 289

adjointoperatorofthegradient:

div = −∇

∗

. Morein detailsif

v

∈ (R

N

×N

_{) × (R}

N

×N

₎

,we 290 have 291

(21)

(divv)

i,j

=











v

1 i,j

+ v

2 i,j

if

i, j

= 1

v

_i,j

1 + v

2 i,j

− v

i

2 −1,j

if

i

= 1, 1 < j < N

v

1 i,j

− v

1 i−1,j

+ v

2 i,j

− v

i−1,j

2

if

1 < i < N, 1 < j < N

−v

1 i

−1,j

+ v

2 i,j

− v

i

2 −1,j

if

i

= N, 1 < j < N

v

_i,j

1 − v

1 i−1,j

+ v

2 i,j

if

1 < i < N, j = 1

v

_i,j

1 − v

1 i

−1,j

− v

2 i

−1,j

if

1 < i < N, j = N

−(v

1 i−1,j

+ v

2 i−1,j

)

if

i, j

= N.

Thenwe an denethedis reteversionoftheLapla ianoperator as

∆I = div(∇I)

. 292 6.2 Dis retization in time 293 We simply repla e

∂U

∂t

and

∂w

∂t

by

U

i,j

n+1

−U

n

i,j

δt

and

w

i,j

n+1

−w

n

i,j

δt

respe tively. Then we write system(14)intheform(forsimpli ityweomitthedependen e on

ε

)











U

₁

n+1

=

−δtΦ

U

1 (U

n

, w

n

)

U

2 n+1

=

−δtΦ

U

2 (U

n

, w

n

)

w

n+1

₌

_−δtΦ

w

(U

n

, w

n

).

Weinitializeouralgorithmwith

U

(0) = ∇f

,where

f

isthesolutionofproblem(15). Todo 294

this,weneedto solveaDiri hletproblem withdatameasure

I

,thereforeweregularizethe 295

image by onvolutionwithaGaussian kernel

G

σ

with verysmall

σ

andthen wesolve,by 296

lassi alnitedieren esmethod, theproblem: 297

(

∆f = I

σ

on

Ω

f

= 0

∂Ω,

(16) where

I

σ

= I ∗ G

σ

. 298

Toinitializeouralgorithm,weneedof aninitialguess on

w

. Sowe hoose

w(0) = 1

. 299

(22)

7 Computer examples 300

7.1 Parameter settings 301

Before running our algorithm all the parameters have to be xed. The most important 302

are

ε

,

β

ε

and

µ

ε

, whi h governtheset

D

approximatingpointswewantto dete t. Those 303

parametersarerelatedbythe onditions

lim

ε

→0

ε| log(ε)|

β

ε

= 0

,

lim

ε

→0

β

ε

µ

ε

= 0

. Furthermore,sin e 304

themesh gridesize is

1

and

β

ε

givesthe radiusofaball entered inthe singularpointwe 305

wanttodete t,from adis retepointofviewthesmallestvaluewe antakeis

√

2

. Thenwe 306

usethevalues

0.1

for

ε

,

0.7

for

β

ε

, and

0.8

for

µ

ε

. Asexponent

p

of thedis repan yterm 307

wealwaystake

p

= 1.5

308

Con erning the parameter

λ

we mainly used the value

λ

= 0.1

, in order to for e the 309

algorithmtoregularizeasmu has possibletheinitialdata

U

0

. 310

Sin e we deal with small values of

ε

, in order to have some stability, we must take a 311

smalldis retizationtimestep. Pra ti allywemainlyusedthevalue

δt

= 1 × 10

−6

. 312

Con erningthestopping riterionweiteratethealgorithmuntilmax

n

kU

1 n+1

−U

n

1 k

1 kU

n

1 k

1 ,

kU

2 n+1

−U

n

2 k

1 kU

n

2 k

1 ,

kw

n+1

−w

n

k

1 kw

n

_k

1 o

313

≤ 1 × 10

−

_2.

314

Inallthe omputerexamplesthepointsaredete tedbymeansof thefun tion

w

ε

. We 315

displaythelevel-set

{w

ε

≃ 0}

. 316

7.2 Commentaries 317

The gure1showshowresistant to the noiseourmodel is. When the noiseis largerthe 318

parameter

ε

mustbeas loseraspossibletotheidealvalue

0

. Besidesitis possibleto see 319

thatforsmallvaluesof

ε

ourdete tionisverynea ordingtothe ontinuoussetting. More 320

in detailsin therstrowwedisplaytheinitial imageobtainedbyaddingaGaussian noise 321

toabinaryimageofvepoints. These ond rowshowsthebehaviorof

w

ε

forsmall values 322

of

ε

and

β

ε

. 323

Looking at the histograms of the gray level of

I

and

w

ε

, one an see that it is easier 324

xingathresholdvaluestartingfrom thefun tion

w

ε

thanfromtheinitialimage

I

. Inthe 325

lastrowwedisplaytheset

{w

ε

≃ 0}

obtainedbyplotting theset

{w

ε

≤ α}

withthreshold 326

value

α

= 0.5

. 327

Ingure2wetestouralgorithmon urvesandpointsatthesametime. Intherstrow 328

we haveasequen e of pointsand a urvewith boundaryinside

Ω

. In these ond rowwe 329

displaythefun tion

w

ε

andthelevelset

{w

ε

≃ 0}

on eagainobtainedbyxingathreshold 330

value

α

= 0.5

. Theresult is that, asdesired, our algorithm is apable of eliminating the 331

urvefromtheinitialimage. A ordingtothe ontinuoussettingwhen

ε

takesvalues lose 332

to

0

theapproximatingenergy (13)behavessimilarly to thelimit energy (1), so that the 333

presen e of the urve is penalized in the minimization pro ess. Then the set

{w

ε

∼

= 0}

334

ontainsnothingelse butpoints. 335

Finallyin gure(a),(b),( ) and(d) wedealwithabiologi alimage. Ourtaskis at h-336

ing the nest stru ture present in the image. In gure (d) the isolated points are quite 337

(23)

well dete ted, while the bran hes the ellule are not. Nevertheless due to the small time 338

dis retization step the omputation time is quite large. To test the image in (a) of size 339

500 × 500

,ouralgorithmtakes

100

iterationsandabout7mm. Certainlythealgorithm an 340

be a eleratedbyusing moresophisti ated te hniques su h asmultigrid methods. Su h a 341

fasteralgorithmisthesubje tofour urrentinvestigation.

Originalimage

I

NoisyimagePSNR

= 5.5

Db NoisyimagePSNR

= 5.5

Db

Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

) Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

)

w

ε

≃ 0

(

ε

= 0.1

,

β

ε

= 0.7

)

Figure1: Syntheti image: wetestouralgorithmonnoisyimages. Whentheparameters

ε

and

β

ε

aresmall asmu haspossiblethedete tionisner.Thedete tionisrenedbyxingathresholdvalue

α

forthefun tion

w

ε

. 342

(24)

Originalimage

I

NoisyImagePSNR

= 7.2

Thefun tion

w

ε

{w

ε

≃ 0}

(

ε

= 0.1

,

β

ε

= 0.7

)

Figure2: Syntheti image: urvepointsandnoisearepresentintheinitialimage. Asexpe tedourmethod is apableofremovingthe urvefromtheimage.

(25)

(a)OriginalImage

(26)

( )Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

)

(27)

8 Con lusion 343

In this work, a new variational method for spot dete tion in biologi al images has been 344

proposed and tested. We emphasize that, a ording to our knolewdge, this is the rst 345

method whi h makes possible isolating the spots from a lament in the observed image. 346

Moreoveritalsopermitsinanoisyimagetoxathresholdvalueinasimpleanddire tway. 347

Moreoverwebelievethat asuitablegeneralizationofthismethodforthedete tionofspots 348

andevenlamentsin

3

-Dbiologi alimages anbeprovided. Thisisasubje tofour urrent 349

investigation. Certainlytherearemanyroomsforimprovementbothfromatheoreti aland 350

numeri alpointofviewsu hasadeepinvestigationofthe

Γ

- onvergen eapproximation,as 351

awellasasigni anta elerationofthealgorithm. 352

(28)

Referen es 353

[1℄ L.Ambrosio,N.Fus o,D.Pallara. Fun tions of boundedvariation and free dis ontinuity 354

problems.OxfordUniversityPress(2000). 355

[2℄ L.Ambrosio,V.M.Tortorelli.Approximationoffun tionalsdependingonjumpsbyellipti 356

fun tionalsvia

Γ

- onvergen e.Communi . PureAppl.Math43(1990),999-1036. 357

[3℄ L.Ambrosio, V.M.Tortorelli. Approximation of fun tionals depending on jumps by 358

quadrati , ellipti fun tionalsvia

Γ

- onvergen e.Boll.Un.Mat.Ital.6-B(1992),105-123. 359

[4℄ G.Anzellotti Pairings between measures and bounded fun tions and ompensated om-360

pa tness.Ann.Mat.PuraAppl.135(1983),293-318. 361

[5℄ G.Aubert, J. Aujol, L.Blan -Feraud Dete ting Codimension-Two Obje ts in an image 362

withGinzurbg-Landau Models. InternationalJournalofComputerVision65(2005), 29-363

42. 364

[6℄ G.Aubert,D.GrazianiVariationalapproximation for dete tingpointslike-targetproblem 365

in

2

-Dbiologi alimagessubmittedtoESAIM:COCV.Preprintavailableon http://www-366

sop.inria.fr/en/publi ations.php. 367

[7℄ F.Bethuel,H.BrezisandF.HéleinGinzburg-LandauVorti es.Birkäuser,Boston(1994). 368

[8℄ A.Braides.

Γ

- onvergen efor beginners.OxfordUniversityPress,Newyork(2000). 369

[9℄ A.Braides, A.Mal hiodi Curvature Theory of Boundary phases: the two dimensional 370

ase.Interfa esFree.Bound.4(2002),345-370. 371

[10℄ A.Braides, R.Mar h. Approximation by

Γ

- onvergen e of a urvature-depending fun -372

tionalin VisualRe onstru tion Comm.PureAppl.Math.59 (2006),71-121. 373

[11℄ G.DalMaso.Introdu tionto

Γ

- onvergen e.Birkhäuser,Boston(1993). 374

[12℄ E.De Giorgi Some Remarks on

Γ

- onvergen e and least square methods, in Com-375

posite Media and Homogenization Theory (G. Dal Maso and G.F.Dell'Antonio eds.), 376

Birkhauser,Boston,1991,135-142. 377

[13℄ E.DeGiorgi,T.Franzoni.Suuntipodi onvergenzavariazionale.AttiA ad.Naz.Lin ei 378

Rend.Cl. S i.Mat.Natur.58(1975),842-850. 379

[14℄ E.De Giorgi, T.Franzoni. Su un tipo di onvergenza variazionale. Rend. Sem. Mat. 380

Bres ia3(1979),63-101. 381

[15℄ L.Modi a, S.Mortola. Un esempio di

Γ

- onvergenza. Boll.Un. Mat. Ital. 14-B(

1977

), 382

285 − 299

. 383

(29)

[16℄ L.Modi a.Thegradienttheoryofphasetransitionsandtheminimalinterfa e riterion. 384

Ar h. RationalMe h. Anal.98(1987),123-142. 385

[17℄ M.Morel,SoliminiVariational models inimage segmentation,Birkäuser,1994. 386

[18℄ M.Röger, R.Shätzle. On a modied onje ture of De Giorgi. Math. Zeits hrift. 254 387

(2006),675-714 388

[19℄ G.Stampa hiaLeproblèmedeDiri hletpourlesequationselliptiques duse ondeordre 389

à oe ientsdis ontinuous. Ann.Inst.Fourier(Grenoble),15(1965),180-258. 390

[20℄ A.Witkin, D.Terzopoulos, M.Kass. Signal math ing trough s ale spa e International 391

JournalofComputerVision12(1987),133-144. 392

(30)