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

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2-D images

Daniele Graziani, Laure Blanc-Féraud, Gilles Aubert

To cite this version:

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

points in 2-D images. [Research Report] RR-6978, INRIA. 2008. �inria-00401570�

a p p o r t

d e r e c h e r c h e

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Thème COG

A Γ-convergence approach for the detection of points

in 2-D images.

Daniele Graziani, Laure BLANC-FÉRAUD, Gilles Aubert

N° 6978

2009

2

Daniele Graziani

∗

, Laure BLANC-FÉRAUD

∗

,Gilles Aubert

†

5

ThèmeCOG Systèmes ognitifs 6

ProjetARIANA 7

Rapport dere her he n° 69782009xxiipages 8

Abstra t: In this report we propose anew variationalmethod to isolate points in a

2

-9

dimensional image. To this purpose we introdu e a suitable fun tional whose minimizers 10

aregivenbythepointswewanttodete t. Theninordertoprovidenumeri alexperiments 11

weapproximate thisenergybymeansofasequen eof moretreatable fun tionalsbyusing 12

a

Γ

- onvergen eapproa h. 13

Key-words: pointsdete tion, urvature-dependingfun tionals,divergen e-measureelds, 14

Γ

- onvergen e,biologi al

2

-Dimages. 15

∗

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

†

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

2

-D images. 17 Résumé: 18 Mots- lés : 19

Contents 20 1 Introdu tion ii 21 2 Preliminaries v 22 2.1 Measuretheory . . . v 23 2.2 Distributionaldivergen e . . . vi 24

3 The Diri hlet problemwith measure data viii 25

4

Γ

- onvergen e ix

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

Referen es xxi

1 Introdu tion 37

Dete tingne stru tures, like pointsor urvesin twoorthree dimensional images respe -38

tively,is animportantissue in image analysis. In biologi alimagesapoint may represent 39

aviralparti lewhosevisibilityis ompromisedbythepresen eof otherstru tureslike ell 40

membranesor somenoise. 41

Fromavariationalpointofviewtheproblemofisolatingpointsisadi ulttask,sin eit 42

isnot learhowthesesingularitiesmustbe lassiedintermofasomedierentialoperator. 43

Indeed,whereastheseareusuallydenedasdis ontinuitywithoutjump,we annotusethe 44

gradientoperator asin the lassi alproblem of dete ting ontours. As a onsequen ethe 45

fun tionalframework wehavetodealwithmaybenot lear. 46

One possible strategy to over omethis obsta leis onsidering these kind of pathology 47

asa

k

- odimensionobje t,meaningthattheyshouldberegardedasasingularityofamap 48

U

: R

k+m

_{→ R}

k

, with

k

≥ 2

and

m

≥ 0

(see[7℄fora ompletesurveyonthis subje t). So 49

thatthedete tingpoint ase orrespondsto the ase

k

= 2

and

m

= 0

. 50

Inthisdire tion,in[5℄,theauthorshavesuggestedavariationalapproa hbasedonthe 51

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

2

-D images are 52

regardedasthetopologi al singularities of amap

U

: R

2

_{→ S}

1

, where

S

1

is aunit sphere 53 of

R

2

. So that it is ru ial to onstru t, starting from the initial image

I

: R

2

_{→ R}

, an 54

initialve toreld

U

0

: R

2

_{→ S}

1

with atopologi al singularityofdegree

1

, wherethereis a 55

pointin theinitial image

I

. How to dothis in arigorousway,it is asubje tof a urrent 56

investigation. 57

Hereourpurposeistoprovidealightervariationalformulation,inwhi hthesingularity 58

in theimageis dire tlygivenin termofof aproperdierentialoperatordened onve tor 59

elds. Another importantdieren e is that in [5℄ points and urves are dete tedboth as 60

singularities,whileinthepresentpaperouraimistoisolatepointsandatsametimeremove 61

anyothersingularities,like urvespre isely,fromtheinitialimage. 62

Inourmodel

Ω ⊂ R

2

isanopenset(the imagedomain),

B

(Ω)

isthe

σ

-algebraoftheall 63

Borelsets of

Ω

,andtheinitial image

I

: B(Ω) → R

isaRadonmeasure. 64

Inordertodete tthesingularitiesoftheimage,wehavetondafun tionalspa ewhose 65

elements generate, in asuitable sense,ameasure on entrated onpoints. Su h aspa e is 66

DM

p

_(Ω)

, the spa e of the ve tor elds

U

: Ω → R

2

whose distributional divergen e is a 67

Radonmeasure introdu edin [4℄. (seeSubse tion2.2fordenitionsandexamples). 68

Unfortunatelyevenifweare apable offabri ating aninitialve toreld

U

0

(see below 69

for su h a fabri ation) belonging to the spa e

DM

p

_(Ω)

, its singular set ould ontains 70

several stru tures, we want to remove from the original image like for instan e urve or 71

pointsgeneratedbysomenoise. Hen e,aftertheinitializationwehaveto learawayallthe 72

stru tureswearenotinterestedinbybuilding up,startingfromtheinitialdata

U

0

, anew 73

ve toreld

U

whosesingularitiesaregivenbythepointsoftheimage

I

wewanttoisolate. 74

Thus,fromonehandwehavetofor ethe on entrationsetofthedivergen emeasureof 75

U

0

to ontainonlythepointswewantto at h,andontheotherhandwehavetoregularize 76

theinitial data

U

0

outsidethepointsof singularities. Tothisend we proposeto minimize 77

anenergy involvinga ompetition between adivergen e term and the ounting Hausdor 78

measure

H

0

. Morepre iselytheenergyisthefollowing 79

Z

Ω\P

|divU |

2

_{+ λ}

Z

Ω

|U − U

0

|

p

+ H

0

(P ).

(1) where

U

∈ L

p,2

_{(div; Ω \ P )}

thespa eofthe

L

p

ve toreld whosedistributionaldivergen e 80

belongsto

L

2

_{(Ω \ P )}

,

P

istheatomi setwewanttotargetand

λ

isapositiveweight. The 81

rstintegralfor es

U

toberegularoutside

P

,whiletheterm

H

0

_{(P )}

penalizesthepresen e 82

ofthesingular urvesandthenoisein theimage. 83

Fromapra ti alpointofview,this hoi eallowsustohandlewitharstorderdierential 84

operator and permits us to formulate the minimization problem in a ommon fun tional 85

framework. 86

Clearly toinitializetheminimization pro essweneedofave toreld

U

0

linkedtothe 87

initialimage

I

,belongingtothespa e

DM

p

_(Ω)

. Su have toreld anbeprovidedbythe 88

gradientofweaksolutionofthe lassi alDiri hletproblemwithmeasure data. 89

(

∆f = I

on

Ω

f

= 0

on

∂Ω.

(2)

In se tion 3 we re all the basi results of the theory of the ellipti equations with data 90

measure. 91

Inordertoprovide omputerexamples,wemustapproximatethefun tional(1)bymeans 92

ofasequen eofmore onvenientfun tionals fromanumeri alpointofview. 93

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

Γ

- onvergen e, 94

thenotion of variational onvergen e introdu ed byDe Giorgi (see [13, 14℄). This theory 95

is designed to approximate a variational problem by a sequen e of dierent variational 96

problems. Themostimportantfeatureofthe

Γ

- onvergen ereliesonthefa tthatitimplies 97

the onvergen e of minimizers of the approximating fun tionals to those of the limiting 98

fun tional. Inthisworkweprovideapossible

Γ

- onvergen eapproa hforthedete tionof 99

points. 100

Classi ally,inimageanalysis,the

Γ

- onvergen ehasbeenusedtodealwith1-dimensional 101

obje tslikesmoothboundaries. Forinstan ein([2,3℄)AmbrosioandTortorellihaveproven 102

thattheMumford-Shahfun tional anbeapproximatedbyasequen eofellipti fun tionals 103

thatarenumeri allymoretreatable. 104

The main di ulty hereis related to thepresen e of a odimension

2

obje t,whi h is nota ontour: theset

P

. Inorder to obtainavariationalapproximation loseto the one providedin([2,3℄),the ru ialstepisthentorepla etheterm

H

0

_{(P )}

ofafun tional(1)by amorehandy,fromavariationalpointofview,fun tionalinvolvingasmoothboundaryand his perimeter measure given by the

1

-dimensionalHausdor measure

H

1

. Following some suggestionfrom the paper of Braides and Mal hiodi and Braides and Mar h (see [9, 10℄) su h afun tionalisgivenby:

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 105

boundary,the onstant

1

4π

isanormalizationfa tor,and

β

ε

innitesimalas

ε

→ 0

. Roughly 106

speakingtheminimumofthisfun tionalsarea hievedontheunionofballsofsmallradius, 107

sothat when

β

ε

→ 0

thefun tionals shrinks to theatomi measure

H

0

_{(P )}

. On theother 108

hand this approa h requires a non trivial and onvenient, for a numeri al point of view, 109

approximation a urvature-dependent fun tional. Su h an approximation is based on a 110

elebrated onje turedue to DeGiorgi (see[12℄. Bymeansof thisargumentitis possible 111

to substitute the urvature-depending fun tional with anintegral fun tional involvingthe 112

Lapla ianoperator ofsmoothfun tions. Then itremains toapproximatethe

H

1

measure 113

andthis anbedonebyretrievinga lassi algradientapproa husedin[15,16℄. Thisstrategy 114

allowstodealwithafun tionalwhoseEuler-Lagrangeequations anbedis retized. Asimple 115

andintuitiveexplanationofthe onstru tionofthe ompleteapproximatingfun tionalswill 116

begivenin se tion

3

. 117

Thepaperis organizedasfollows: se tion

1

is intendedto remindthereaderof mathe-118

mati altoolsusefulin thefollowing. Inse tion

2

were allthenotionof

Γ

- onvergen eand 119

weillustratethetwo

Γ

- onvergen eresultsweneedinthesequel. Inse tion

3

weaddressthe 120

onstru tionoftheapproximatingfun tionals. Inse tion

4

wepresentthedis rete model. 121

In se tion

5

we explain the dete tion strategy. Finally in the last se tion we give some 122

omputerexamples. 123

2 Preliminaries 124

2.1 Measure theory 125

Westartby lassi aldenitionsofmeasuretheory. Forageneralsurveyonmeasuretheory 126

werefer,amongothers,to[1℄. 127

Denition 2.1. Let

(X, Σ)

be a measure spa e and

µ

: Σ → [0, ∞).

We say that

µ

is a positive measureif

µ(∅) = 0

andfor anysequen e

{E

n

}

of pairwisedisjoints elements of

Σ

µ(

∞

[

n=0

E

n

) =

∞

X

n=0

µ(E

n

).

Wesaythat

µ

isnite if

µ(X) < +∞

andthat

µ

is

σ

-niteif

X

isthe unionof in reasing 128

sequen e ofsetswith nitemeasures. 129

Denition2.2. Let

(X, Σ)

bea measurespa eandlet

N

∈ N

,

N

≥ 1

. 130

(i)

We say that

µ

: Σ → R

N

is a measure, if

µ(∅) = 0

and for any sequen e

{E

n

}

of pairwisedisjoints elements ofthe

σ

-algebra

Σ

µ(

∞

[

n=0

E

n

) =

∞

X

n=0

µ(E

n

).

If

N

= 1

wesaythat

µ

isarealmeasure, if

N >

1

wesay that

µ

isave tormeasure. 131

(ii)

If

µ

isameasure, wedeneitstotal variation

|µ|

asfollows

∀E ∈ Σ

|µ|(E) := sup{

∞

X

n=0

|µ(E

n

)| : E

n

∈ Σ

pairwisedisjoints,

E

=

∞

[

n=0

E

n

}.

(iii)

Wesaythat

µ

is niteis

|µ|(X) < +∞

. 132

It an be proven that the total variation

|µ|

is apositive measure, more pre isely, it is the 133

smallest positive measure su h that

|µ|(E) ≥ |µ(E)|

for every

E

∈ Σ

. Clearly, if

µ

is a 134

positive measure, then

|µ| = µ

. 135

An important lass of measures is the Radon measures. In order to introdu e su h a 136

on ept,thenotionofBorelsetisne essary. 137

Denition 2.3. Let

X

bea lo ally ompa t and separable metri spa e. Let

A(X)

be the 138

olle tion of all open sets of

X

. We all the Borel

σ

-algebra on

X

the smallest

σ

-algebra 139

ontaining

A(X)

andwedenoteditby

B

(X)

. We allitselements the Borelsets of

X

. 140

Denition 2.4. Let

X

be a lo ally ompa t and separable metri spa e and

B

(X)

be the 141

Borel

σ

-algebra of

X

. Consider themeasurespa e

(X, B(X))

. 142

(i)

A positivemeasureon

(X, B(X))

isaBorelmeasure. 143

(ii)

A positive measure on ea h ompa t subset

K

of

X

will be alled a positive Radon 144

measure. If

µ

is an

R

N

-valued measure dened on all the Borel subset of

X

s.t. 145

|µ|

is a Radon measure and

|µ|(X) < +∞

, we say that

µ

is an

R

N

-valued nite 146

Radon measure. Wedenote by

M(X; R

N

₎

thespa eofthe all

R

N

-valuednite Radon 147

measures. 148

2.2 Distributional divergen e 149

Inthissubse tionwere allthedenitionofthespa e

L

p,q

_{(div; Ω)}

and

DM

p

_(Ω)

,introdu ed 150 in[4℄. 151 Let

Ω ⊂ R

2

beanopenset andlet

U

: Ω ⊂ R

2

_{→ R}

2

beave toreld. 152

Denition 2.5. We say that

U

∈ L

p,q

_{(div; Ω)}

if

U

∈ L

p

_{(Ω; R}

2

₎

and if its distributional 153 divergen e

divU ∈ L

q

_(Ω)

. If

p

= q

the spa e

L

p,q

_{(div; Ω)}

willbe denotedby

L

p

_{(div; Ω)}

. 154 If

p

= q = 2

the spa e

L

2

_{(div; Ω)}

equippedwiththe s alarprodu t

(U, V ) =

Z

Ω

U

· V dxdy +

Z

Ω

divU divV dxdy;

isanHilbertspa e. 155 Wesay that

U

∈ L

p,q

loc

(div; Ω)

if

U

∈ L

p,q

_{(div; A)}

for everyopen set

A

⊂⊂ Ω

. 156

Denition2.6. For

U

∈ L

p

_{(Ω; R}

2

₎

,

1 ≤ p ≤ +∞

,set

|divU|(Ω) := sup{hU, ∇ϕi : ϕ ∈ C

0

1

(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

. 157

Remark2.1. If

U

∈ DM

p

_(Ω)

thenviaRieszTheorem itispossibletorepresentthe distri-158

butionaldivergen e of

U

by aRadonmeasure. Morepre isely thereexistsaRadon measure 159

µ

su hthat for every

ϕ

∈ C

1

0

(Ω)

the following equalityholds: 160

Z

Ω

U

· ∇ϕdxdy = −

Z

Ω

ϕdµ.

Forinstan ethe eld

U

(x, y) = (

x

x

2

_+y

2

,

y

x

2

_+y

2

)

belongsto

DM

1

loc

(R

2

)

anditsdivergen e 161

measureisgiven by

−2πδ

0

,where

δ

0

isthe the Dira mass. 162

Su h aresult an beprovenby approximation. Letus denethe following map: 163

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 164

2

ε

2

χ

B(0,ε)

. 165

Then for every testfun tion

ϕ

∈ C

1

0

(R

2

)

wehave 166

Z

U

ε

· ∇ϕdxdy = −

Z

₂

ε

2

χ

B(0,ε)

ϕdxdy.

Byapplying the hanging variable

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

,bydominated onvergen etheorem weobtain

Z

Ω

U

· ∇ϕdxdy = −2πϕ(0, 0) = −2π

Z

Ω

ϕdδ

0

3 The Diri hlet problem with measure data 167

Fortheinitializationofouralgorithmwemustbuiltave toreld

U

0

whi hshouldbesu h 168

thatitsdivergen e issingularonpointsoftheimage

I

. Thereforewewilluse thegradient 169

ofthesolutionofthefollowingDiri hletproblem(appliedwith

µ

= I

) 170 171

(

∆f = µ

on

Ω

f

= 0

on

∂Ω

(3)

where

µ

isaRadonmeasure 172

The study of a PDE with data measure is avery lassi al resear h topi in Analysis 173

andthere isavast literatureaboutthis subje t. Inwhat followswe re alljust somebasi 174

denitionsand ru ialresultsforourpurpose. 175

Denition3.1. Wesaythat

f

isaweaksolutionofthe(3 )if

f

∈ W

1,1

0

(Ω)

andthefollowing equalityholds forevery test

ϕ

∈ C

1

0

(Ω)

:

Z

Ω

∇f · ∇ϕ = −

Z

Ω

ϕdµ.

Note that if

f

isa weak solution,then itsweakgradientis adivergen e measure eld, 176

i.e.

U

= ∇f ∈ DM

p

_(Ω)

,anditsdivergen eisequaltothemeasuredata

µ

. 177

A lassi al example of a weak solution is provided by a fundamental solution of the 178

Lapla eequation. Letus onsidertheproblem: 179

(

∆f = δ

0

on

Ω

f

= 0

∂Ω.

(4)

Then,bythesameargumentusedinRemark2.1,one aneasily he kthatthedistributional 180

divergen eof

∇f

isgivenby

δ

0

. 181

Moregenerally lassi alresults(see[17℄)guaranteetheexisten eofauniquesolutionof 182

theproblem(3). Con erningtheregularityitisknownthen

f

∈ W

1,p

with

p <

2

. 183

4

Γ

- onvergen e 184

Thekeypointof ourstrategyis theapproximationofthefun tional(1)bymeansof more 185

regularfun tionalsvia

Γ

- onvergen e. 186

Thereforethisse tionisdevotedtoaverysimplepresentationofthenotionof

Γ

onver-187

gen eandthetwo ru ialresultsforourgoal: theModi a-Mortola'stheorem (see[15,16℄) 188

on erningtheapproximationof theperimetermeasure and theDeGiorgi onje ture(see 189

[12℄)abouttheapproximationof urvaturedepending fun tionals. 190

Westartbyre allingthedenitionofthe

Γ

- onvergen ejustinthespe ial aseweneed 191

of. Fora ompletesurveyonthissubje twereferto[8,11℄andreferen estherein. Inwhat 192

follows

X

standsforametri spa e

(X, d)

193

Denition4.1. Wesay that asequen e

F

n

: X → [−∞, +∞] Γ

- onvergesto afun tional 194

F

: X → [−∞, +∞]

as (

n

→ +∞

), with respe t to the

X

-distan e, if the following two 195

properties hold: 196

for every

u

∈ X

andevery

{u

n

} ∈ X

,su hthat

u

n

→ u

with respe tto

X

-distan e, 197

F(u) ≤ lim inf

n

→+∞

F

n

(u

n

),

(5)

for every

u

∈ X

thereexists

{u

n

} ⊂ X

,su hthat

u

n

→ u

with respe t to

X

-distan e 198

F

(u) = lim

n

→+∞

F

n

(u

n

).

(6)

The fun tional

F

is alled the

Γ

-limit of

F

(with respe t to the

X

-distan e) and we write 199

F

= Γ − lim

n

F

n

. 200

Therelevan eofthisnotionisexplainedbythefollowingfundamental theorem 201

Theorem 4.1. Let us suppose that

F

= Γ − lim

n

F

n

and let a ompa t set

K

⊂ X

exist su hthat

inf

X

F

n

= inf

K

F

n

for all

n

. Then

min

X

F

= lim

n

inf

X

F

n

. 202

Moreover, if

u

n

isa onverging sequen esu hthat

lim

n→+∞

F

ε

(u

n

) = lim

n→∞

inf

X

F

n

,thenits 203

limitisaminimum pointfor

F

. 204

Itispossibletoextendthedenitionof

Γ

- onvergen etoafamilydependingonareala 205

parameter. Sothatwe an dealwith

Γ

-limitsoffamilies

{F

ε

}

as

ε

→ 0

. 206

Denition 4.2. We saythat a sequen e

F

ε

: X → [−∞, +∞] Γ

- onvergesto afun tional 207

F

: X → [−∞, +∞]

as (

ε

→ 0

), if for every sequen e

{ε

n

}

onverging to

0

, as

n

→ +∞

, 208

the sequen e

F

ε

n

Γ

- onvergesto

F

. 209

4.1 Modi a Mortola's approa h 210

TheModi a-Mortolatheoremstatesthatitispossibletoapproximate,inthe

Γ

- onvergen e 211

sense,aperimetermeasure bymeansofthefollowingsequen eoffun tionals 212

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 a double well potential. Besides in order to avoid that the 213

minimizers of the fun tional

F

1

ε

are trivial, some onstraint on the fun tions

u

ε

must be 214

added. Intheoriginal versionof the Modi a-Mortola'spaperavolume onstrained ofthe 215

type

R

Ω

udxdy

= m

,wasassumed. 216

Letusgiveanintuitiveexplanationofsu haresult. Sin e

V

hastwoabsoluteminimizers 217

at

u

= 0, 1

,when

ε

issmall,alo al minimizer

u

ε

is losedto

1

onapartof

Ω

and loseto 218

0

ontheotherpart,makingarapidtransitionoforder

ε

between

0

and

1

. When

ε

→ 0

the 219

transitionset shrinkstoasetof dimension

1

,sothat

u

ε

goestoafun tion takingvalues

u

220

into

{0, 1}

andthefun tionals onvergetothemeasureoftheperimeterofthedis ontinuities 221

setof

u

. 222

BythewaytheModi a-Mortola'stheoremstillholdsevenwithoutassumingthevolume 223

onstrain. It meansthat in general the

Γ

-limit ould begiven bya fun tional identi ally 224

equalto

0

. TheModi a-Mortola'sTheorem isthefollowing. 225

Theorem 4.2. The fun tionals

F

1

ε

: L

1

(Ω) → [0, +∞] Γ−

onverge with respe t to the 226

L

1

-distan etothefollowing fun tional 227

F

1

(u) =

(

C

V

H

1

(S

u

)

if

u

∈ {0, 1}

+∞

otherwise

where,asusual,

S

u

denotesthe setofthe dis ontinuitiesof

u

and

C

V

isasuitable onstant 228

dependingon thepotential

V

. 229

4.2 De Giorgi's onje ture 230

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

. 231

Sin e

∂D

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

u

0

= 1 − χ

D

, 232

by Modi a-Mortola's Theorem it follows that there is a sequen e of non onstant lo al 233

minimizerssu hthat

u

ε

→ u

0

withrespe tto the

L

1

- onvergen esu hthat 234

lim

ε→0

F

1

FurthermorelookingattheEuler-Lagrangeequationasso iatedtoa ontourlengthterm, 235

yields a ontour urvature term

κ

, while the Euler-Lagrange equations for the fun tional 236

F

1

ε

(u)

ontainsaterm

2ε∆u −

V

′

(u)

ε

. 237

Then DeGiorgi suggestedtoapproximatethe fun tional

F

2

byadding to the Modi a-238

Mortolaapproximatingfun tionalsthefollowingterm 239

F

_ε

2

(u) =

Z

Ω

(2ε∆u −

V

′

(u)

ε

)

2

dx.

In ([6℄) Bellettini and Paolini proved the

lim sup

inequality (6), while the validity of 240

lim inf

inequality(5)remainsanopenproblem. 241

5 The approximating fun tionals 242

Inthisse tionwepresenttheenergywedealwithandthe onstru tionoftheapproximating 243

sequen e. 244

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 245

nitenumber

N

ofpoints,i.e.

P

= {x

1

, ..., x

N

}

. 246

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 247

x

i

∈ D

i

,

D

i

T D

j

= ∅

for

i

6= j

.

κ

is the urvature of the boundary of the sets

D

i

, the 248

onstant

1

4π

isanormalizationfa torand

β

ε

isinnitesimalas

ε

→ 0

. 249

Tounderstand why we an approximate

H

0

_{(P )}

with

G

β

ε

(D)

one shouldnote that the 250

solutionofthefollowingminimumproblem 251

min

D

⊃P

G

β

ε

(D)

(7) isgivenby

D

=

S

N

i

B(x

i

, β

ε

),

where

x

i

arethepointsof

P

. Wesket htheargumentinthe 252

singlepoint ase. 253

BytheYoung'sinequalitywehave 254

G

β

ε

(D) ≥

1

4π

Z

∂D

2κdH

1

(x)

andbyapplyingtheGauss-BonnetTheorem 255

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

)

, 256

weobtainthe value

1

, i.e. thenumberof pointsin

P

, i.e.

H

0

_{(P )}

. The

N

points ase an 257

bere overedwithminor hangesbythesameargument. 258

Forwhatfollowsitis onvenientto splitthefun tional

G

β

ε

intwoterms:

G

β

ε

(D) = G

1

βε

(D) + G

2

βε

(D)

where 259

G

1

_β

_ε

(D) =

1

4π

Z

∂D

1

βε

dH

1

_(x);

and 260

G

2

_β

_ε

(D) :=

1

4π

Z

∂D

β

ε

κ

2

(x)dH

1

(x).

We anwrite anintermediateapproximationoftheenergy(1): 261

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

.

(8) Theadvantageofsu h aformulationis that weknowhow to provideavariational ap-262

proximation of theperimeter measure

H

1

_⌊∂D

. Following the Modi a-Mortola's approa h 263

su h anapproximation anbeobtainedbyusingthefollowingmeasure: 264

µ

ε

(w, ∇w)dx = ε|∇w|

2

+

V

(w)

ε

dx,

where

V

(w) = w

2

_{(1 − w)}

2

isadoublewellfun tional. 265

Next stepis expressingthe urvature termbymeansofthe fun tion

w

. Thankstothe 266

DeGiorgi's onje turewe an repla etheterm

κ

bytheterm

2ε∆w −

V

′

(w)

ε

.

267

Sothatwe anformallywritethe 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,

(9) where

U

∈ L

p,2

_{(div; Ω)}

and

w

is smooth fun tion equal to

1

on the boundary, i.e. 268

1 − w ∈ C

∞

0

(Ω)

. We add the volume onstraint

1

L

2

_(Ω)

R

Ω

wdx

= 1

. This last ondition 269

prevents

w

from onvergingtothefun tion onstantlyequalto

0

as

ε

→ 0

. 270

Thenif

(U

ε

, w

ε

)

isaminimizingsequen eof

Φ

ε

,then

w

ε

mustbevery losetothevalues 271

1

when

ε

goesto

0

, sin ethe double well potential is positive ex eptfor

w

ε

= 0, 1

and

w

272

mustbeequalto

1

onthe

∂Ω

. Ontheotherhand,nearthepointswhere thedivergen eis 273

verybig

w

ε

mustbe lose to

0

. Besides when

ε

→ 0

,

β

ε

→ 0

goesto

0

aswell,so thatthe 274

singularset

D

is givenbyanunionofballsof asmallradius

β

ε

. 275

Therefore,whilethefun tions

U

ε

approximateaminimizer

U

oftheoriginalfun tional, 276

thelevelset

{w

ε

= 0}

approximatetheoriginalsingularset

P

. 277

Therstvariationofthis fun tionalleadstothefollowinggradientowsystem

∂U

∂t

=

∇(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

(10)

where

h

isgivenbytheequation 278

h

= 2ε∆w −

1

ε

V

6 Dete tion 279

Inourmodeltheimage ontainsanatomi Radonmeasure. Thus,inordertondaninitial 280

ve tor eld whi h opies the singularities of the initial image, we use the gradient of the 281

solutionofthefollowingDiri hletproblem: 282

(

∆f = I

on

Ω

f

= 0

on

∂Ω.

(11)

In this way we obtain a ve tor eld whose divergen e is singular on a proper set whi h 283

ontainsthe points we wantto dete t. In generalthis set ould ontain other stru tures. 284

For instan e if the initial image is a Radon measure on entrated both on points and on 285

urves,thedivergen eof

∇f

issingularonpointsandon urvesat thesametime. Besides 286

ifthere is somenoisein theimage, it ould be not lear how to dierentiate thesingular 287

points due to the noise, from those we want to at h. As a onsequen e, by solvingthe 288

problem(11),wegetjustapredete tion,whi hhastoberened. 289

Tothis purposewesear hfor aminimizerof theenergy

Φ

ε

(U, w)

viasolvingequations 290

(10)withinitialdata

U

0

givenby

∇f

. Sothat weobtainave toreld

U

whosedivergen e 291

isrelevantonlyontheset

P

and afun tion

w

whosezerosaregivenbytheset

P

. 292

6.1 Dis retization 293

Theimageisan

N

×N

ve tor. Weendowedthespa e

R

2N

withthestandards alarprodu t 294

andstandardnorm. 295

Thedis reteversionofgradientoperator isthefollowing. 296

Let

I

∈ R

2N

. Thenthegradient

∇I

isanelementofthespa e

R

2N

_{× R}

2N

givenby:

(∇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 297

adjointoperatorofthegradient:

div = −∇

∗

. Moreindetailsif

v

∈ R

2N

wehave 298

(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)

. 299 6.2 Dis retization in time 300 Wesimplyrepla e

∂U

∂t

and

∂w

∂t

by

U

_i,j

n+1

−U

n

i,j

δt

and

w

n+1

_i,j

−w

n

i,j

δt

respe tively. Thenwewritethe system(10)intheform(forsimpli ityweomitthedependan 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

).

Weinitialize ouralgorithm with

U(0) = ∇f

, where

f

isthe solutionof theproblem (11). 301

Todothis, weneedto solvetheDiri hletproblem (11)withmeasure data

I

, thereforewe 302

regularizetheimageby onvolutionwithaGaussiankernel

G

σ

withverysmall

σ

andthen 303

wesolve,by lassi alnitedieren esmethod,theproblem: 304

(

∆f = I

σ

on

Ω

f

= 0

∂Ω,

(12) where

I

σ

= I ∗ G

σ

. 305

Toinitializeouralgorithm,weneedof aninitialguess on

w

. Sowe hoose

w(0) = 1

. 306

7 Computer examples 307

7.1 Parameter settings 308

Beforerunningouralgorithmalltheparametershavetobexed. Themostimportantare

ε

309

and

β

ε

,whi hgoverntheset

D

approximatingpointswewanttodete t. Thoseparameters 310

arerelatedbythe ondition

lim

ε→0

ε

β

ε

= 0

. Furthermore,sin ethemeshgridesizeis

1

and

β

ε

311

givestheradiusof aballs enteredinthesingularpointwewanttodete t,from adis rete 312

point of viewthe smallestvaluewe antake is

√

2

2

. Then weuse valuesranging from

0.1

313

and

0.5

for

ε

,andfrom

1.2

to

0.7

for

β

ε

. 314

Con erning theparameter

λ

wemainly used thesmall value

λ

= 0.1

, in order to for e 315

thealgorithmtoregularizeasmu haspossibletheinitial data

U

0

. 316

Sin e we deal with small values of

ε

, in order to have some stability, we must take a 317

smalldis retizationtimestep,Pra ti allywemainly usedthevalue

δt

= 1 × 10

−8

. 318

Con erningthestop riterionweiteratethealgorithmuntilmax

n

_|U

n+1

1

−U

2

n

|

|U

n

1

|

,

|U

n+1

2

−U

2

n

|

|U

n

2

|

,

|w

n+1

−w

n

|

|w

n

_|

o

319

≤ 0.001.

Finallyweset

p

= 1.5

. 320

Inallthe omputerexamplesthepointsaredete tedbymeansof thefun tion

w

ε

. We 321

displaythelevel-set

{w

ε

≃ 0}

. 322

7.2 Commentaries 323

The gure1showshowresistant to the noiseourmodel is. When the noiseis largerthe 324

parameter

ε

mustbeas loseraspossibletotheidealvalue

0

. Besidesitis possibleto see 325

thatforsmallvaluesof

ε

ourdete tionisnera ordingtothe ontinuoussetting. Morein 326

detailsintherstrowwedisplaytheinitialimageobtainedbyaddingagaussiannoisetoa 327

binaryimageof vepoints. Thein reasinglevelof thenoise orrespondstothein reasing 328

P SN R

betweenthebinaryimageandthenoisyimage. These ondrowshowsthebehavior 329

of

w

forintermediate valueofthe parameters

ε

and

β

ε

. Inthelast rowone annote that 330

thesmaller

ε

and

β

ε

arethenerthedete tionis. 331

Ingure2in therst tworowswedisplaythe histogramsofthegraylevelof

I

and

w

332

orrespondingtothedierentvaluesofPSNR.One anseethatitiseasierxingathreshold 333

valuestartingfromthefun tion

w

ε

thanfromtheinitialimage

I

. Inthelastrowwedisplay 334

theset

{w

ε

≃ 0}

obtainedbyplottingtheset

{w

ε

≤ α}

with thresholdvalue

α

= 0.5

335

In gure 3we test our algorithm on urves and points at thesame time. In the rst 336

olumn we havea sequen eof points and a urvewith no boundary on

Ω

. Inthe se ond 337

olumn we onsider the same urve with boundary in

Ω

. The se ond olumn we display 338

the fun tion

w

ε

and in the last olumn the level set

{w

ε

∼

= 0}

on e again obtained by 339

xingathresholdvalue

α

= 0.5

. Theresultisthat, asdesired,ouralgorithm is apableof 340

eliminating the urvefrom theinitial image. A ording to the ontinuous setting when

ε

341

takesvalues loseto

0

the approximatingenergy (9)behavessimilarly tothe limitenergy 342

(1),sothatthepresen eofthe urveispenalizedintheminimizationpro ess. Thentheset 343

{w

ε

∼

= 0}

ontainsnothingelsebutpoints. 344

Finally in gure 4 we deal with a biologi al image. Our task is at hing the nest 345

stru turepresentintheimage. Inboth asesofgure4theisolatedpointiswelldete ted, 346

whilethebran handthedashesofthe elluleisnot.

Noisyimage

I

PSNR

= 9.9

Db Noisyimage

I

PSNR

= 7.6

Db Noisyimage

I

PSNR

= 5.5

Db

Thefun tion

w

(

ε

= 0.5

,

β

ε

= 1.2

) Thefun tion

w

(

ε

= 0.5 β

ε

= 1.2

) Thefun tion

w

(

ε

= 0.5

,

β

ε

= 1.2

)

Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

) Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

) Thefun tion

w

ε

(

ε

= 0.1 β

ε

= 0.7

)

Figure1: Syntheti image: wetestouralgorithmonnoisyimages. Whentheparameters

ε

and

β

ε

aresmall asmu haspossiblethedete tionisner.

Noisyimage

I

PSNR

= 9.9

Db Noisyimage

I

PSNR

= 7.6

Db Noisyimage

I

PSNR

= 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

) Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

)

{w

ε

≃ 0}

(

ε

= 0.1

,

β

ε

= 0.7

)

{w

ε

≃ 0}

(

ε

= 0.1

,

β

ε

= 0.7

)

{w

ε

≃ 0}

(

ε

= 0.1

,

β

ε

= 0.7)

Originalimage

I

Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

)

{w

ε

≃ 0}

(

ε

= 0.1

,

β

ε

= 0.7

)

Figure 3: Syntheti image: urveand points arepresentintheinitial image. As expe tedour methodis apableofremovingthe urvefromtheimage.

Originalimage

I

Thefun tion

w

ε

(

ε

= 0.1

,

β

ε

= 0.7

)

{w

ε

≃ 0}

(

ε

= 0.1

,

β

ε

= 0.7

)

Figure 4: Biologi alimage: oneisolatedpoint,dashesandlament arepresentintheinitialimage. After thealgorithmpro esswekeepnothingelsebuttheisolatedpoint.

Referen es 348

[1℄ L.Ambrosio,N.Fus o,D.Pallara. Fun tions of boundedvariation and free dis ontinuity 349

problems.OxfordUniversityPress(2000). 350

[2℄ L.Ambrosio,V.M.Tortorelli.Approximationoffun tionalsdependingonjumpsbyellipti 351

fun tionalsvia

Γ

- onvergen e.Communi . PureAppl.Math43(1990),999-1036. 352

[3℄ L.Ambrosio, V.M.Tortorelli. Approximation of fun tionals depending on jumps by 353

quadrati , ellipti fun tionalsvia

Γ

- onvergen e.Boll.Un.Mat.Ital.6-B(1992),105-123. 354

[4℄ G.Anzellotti Pairings between measures and bounded fun tions and ompensated om-355

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