Object extraction using a stochastic birth-and-death dynamics in continuum

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dynamics in continuum

Xavier Descombes, Robert Minlos, Elena Zhizhina

To cite this version:

Xavier Descombes, Robert Minlos, Elena Zhizhina. Object extraction using a stochastic

birth-and-death dynamics in continuum. [Research Report] RR-6135, INRIA. 2007, pp.27. �inria-00133726v3�

inria-00133726, version 3 - 1 Mar 2007

a p p o r t

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Thème COG

Object extraction using a stochastic birth-and-death

dynamics in continuum

Xavier Descombes — Robert Minlos — Elena Zhizhina

N° 6135

Xavier Des ombes, Robert Minlos , Elena Zhizhina

∗

ThèmeCOG Systèmes ognitifs ProjetAriana

Rapportdere her he n° 6135Mar h200727pages

Abstra t: Wedeneanewbirthanddeathdynami sdealingwith ongurationsofdis sin theplane. Weprovethe onvergen eofthe ontinuouspro essandproposeadis retes heme onvergingtothe ontinuous ase. Thisframeworkisdevelopedtoaddressimagepro essing problems onsisting in extra tingobje ts. Thederived algorithmis applied fortree rown extra tionandbirddete tionfromaerialimages. Theperforman eofthisapproa hisshown onrealdata.

Key-words: Obje textra tion,Sto hasti modeling,Birthanddeathdynami s

ThisworkwaspartiallysupportedbyEGIDEwithintheECO-NETproje t10203YKnad byINRIA COLORS Flamants. Wewouldliketo thankthe Fren h NationalForestInventory(IFN) and Arnaud Bé hetfromLaStationBiologiqueTourduValatforkindlyprovidingthedata.

∗

R.MinlosandE.ZhizhinaarewithinLab.Dobrushin-IITP,Bol'shoiKaretnyiper.,19,127994,GSP-4, Mos ow,Russia.

Résumé: Nousdénissonsunedynamiquedenaissan eetmorts'appliquantàdes ongurations dedisquesdansleplan. Nousprouvonsla onvergen edupro essus ontinuetproposonsune dis rétisationduproblème onvergeantversle as ontinu. Cetteappro heest développée pourrésoudredesproblèmesd'analysed'imageliésàl'extra tiond'objets. L'algorithmequi s'endéduitestappliquéauxproblèmesdel'extra tiondehouppiersetàladéte tiond'oiseaux àpartird'imagesaériennes. Lesperforman esdel'appro hedéveloppéesontmontréessur desimagesréelles.

1 Introdu tion

Weproposeanewsto hasti algorithmtosolveobje textra tionproblemsfromimages. The algorithmisbasedonanevolutionofma ro-obje tsin ontinuum. We onsiderhereamodel ofpossiblypartiallyoverlappingdis s. Ea hdis inthenal ongurationisasso iatedwith agivenobje tin theimage, forexampleatreeorabird. Wedeneasto hasti evolution ofaset ofobje ts onvergingtothesetofobje tsofinterestin theimage.

The evolution under onsideration is a birth-and-death equilibrium dynami s on the ongurationspa e of dis s (or the ongurationspa e of points) with agiven stationary Gibbs measure(see [1, 2℄). Wedenebirth anddeathratesmeeting theso- alled detailed balan e onditions. In ours hemethe intensity of birth is a onstant,whereas intensities ofdeathdepend ontheenergyfun tion andthe urrent onguration. This hoi eofrates hasbeenmadetooptimizethe onvergen espeed. Indeed,thevolumeofthespa eforbirth ismu hbiggerthanthenumberofdis sin the onguration. Ea hdis anbekilledwith a ertainintensity dependingon itsneighborhoodand theenergy fun tion. It istherefore fastertoupdatethedeathmapthanthebirthmap.

We then embed thedened stationary dynami s into asimulated annealing pro edure wherethetemperatureofthesystemtendstozerointime. Wethusobtainanon-stationary sto hasti pro ess, su hthat allweaklimitmeasures haveasupporton ongurations giv-ing the global minimum of the energy fun tion under a minimal number of dis s in the onguration. The nal step is the dis retization of this non-stationary dynami s. The dis retization is a non-homogeneous (in time and in spa e) Markov hain with transition probabilitiesdependingonatemperature,theenergyfun tionandadis retizationstep. We provethat:

1)thedis retization pro ess onvergestothe ontinuoustimepro essunder xed tempera-tureasthestepofdis retizationtendsto zero;

2) if we apply the dis retization pro ess to any initial measure with a ontinuous density w.r.t. theLebesgue-Poissonmeasure, then in thelimit whenthe dis retization steptends to0,timetendstoinnityandthetemperaturetendsto0,wegetameasure on entrated ontheglobalminimaoftheenergyfun tionwithaminimalnumberofdis s.

These results onrm that theproposed algorithm based on the dis retization s heme togetherwiththe oolingpro edure anbeappliedtoproblemsofsear hing ongurations givingglobalminimaoftheenergyfun tion.

Weapply thisframeworkto obje tdete tion fromimages. Insomepreviousworks,we haveshown that markedpoint pro esses,dened byaGibbs measure againstthePoisson pro ess, are adapted to su h problems by modeling simple geometri obje ts dened by themarksasso iatedto ea h point. Moreover,intera tionsbetweenpointsallowto model someaprioriinformationontheobje t onguration. Thisapproa hhavebeenapplied to dete tdierentfeaturessu hasroadnetworks[3,4℄,buildings[5℄ortrees[6℄. Inthese refer-en es,theoptimizationofthedenedmarkedpointpro essisperformedusingaRJMCMC s heme[7℄. IntheRJMCMCs heme,ea hiteration onsistsinperturbatingoneora ouple of obje ts. Besides the reje tion rate indu es a huge omputation time. In theproposed approa h, ea h step on ernsthe whole onguration and there is no reje tion. We thus

obtainedbetterperforman esin termof omputationaltime,whi hallowtodealwithreal imagesofseveralmillionsofpixels. Webuildanenergyfun tionwhi hembedssomeapriori knowledgeontheobje t ongurationsu haspartial nonoverlappingbetweenobje tsand adatatermwhi hallowstheobje tstottheimageunderstudy. Theoptimizationisthen performedbyusingtheproposedbirthanddeathdynami s. Someresultsareshownonreal datafortheproblems oftreeand birddete tion.

2 Des ription of the model 2.1 Conguration spa e

We onsider nite systems of dis s

{d

x

1

, . . . , d

x

k

}

of the same radius

r

with a hard ore distan e

ǫ

betweenanytwoelements,lyingin aboundeddomain

V ⊂ R

2

. Let

γ = {x

i

} ∈ Γ

d

(V ),

x

i

∈ V ⊂ R

2

,

bea ongurationofthe entersofdis sandwhere

Γ

d

(V )

denotesthe ongurationspa eof thedis s enterin

V

. Sin ethedomain

V

isboundedthenumberofdis sinany onguration isuniformlybounded

|γ| < N =

4|V |

πǫ

2

,

where

|V |

isthevolumeof

V

. Theset

Γ

d

(V )

anbede omposed intostrata:

Γ

d

(V ) =

N

[

n=0

Γ

d

(V, n),

whereea hstratum

Γ

d

(V, n)

istheset of ongurations ontaining

n

dis s,and

Γ

d

(V, 0) =

{∅}

. Sin e we onsider unordered sets of dis s, the set

Γ

d

(V, n)

for any

n > 0

an be representedasafa torset

Γ

d

(V, n) = V

d

n

/ S

n

,

where

V

n

d

= {(x

1

, . . . , x

n

) ∈ V

n

: |x

i

− x

j

| ≥ ǫ, i, j = 1, . . . , n, i 6= j},

and

S

n

isthepermutation groupintheset

(x

1

, . . . , x

n

)

. Wedeneamapping

Π

n

: V

d

n

→ Γ

d

(V, n),

Π

n

(x

1

, . . . , x

n

) ∈ Γ

d

(V, n)

(1) Ea hfun tion

F (γ), γ ∈ Γ

d

(V )

onthespa e

Γ

d

(V )

anberepresentedasafun tioninthe Fo kspa e:

F

0

, F

1

(x

1

), F

2

(x

1

, x

2

), . . . , F

N

(x

1

, . . . , x

N

),

(2) where

F

0

= F (∅), F

n

(x

1

, . . . , x

n

) = F (Π

n

(x

1

, . . . , x

n

))

isasymmetri alfun tion on

V

n

d

. A fun tion

F

onthespa e

Γ

d

(V )

issaidtobeasmooth( ontinuous)fun tionifea hfun tion in(2)isasmooth( ontinuous)fun tionon

V

n

d

.

Letus onsiderameasureinthespa e

Γ

d

(V, n)

:

λ

n

(A) =

Π

−1

n

(A)

n!

,

A ⊂ Γ

d

(V, n), λ

0

(∅) = 1.

(3) Here

|Π

−1

n

(A)|

is the

2n

-dimensional Lebesgue volume of the domain

Π

−1

n

(A) ⊂ V

d

n

(a ompletepreimageof

A

undermapping

Π

n

). Themeasure

λ

onthespa e

Γ

d

(V )

,su hthat therestri tionof

λ

onea h stratum

Γ

d

(V, n)

isgivenby

λ

n

,is alled theLebesgue-Poisson measure.

2.2 Energy fun tion

Wedeneonthespa e

Γ

d

(V )

areal-valuedsmoothandboundedfrombelowfun tion

H(γ)

whi h is alled the energy fun tion. We set

H(∅) = 0

. Then the orresponding Fo k representationfor

H

hastheform

H = (0, H

1

(x

1

), H

2

(x

1

, x

2

), . . . , H

N

(x

1

, . . . , x

N

)) .

(4) TheGibbsdistribution

µ

V

β

onthespa e

Γ

d

(V )

generatedbytheenergy

H(γ)

isdenedby thedensity

p

V

(γ) =

dµ

V

β

dλ

(γ)

withrespe tto theLebesgue-Poissonmeasure

λ

:

p

V

(γ) =

z

|γ|

Z

β,V

exp{−βH(γ)},

(5) withpositiveparameters

β > 0, z > 0

andanormalizingfa tor

Z

β,V

:

Z

β,V

=

Z

Γ

d

(V )

z

|γ|

exp{−βH(γ)}dλ(γ) = 1 +

N

X

n=1

z

n

n!

Z

V

n

d

e

−βH

n

(x

1

,...,x

n

)

_dx

1

. . . dx

n

.

Weformulatenowsomeassumptionsontheenergyfun tion

H

V

(γ)

. Denoteby

¯

H =

min

γ∈Γ

d

(V )

H(γ),

where

Γ

d

(V )

isthe losureof

Γ

d

(V )

,andlet

T

V

= {¯

γ ∈ Γ

d

(V ) : H(¯

γ) = ¯

H}

beaset ofallpointsfrom

Γ

d

(V )

givingtheglobalminimum

H

¯

ofthefun tion

H(γ)

. The set

T

V

anbewrittenas

T

V

=

N

[

n=0

T

V,n

,

where

T

V,n

is aset of ongurationsfrom

T

V

whi h arealso ongurationsfrom

Γ

d

(V, n)

, i.e. ontainexa tly

n

dis s.

Inpra ti e,thisenergy ontainsarsttermrepresentingaprioriknowledgeonthedis s ongurationand whi his denedbyintera tionsbetweenneighboringdis s,andase ond term,obtainedfromdata, whi hisdened forea hobje tandwhi h anbenegative.

3 Convergen e of measures Weassumethat

1)theset

T

V

isniteand situatedin

Γ

d

(V )

,

2) for any onguration

γ ∈ T

¯

V,n

any preimage

(¯

x

1

, . . . , ¯

x

n

) ∈ Π

−1

n

(¯

γ)

of

γ

¯

is a non-degenerated riti alpointofthefun tion

H

n

,i.e.:

∂H

n

∂y

m

i

(¯

x

1

, . . . , ¯

x

n

) = 0,

forany

i = 1, . . . , n, ¯

x

i

= (y

1

i

, y

i

2

) ∈ V, m = 1, 2

,andthematrix

A

_(¯

_{γ) =}

(

∂

2

_H

n

∂y

m

1

i

∂y

m

2

j

(¯

x

1

, . . . , ¯

x

n

)

)

atpoint

(¯

x

1

, . . . , ¯

x

n

)

isstri tlypositive-denite(thismatrixisthesameforallpreimagesof the onguration

γ

¯

). Wedenote

B(¯

γ) = det A(¯

γ).

(6)

Theorem 1. Let

n

0

∈ [0, . . . , N ]

be the minimal index for whi h the set

T

V,n

is not empty. Then the Gibbsdistributions

µ

β

onvergeweakly as

β → ∞

toadistribution

µ

∞

on

Γ

d

(V )

of the form

µ

∞

=

X

¯

γ∈T

_V,n0

C

¯

γ

δ

γ

¯

if

n

0

> 0,

and

µ

∞

= δ

{∅}

if

n

0

= 0.

(7)

Here

δ

γ

¯

istheunitmeasure on entratedonthe onguration

¯

γ

,andthe oe ients

C

γ

¯

hold the equality

X

¯

γ∈T

_V,n0

C

γ

¯

= 1.

Proof. Let

F (γ)

beasmoothfun tionon

Γ

d

(V )

. Then

hF i

µ

β

=

I(F )

Z

β,V

Z

β,V

−1







F

0

+

N

X

n=1

z

n

n!

Z

V

n

d

F

n

(x

1

, . . . , x

n

) e

−βH

n

(x

1

,...,x

n

)

dx

1

. . . dx

2







.

Letus onsiderea h integralin(8)

I

n

(F

n

) =

Z

V

n

d

F

n

(x

1

, . . . , x

n

) e

−βH

n

(x

1

,...,x

n

)

dx

1

. . . dx

2

separately. Iftheset

T

V,n

is empty,thentheintegral

I

n

(F

n

)

meets theestimate

|I

n

(F

n

)| < e

−βh

n

|V

d

n

|,

(9)

where

h

n

= min H

V,n

(x

1

, . . . , x

n

) > ¯

H.

If

T

V,n

isnotempty,thenthefollowingasymptoti sholds,seeforexample[8℄

I

n

(F

n

) = e

−β ¯

H





X

¯

γ∈T

V,n

F (¯

γ)R(¯

γ)β

−n/2

+ β

−n/2−1/2

S(F, β)





,

(10)

where

S(F, β)

isbounded as

β → ∞

,and

R(¯

γ) =

1

(2π)

n/2

B

−1/2

_(¯

_γ),

where

B(¯

γ)

is denedin (6). Then bound (9) andasymptoti s(10)imply thatfor

n

0

> 0

and

β → ∞

weget

I(F ) = F

0

+

N

X

n=1

z

n

n!

I

n

(F

n

) =

e

−β ¯

H

β

n

0

/2





X

¯

γ∈T

_V,n0

F (¯

γ)R(¯

γ) + o(1)





.

(11) Analogously,

Z

β,V

= I(1) =

e

−β ¯

H

β

n

0

/2





X

¯

γ∈T

_V,n0

R(¯

γ) + o(1)





.

(12)

Finallyfrom(8),(11)and(12)weget(7)with

C(¯

γ) =

P

R(¯

γ)

¯

γ∈T

_V,n0

R(¯

γ)

.

The ase

n

0

= 0

an be studied in the same way. Sin e the spa e of smooth fun tions is dense in the spa e of bounded fun tions

B(Γ

d

(V ))

, we prove the weak onvergen e of measures

µ

β

→ µ

∞

onthespa e

B(Γ

d

(V ))

.

2

4 A ontinuous-time equilibrium dynami s

We onsideranoperatorin thespa e

C(Γ

d

(V ))

of ontinuousbounded fun tionson

Γ

d

(V )

oftheform:

(L

β

f )(γ) =

X

x∈γ

e

βE(x,γ\x)

(f (γ\x) − f (γ)) + z

Z

V (γ)

(f (γ ∪ y) − f (γ)) dy,

(13) where

V (γ) = V \D(γ),

D(γ) = (∪

x∈γ

B

x

(ǫ)) ∩ V,

where

B

x

(ǫ)

isthedisk with enteratpoint

x ∈ V

andradius

ǫ

,and

E(x, γ\x) = H(γ) − H(γ\x).

Theoperator dened in equation (13) isthe generatorofabirth-and-death pro essin the domain

V ⊂ R

2

withbirthintensity

b(γ, x)

intheunordered onguration

γ

at

x

anddeath intensity

d(γ\x, x)

fromthe onguration

γ

atposition

x

respe tivelygivenby:

b(γ, x) dx = z dx,

d(γ\x, x) = e

βE(x,γ\x)

_.

Underthis hoi e ofthebirth anddeathintensities, thedetailedbalan e onditionholds:

b(γ, x)

d(γ\x, x)

=

p

V

(γ)

p

V

(γ\x)

= z e

−βE(x,γ\x)

,

and onsequently,seeforexample[9℄,the orrespondingbirth-and-deathpro essasso iated withthesto hasti semigroup

T

β

(t) = e

tL

β

istimereversible,and itsequilibrium distribu-tionis theGibbs stationarymeasure

µ

V

β

withdensity(5).

Theorem2. 1)Theoperator

L

β

isaboundedoperatorinthespa eofboundedfun tions

B(Γ

d

(V ))

andin

L

2

(Γ

d

(V ), µ

β

)

,moreover

L

β

isaself-adjoint operator in

L

2

(Γ

d

(V ), µ

β

)

. 2) The family of operators

T

β

(t) = e

tL

β

, t ≥ 0

forms a self-adjoint Markov semigroup in

L

2

(Γ

d

(V ), µ

β

)

,i.e.

e

tL

β

_{1 = 1,}

and

e

tL

β

_{F ≥ 0}

for anynon-negative

F ∈ L

2

(Γ

d

(V ), µ

β

).

(14) 3)The semigroup

T

β

(t)

isa ontra tion semigroupin

B(Γ

d

(V ))

.

4)Thesemigroup

T

β

(t)

meetsthe onditionofimprovingpositivity,i.e. foranynon-negative fun tion

F ≥ 0

weget

T

β

(t)F > 0

Fortheproof,seeAppendix A.

Corollary. Theimprovingpositivitypropertyimpliesthatthere existsauniquexed ve toroftheoperators

T

β

(t) = e

tL

β

in

L

2

(Γ

d

(V ), µ

β

)

(whi hisequalto 1),see[10℄. The onvergen eto thestationarymeasure

µ

β

isguaranteedbythegeneralresultgiven byC. Preston in [11℄. We onsider afamily

B(λ)

of measures

ν

on the spa e

Γ

d

(V )

with

abounded density

p

˜

ν

(γ)

withrespe t to Lebesgue-Poisson measure

λ

. This, in parti ular, impliesthatadensity

p

ν

(γ)

ofthemeasure

ν ∈ B(λ)

w.r.t. aGibbsmeasure

µ

β

(forany

β

):

p

ν

(γ) =

dν

dµ

β

(γ)

isalsobounded,and onsequently,

p

ν

(γ) ∈ L

2

(Γ

d

(V ), µ

β

)

. Thenwe andenetheevolution

ν

t

≡ T (t)ν

of themeasure

ν ∈ B(λ)

asfollows:

hν

t

, F i = (T

β

(t)p

ν

, F )

µ

β

.

Noti ethat property2)ofTheorem2impliesthat

T

β

(t)p

ν

isagainadensityw.r.t. aGibbs measure,i.e.

T

β

(t)p

ν

≥ 0,

hT

β

(t)p

ν

i

µ

β

= hp

ν

i

µ

β

= 1.

Theorem3. Let

ν ∈ B(λ)

. Then for any

F ∈ L

2

(Γ

d

(V ), µ

β

)

weget

hT

β

(t)ν, F i ≡ hν

t

, F i = (T

β

(t)p

ν

, F )

µ

β

→ hF i

µ

β

.

(15) Theproofoftheorem3followsfrom thegeneraltheoremsbyC.Preston[11℄.

5 Approximation pro ess

Inthis se tion, we dene a dis rete time approximationof the proposed ontinuous birth anddeathpro ess.

We onsiderMarkov hains

T

β,δ

(n), n = 0, 1, 2, . . .

onthesamespa e

Γ

d

(V )

. Thepro ess

T

β,δ

(n)

an be des ribed as follows: a onguration

γ

is transformed to a onguration

γ

′

=

γ

1

∪ γ

2

, where

γ

1

⊆ γ

, and

γ

2

is a onguration of enters of dis s su h that

γ

1

∩ γ

2

= ∅

andisdistributedw.r.t. thePoissonlawwithintensity

z

.

Thistransformationembedabirthpartgivenby

γ

2

andadeathpartgivenby

γ\γ

1

. Thetransitionprobabilityforthedeathofaparti leat

x

(i.e. adis with the enterat

x

)fromthe onguration

γ

isgivenby:

p

x,δ

=











e

βE(x,γ\x)

_δ

1 + e

βE(x,γ\x)

_δ

=

a

x

δ

1+a

x

δ

,

if

γ → γ\x,

1

1+a

x

δ

,

if

γ → γ

(x

survives

).

(16) with

a

x

= a

x

(γ) = e

βE(x,γ\x)

. Moreover,alltheparti lesarekilledindependently,andboth ongurations

γ

1

and

γ

2

areindependent.

Thetransitionsasso iatedwiththebirthofanewparti leinasmalldomain

∆y ⊂ V (γ)

havethefollowingprobabilitydistribution:

q

y,δ

=







z∆yδ,

if

γ → γ ∪ y,

1 − z∆yδ,

if

γ → γ

(

nobirth in

∆y).

Finally,thetransitionoperator

P

β,δ

forthepro ess

T

β,δ

(n) = P

β,δ

n

hasthefollowingform:

(P

β,δ

f ) (γ) =

X

γ

1

⊆γ

Y

x∈γ

1

1

1 + a

x

δ

Y

x∈γ\γ

1

a

x

δ

1 + a

x

δ

(18)

Ξ

−1

_δ

(γ

1

)

∞

X

k=0

Z

V

k

(γ

1

)

(zδ)

k

k!

f (γ

1

∪ y

1

∪ . . . ∪ y

k

) dy

1

. . . dy

k

,

where

Ξ

δ

(γ) = Ξ

δ

(V (γ), z, δ)

isthenormalizingfa torforthe onditionalLebesgue-Poisson measure under a given ongurationof dis s

γ

1

. We provebelow that theapproximation pro ess

T

β,δ

(t) ≡ T

β,δ



t

δ



onverges to the ontinuous time pro ess

T

β

(t)

uniformly on boundedintervals

[0, ¯

t]

asthedis retizationstep

δ

tendsto0.

Letusdenote

L

= B(Γ

d

(V ))

aBana hspa eofboundedfun tions on

Γ

V

withanorm

kF k =

sup

γ∈Γ

d

(V )

|F (γ)|.

Theorem4. Forea h

F ∈ L

kT

β,δ

(t)F − T

β

(t)F k

L

= sup

γ

|(T

β,δ

(t)F )(γ) − (T

β

(t)F )(γ)| → 0,

(19) as

δ → 0

for all

t ≥ 0

uniformlyon boundedintervals of time.

SeeAppendixBfortheproof.

Corollary. Theresultof theorem4impliesthatforany

F, G ∈ B(Γ

d

(V ))

weget

(G, T

β,δ

(t)F )

µ

β

→ (G, T

β

(t)F )

µ

β

as

δ → 0.

(20)

Wedenoteby

S

β,δ

(n)

anadjointto

T

β,δ

(n)

semigroupa tingonmeasures,su hthatfor any

ν ∈ B(λ)

:

hS

β,δ

(n)ν, F i = (p

ν

, T

β,δ

(n)F )

µ

β

.

Wenowformulatethemainresultabout onvergen e.

Main theorem. Let

F ∈ B(Γ

d

(V ))

and an initial measure

ν ∈ B(λ)

. Then under relation

δ e

βb

< const

(21)

with

b = sup

γ∈Γ

d

(V )

sup

x∈γ

E(x, γ\x)

we have

lim

β→∞, t→∞, δ→0

hF i

S

β,δ

([

t

δ

])ν

= hF i

µ

∞

,

wheremeasure

µ

∞

isdenedin Theorem1,and

hF i

S

β,δ

([

t

δ

])ν

= hS

β,δ

([

t

δ

])ν, F i.

Proof. We anwriteas follows

hF i

S

β,δ

([

δ

t

])ν

− hF i

µ

∞

= (p

ν

, T

β,δ

([

t

δ

])F )

µ

β

− (p

ν

, T

β

(t)F )

µ

β

+

+(T

β

(t)p

ν

, F )

µ

β

− hF i

µ

β

+ hF i

µ

β

− hF i

µ

∞

.

Then,usingtheresultsoftheorem 1andthelimitrelations(7),(15)and (20)weget(22). Inaddition,relation(21)followsfrom theapproximationte hnique(seeappendixB, equa-tions(43)and(50)).

Remark. Relation(22)determinesthelimitoverthreequantities:

β → ∞, t → ∞, δ →

0

. Intheapproximationte hniqueweused therelation(21)between

δ

and

β

:

δ = φ(β) e

−βb

with

φ(β) = O(1)

as

β → ∞.

Unfortunately we ould not nd the relation between

t

and

β

. If we have the relation

t = ψ(β)

in anexpli itform,then(22) anberewrittenasalimitwhen

t → ∞

under two relations

β(t) = ψ

−1

_(t)

and

δ(t) = φ(β(t))e

−β(t)b

.

6 Appli ation to obje t dete tion from numeri al images 6.1 Model

Let onsider anumeri alimageonthelatti e

I ⊂ Z

2

,denedasfollows:

Y : I

→ Λ ⊂ N

s

7→ y

s

(23)

Ea h

y

s

referstothegreylevelatpixel

s

,onthelatti e

I = {1, · · · , N L}×{1, · · · , N C}

,

N L

(resp.

N C

)beingthenumberof lines(resp. olumns)oftheanalyzedimage. We onsider ongurationsof entersofdis s

γ = {x

i

} ∈ Γ

d

(V )

,where

V = [1/2, N L + 1/2] × [1/2, N C +

1/2]

. Ea h diskin thenal ongurationrepresentsanobje tin theimage. Thehard ore distan e

ǫ

isnaturallytakento beequalto thedata resolution:

ǫ = 1

pixel. Todenethe energy

H

,werst onsidersomepriorknowledge. Wewanttominimizetheoverlapbetween obje ts. However,toobtainamoreexiblemodel w.r.t. thedataandthekindof obje ts, wedonotforbidbutonlypenalizeoverlappingobje ts. Wedeneapairwise intera tionas follows:

∀{x

i

, x

j

} ∈ γ × γ, H

2

(x

i

, x

j

) = max



0, 1 −

||x

j

− x

i

||

2r



(24) where

||.||

istheEu lidean normand

r

istheradiusoftheunderlying dis .

Arstordertermisthenaddedforea hobje ttotthedis ongurationontothedata. We onsider that there is an obje t, modeled by adis entered at pixel

s

, in the image,

if the grey levelvalues of the pixels inside the proje tionof the dis onto the latti e are statisti allydierentfromthose ofthepixels intheneighborhood ofthedis . Toquantify thisdieren ewe omputetheBhatta haryadistan ebetweentheasso iateddistributions. Denote

D

1

(s)

, theproje tionof the dis with radius

r

entered at

s

onto thelatti e, and

D

2

(s)

thesurrounding rown:

D

1

(s) = {t ∈ I : ||t − s|| ≤ r}

and

D

2

(s) = {t ∈ I : ||t − s|| ≤ r + 1}\D

1

(s).

(25) We onsiderthemeanandthevarian eofthedataofthese twosubsets:

µ

1

(s) =

P

t∈D

1

(s)

y

t

P

t∈D

1

(s)

1

and

µ

2

(s) =

P

t∈D

2

(s)

y

t

P

t∈D

2

(s)

1

σ

1

2

(s) =

P

t∈D

1

(s)

y

2

t

P

t∈D

1

(s)

1

− µ

1

(s)

2

and

σ

2

2

(s) =

P

t∈D

2

(s)

y

2

t

P

t∈D

2

(s)

1

− µ

2

(s)

2

(26)

Assuming Gaussian distributions, the Bhatta haryadistan e betweenthe distributions in

D

1

(s)

andin

D

2

(s)

isthengivenby:

B(s) =

1

4

(µ

1

(s) − µ

2

(s))

2

q

σ

2

1

(s) + σ

2

2

(s) −

1

2

log

2σ

1

(s)σ

2

(s)

σ

2

1

(s) + σ

2

2

(s)

.

(27)

Fromthisdistan ebetweenthetwodistributions,arstorderenergytermis built:

∀x

i

∈ γ, H

1

(x

i

) =









1 −

B(i)

_T



if

B(i) < T



exp −

B(i)−T

_3B(i)

− 1



if

B(i) ≥ T

(28)

where

i

is the losestpoint to

x

i

onthe latti e, and

T

is a thresholdparameter. Finally, underthehard ore onstraint,theglobalenergyiswritten asfollows:

H(γ) = α

X

x

i

∈γ

H

1

(x

i

) +

X

{x

i

,x

j

}∈γ×γ,i6=j

H

2

(x

i

, x

j

)

(29)

where

α

isaweightingparameterbetweenthedatatermandtheprior. 6.2 Algorithm

Thealgorithm simulatingthepro essisdened asfollows:

ˆ Computation of the rst order term: Forea hsite

s ∈ I

ompute

H

1

(s)

from thedata

ˆ Computation of the birth map: To speed up the pro ess, we onsider a non homogeneousbirthratetofavorbirthwheretherstordertermislow(i.e. wherethe datatendto deneanobje t):

∀s ∈ I, b(s) = 1 + 9

max

t∈I

H

1

(t) − H

1

(s)

max

t∈I

H

1

(t) − min

t∈I

H

1

(t)

.

Thenormalizedbirth rateisthen givenby:

∀s ∈ I, B(s) =

P

zb(s)

t∈I

b(s)

(31) Thisnonhomogeneousbirthratereferstoanonhomogeneousreferen ePoisson mea-sure. Ithasnoimpa tonthe onvergen etotheglobalminimaoftheenergyfun tion but dohave an impa t onthe speedof onvergen e in pra ti e by favoring birth in relevantlo ations.

ˆ Main program: initialize the inverse temperature parameter

β = β

0

and the dis- retizationstep

δ = δ

0

andalternatebirth anddeathsteps

 Birth step: forea h

s ∈ S

, if

x

s

= 0

(nopointin

s

)add apointin

s

(

x

s

= 1

) with probability

δB(s)

(note that the hard ore onstraint with

ǫ = 1

pixel is satised).

 Death step: onsider the ongurationof points

x = {s ∈ I : x

s

= 1}

andsort it from the highest to the lowest value of

H

1

(s)

. For ea h point taken in this order, omputethedeathrateasfollows:

d

x

(s) =

δa

x

(s)

1 + δa

x

(s)

,

(32) where:

a

x

(s) = exp −β (H(x/{x

s

}) − H(x)) .

(33) andkill

s

(

x

s

= 0

)withprobability

d(s)

.

 Convergen etest: ifthepro esshasnot onverged,de reasethetemperature andthedis retization stepbyagivenfa torandgoba ktothebirthstep. The onvergen e is obtained when all the obje tsadded during the birth step, and onlytheseones,havebeenkilled duringthedeathstep.

6.3 Results

The rst appli ation we address on erns tree rown extra tion from aerial images. We onsider 50 m resolution images of poplars. Some examples of the obtained results are givenongures1and2. Theresultsaresatisfa tory. One anremarkafewfalsealarmson gure1,ontheborderontheplantation,duetoshadowsandafewmisdete tionongure2 onsmalltreesforwhi hthe hosenradius(

3

pixels)istoobig.

These ondappli ation on ernsthe ountingofamingopopulation. An extra tofthe obtainedresultisgivenongure3fortheinitialimageandongure4forthedete tedbirds. Almostallthebirdshavebeen orre tlydete ted. Thefullimage ontains

6128×3920

pixels andhasbeenanalyzedin tenminutesonabi-pro essor2GHzPC. Thisrepresentsamain advantage with respe t to more standard optimization te hniques based on a RJMCMC sampler [6, 12℄. Indeed, thespeedof onvergen e and the omputationale ien y of the proposedalgorithm allowusto dealwithwithhugesets ofdatainareasonabletime.

7 Con lusion

Inthis report, we haveproposed anew approa h for dete tingobje tsin animage. This approa h is basedon a birth and deathpro ess. We haveproven the onvergen e of the ontinuous pro ess. We then have des ribed adis retization s heme and proven its on-vergen e tothe ontinuouspro ess. Fromthisgeneralframework,wehaveproposedadis modelwhi hpermitsthedete tionofobje tsinagivenimage. Twoappli ations, on erning treeandbirddete tion,haveshownthereleven eoftheproposedapproa h. Thetwomain advantagesofthiste hniqueareitsgeneralityandits omputationale ien y.

Next steps will on ern the generalization of the model to a broader lass of obje ts. Taking intoa ountotherkindsofobje tssu hasellipsesorre tanglesis straightforward. However, it will be interesting to embed some randomness in the denition of obje ts. Dealingwithrandomradiusormoregenerallyrandommarksasso iatedwiththepointsin the ongurationwillin reasetheappli ationdomainofthispromisingapproa h. Tota kle this new generation of models, we are urrently working of new dynami s for addressing geometri hangesinthe ongurationsu hasobje tdilation,translation,rotationorobje t splittingandmerging.

Appendix A. Proof of theorem 2.

1) The boundness of

L

β

is obvious, and then we an dene the semigroup

T

β

(t)

as the exponentof the operator

L

β

using the usual espansionfor the exponent. Self-adjointness followsfromthedetailedbalan e onditionandtheboundness oftheoperator

L

β

.

2)Sin e

L

β

1 = 0

weget

e

tL

β

_{1 = 1}

. These ond onditionin(14)followsfromtheanalogous propertyoftheoperators

P

n

β

andthe onvergen e oftheapproximationpro essasso iated withtransitionoperator

P

β

(seeappendix B).

3)Usingrelations

T

β

(t) = e

tL

β

= lim

n→∞



E +

t

n

L

β



n

(34) and



E +

t

n

L

β



f (γ) =

1 −

t

n

X

x∈γ

e

βE(x,γ\x)

+ zV (γ)

!!

f (γ) +

t

n

X

x∈γ

e

βE(x,γ\x)

_{f (γ\x) +}

t

n

z

Z

V (γ)

f (γ ∪ y) dy

(35)

weobtainthat, forany

t > 0

and forlargeenough

n

, all oe ientsin the de omposition (35)arepositive,andmoreover,











E +

t

n

L

β



f (γ)









≤











E +

t

n

L

β



|f |(γ)









≤ sup

γ

|f (γ)|.

Thus,

sup

γ











E +

t

n

L

β



f (γ)









≤ sup

γ

|f (γ)|,

(36) andapplyinginequality(36)

n

timeswestillkeepthesamebound:

sup

γ











E +

t

n

L

β



n

f (γ)









≤ sup

γ

|f (γ)|

foralllargeenough

n ∈ N.

Consequently,

T

β

(t)

isa ontra tionsemigroupin

B(Γ

d

(V ))

,i.e.

sup

γ

|(T

β

(t) f )(γ)| ≤ sup

γ

|f (γ)|.

4)Using(35)wehave:



E +

t

n

L

β



f = B

0

f +

t

n

B

+

f +

t

n

B

−

f,

where

(B

0

f )(γ) =

1 −

t

n

X

x∈γ

e

βE(x,γ\x)

+ zV (γ)

!!

f (γ) >



1 −

t

n

R



f (γ),

(B

−

f )(γ) =

X

x∈γ

e

βE(x,γ\x)

f (γ\x),

(B

+

f )(γ) = z

Z

V (γ)

f (γ ∪ x)dx,

and

R = max

γ

X

x∈γ

e

βE(x,γ\x)

+ zV (γ)

!

< ∞.

For any given

t > 0

, if

n

is large enough, then

1 −

tR

n

> 0

. The following de omposition holds:



E +

t

n

L

β



n

f =

X

(σ1,...,σn)

σi=0,+,−

B

σ

1

B

σ

2

. . . B

σ

n

 t

n



n

+

_{ t}

n



n

−

f,

(37)

where

n

±

isthenumberof

” + ”

or orrespondingly

” − ”

in thesequen e

(σ

1

, . . . , σ

n

)

,and the sum is taken over all sequen es

(σ

1

, . . . , σ

n

)

with

σ

i

= 0, +, −

. Ea h term in (37) is non-negativeif

f

isanon-negativefun tionandif

n

islargeenough.

Letus onsidertwo ases. If

f (∅) > 0

,thenweshowthat forany

γ 6= ∅

thesum







X

(σ1,...,σn)

n−=k

B

σ

1

B

σ

2

. . . B

σ

n

 t

n



k

f







(γ) ≡ (I

(k)

n

f )(γ) > c

k

f (∅),

(38)

where

|γ| = k

anda onstant

c

k

> 0

doesnotdependon

n

. Thesumin(38)istakenoverall sequen es

(σ

1

, . . . , σ

n

)

freefrompluseswith

n

−

= k

. Indeed,wehaveforalllargeenough

n

(I

n

(k)

f )(γ) >



1 −

t

n

R



n−k

B

−

k

(γ, ∅) C

n

k

 t

n



k

f (∅),

withthe orrespondingmatrixelementoftheoperator

B

k

−

B

₋

k

(γ, ∅) = k! e

βH(γ)

.

Sin eforanyxed

t, R, k



1 −

tR

n



n−k

C

n

k

 t

n



k

→

t

k

k!

e

−tR

as

n → ∞,

wehaveforlargeenough

n



1 −

tR

n



n−k

C

k

n

 t

n



k

>

t

k

2 k!

e

−tR

_.

Consequently,forany

γ

with

|γ| = k

(I

n

(k)

f )(γ) >

1

2

t

k

_e

βH(γ)−tR

_{f (∅) ≡ c}

andinthelimit(34)as

n → ∞

relations(37)-(39)implythat

e

tL

β

_f

(γ) > c

k

f (∅) > 0.

Assume now that

f (∅) = 0

and

f (γ) > 0

on a set

Q ⊂ Γ

d

(V, k)

of a positive measure

λ(Q) > 0

. We onsiderin thesum(37)thefollowingterm

X

(σ1,...,σn)

n+=k

B

σ

1

B

σ

2

. . . B

σ

n

 t

n



k

≡ J

(k)

n

,

wherethesumistakenoverallsequen es

(σ

1

, . . . , σ

n

)

freefromminuseswith

n

+

= k

. Then asabovewehaveforalllargeenough

n

(J

n

(k)

f )(∅) >



1 −

tR

n



n−k

C

n

k

 t

n



k

Z

Q

(B

+

k

)(∅, γ)f (γ)dλ >

t

k

2 k!

e

−tR

Z

Q

(B

+

k

)(∅, γ)f (γ)dλ.

Taking thelimit(34)as

n → ∞

wehave

e

tL

β

_f

(∅) >

t

k

2 k!

e

−tR

Z

Q

(B

+

k

)(∅, γ)f (γ)dλ > 0.

Thususingresultsoftheprevious asewegetthatforany

γ

e

2tL

β

_f

(γ) > 0.

Theorem2isproved ompletely.

Appendix B. Convergen e of the approximation pro esses. Proof of theorem 4.

Toprovethe onvergen eofthe orrespondingsemigroup

T

β,δ

(t) = P

[

t

δ

]

β,δ

→ T

β

(t),

δ → 0

uniformlyonboundedintervalsoftime

t

weuseherethefollowingapproximationtheorem: Theorem [13℄. For

n = 1, 2, . . .

let

T

n

be a linear ontra tion on a Bana h spa e

L

, andlet

δ

n

bepositive numbers. We set:

L

n

=

1

δ

n

(T

n

Assumethat

lim

n→∞

δ

n

= 0

.

Let

{T (t)}

be astrongly ontinuous ontra tion semigroup on the Bana h spa e

L

with generator

L

,andlet

C

bea orefor

L

. Then thefollowing propositions areequivalent: a)for ea h

f ∈ L

kT

n

(t)f − T (t)f k

L

→ 0,

as

δ

n

→ 0

for all

t ≥ 0

uniformlyon boundedintervals;

b)for ea h

f ∈ C

kL

n

f − Lf k

L

→ 0,

as

δ

n

→ 0.

Wedenote by

L

β,δ

the generatorofthepro ess

T

β,δ

denedby transitionprobabilities (18)(homogeneousintime):

(L

β,δ

f )(γ) =

1

δ

n

((P

β,δ

f ) (γ) − f (γ)) =

(40)

1

δ

n





X

γ

1

⊆γ

Y

x∈γ\γ

1

(a

x

δ

n

)

Y

x∈γ

1

1 + a

x

δ

n

Ξ

−1

_δ

(γ

1

)

∞

X

k=0

Z

V

k

(γ

1

)

(zδ

n

)

k

k!

f (γ

1

∪ y

1

∪ . . . ∪ y

k

) dy

1

. . . dy

k

− f (γ)







=

1

δ

n

Ξ

−1

_δ

(γ)

Y

x∈γ

1

1 + a

x

δ

n

f (γ) − f (γ)

!

+

1

δ

n

Ξ

−1

_δ

(γ\x)

Y

y∈γ

1

1 + a

y

δ

n

X

x∈γ

a

x

δ

n

f (γ\x) +

1

δ

n

Ξ

−1

_δ

(γ) zδ

n

Y

y∈γ

1

1 + a

y

δ

n

Z

V (γ)

f (γ ∪ ˜

y) d˜

y +

1

δ

n

X

˜

γ⊆γ

Ξ

−1

_δ

(γ\˜

γ)

X

k:|˜

γ|+k≥2

(zδ

n

)

k

k!

Y

y∈γ

1

1 + a

y

δ

n

Y

x∈˜

γ

a

x

δ

n

Z

V

k

(γ\˜

γ)

f ((γ\˜

γ) ∪ y

1

∪ . . . ∪ y

k

) dy

1

. . . dy

k

.

We take here asa ore

C = B(Γ

d

(V ))

a whole set of bounded fun tions on

Γ

V

. Let us onsiderthefollowingtheorem,provedinappendixC:

Theorem5. Let usdenote

Then

sup

γ

|∆

δ

f (γ)| → 0

as

δ → 0

(41)

for ea h

f (γ) ∈ B(Γ

V

)

.

Finally, relation (41) immediately implies onvergen e (19) of the semigroups in the uniformnormofthespa e

L

bytheaboveapproximationtheorem. Theorem4isproved.

Appendix C. Proof of theorem 5. Somepreliminary expansions.

Remark. Wehaveforany

x ∈ γ

E(x, γ\x) ≤ b,

(42) sothat

a

x

≡ e

βE(x,γ\x)

≤ e

βb

(43) andlet

a = a(β) =

sup

γ∈Γ

d

(V )

sup

x∈γ

a

x

< ∞.

Lemma1. The normalizing fa tor

Ξ

−1

δ

(γ)

from(18 ) anbewritten as

Ξ

−1

δ

(γ) = 1 − zδ|V (γ)| + O z

2

δ

2

as

δ → 0.

(44) Proof. Sin e

Ξ

δ

(γ) = 1 +

∞

X

m=1

(zδ)

m

m!

Z

V

m

(γ)

dy

1

. . . dy

m

= 1 + zδ|V (γ)| +

∞

X

m=2

(zδ)

m

m!

V

m

with

V

m

< |V (γ)|

m

_{< |V |}

m

_{, m ≥ 2}

,we anwrite

Ξ

δ

(γ) = 1 + zδ|V (γ)| + O(z

2

δ

2

).

(45) Thus,(45)implies(44).

2

UsingtheTaylorexpansions

ln(1 + x) = x −

x

2

2

ξ

′

_,

_ξ

′

_{∈ (4/9, 4)}

as

|x| <

1

2

,

(46) and

e

x

= 1 + x +

x

2

2

e

ξ

_,

_{ξ ∈ (0, x),}

(47)

wehaveforsmallenough

δ

:

1

1 + a

x

δ

= e

− ln(1+a

x

δ)

_{= e}

−a

x

δ+

ξ1

2

a

2

x

δ

2

,

Y

x∈γ

1

1 + a

x

δ

= e

−δ

P

x∈γ

a

x

+

ξ1

2

δ

2

P

x∈γ

a

2

x

.

(48)

Thenusing(44),(48)andrelation

ξ

1

2

δ

2

P

x∈γ

a

2

x

= O(a

2

δ

2

)

wehaveforallsmallenough

δ

:

Ξ

−1

_δ

(γ)

Y

x∈γ

1

1 + a

x

δ

−

1 − δ

X

x∈γ

a

x

− z δ |V (γ)|

!

=

(49)

1 − zδ|V (γ)| + O (zδ)

2

e

−δ

P a

x

+δ

2 ξ1

2

P a

2

x

−



1 − δ

X

a

x

− δz|V (γ)|



= O δ

2

(a

2

+ z

2

)
.

Weassumeherethat

δ a = δ a(β) < const,

(50)

whi hisof oursetrueforanyxed

β

andsmallenough

δ

. Letuswritenowtheexpression for

∆

δ

f (γ)

using(40)and (13):

(∆

δ

f )(γ) =

1

δ

((P

δ

f )(γ) − f (γ)) −

(51)

X

x∈γ

a

x

(γ) (f (γ\x) − f (γ)) − z

Z

V (γ)

(f (γ ∪ y) − f (γ)) dy =

1

δ

Ξ

−1

δ

(γ)

Y

x∈γ

1

1 + a

x

δ

f (γ) − f (γ)

!

+

+

X

x∈γ

a

x

(γ) f (γ) + z |V (γ)| f (γ) +

1

δ

X

x∈γ

a

x

(γ)δ Ξ

−1

δ

(γ\x)

Y

y∈γ

1

1 + a

y

δ

f (γ\x) −

X

x∈γ

a

x

(γ)f (γ\x) +

1

δ

Ξ

−1

δ

(γ) z δ

Y

y∈γ

1

1 + a

y

δ

Z

V (γ)

f (γ ∪ y) dy − z

Z

V (γ)

f (γ ∪ y) dy +

1

δ

X

˜

γ⊆γ

Ξ

−1

_δ

(γ\˜

γ)

X

k:|˜

γ|+k≥2

(zδ)

k

k!

Y

x∈˜

γ

a

x

δ

Y

y∈γ

1

1 + a

y

δ

Z

V

k

(γ\˜

γ)

f (γ\˜

γ ∪ y

1

∪ . . . ∪ y

k

) dy

1

. . . dy

k

.

Estimationof the oe ientsin (51).

Letusestimatethetermswith

f (γ), f (γ\x), f (γ ∪ y)

in (51)separately. Using(49)andboundedness of

f (γ)

wehave:











1

δ

Ξ

−1

δ

(γ)

Y

x∈γ

1

1 + a

x

δ

f (γ) − f (γ)

!

+

X

x∈γ

a

x

(γ) + z|V (γ)|

!

f (γ)











= O(δ(a

2

+ z

2

)), δ → 0.

(52)

Analogously,we an estimatethe oe ientsbefore

f (γ\x)

and

f (γ ∪ y)

:











X

x∈γ

a

x

Ξ

−1

δ

(γ\x)

Y

y∈γ

1

1 + a

y

δ

− 1

!

f (γ\x)











=











X

x∈γ

a

x



1 − zδ|V (γ\x)| + O(z

2

δ

2

)

e

−δ

P a

x

+

ξ1

2

δ

2

_{P a}

2

x

− 1



f (γ\x)











≤

a|γ| δ(a|γ| + z|V |) + δ

2

(A

2

|γ| + A

3

z

2

|V |

2

)

K

f

= O(δa(z + a)).

(53) And

z















Z

V (γ)

Ξ

−1

δ

(γ)

Y

x∈γ

1

1 + a

x

δ

− 1

!

f (γ ∪ y) dy















≤

z|V | δ(a|γ| + z|V |) + δ

2

(A

2

|γ| + A

3

z

2

|V |

2

)

K

f

= O(δz(z + a)).

(54) Letusestimatenowthelast termin (51). Usingthat

Ξ

−1

_δ

(γ) ≤ 1,

Y

y∈γ

1

1 + a

y

δ

≤ 1,

and

sup

γ

|f (γ)| < K

f

wehave

1

δ

X

˜

γ⊆γ

Ξ

−1

δ

(γ\˜

γ)

X

k:|˜

γ|+k≥2

(zδ)

k

k!

Y

x∈˜

γ

a

x

δ

Y

y∈γ

1

1 + a

y

δ

Z

V

k

(γ\˜

γ)

f (γ\˜

γ ∪ y

1

∪ . . . ∪ y

k

) dy

1

. . . dy

k

≤

K

f

δ

X

˜

γ⊆γ

(aδ)

|˜

γ|

X

k≥0:|˜

γ|+k≥2

(z|V |δ)

k

k!

≤

K

f

δ

a

N

X

m=0,...,|γ|

k≥0:m+k≥2

C

_|γ|

m

δ

m

(z|V |δ)

k

k!

=

K

f

δ

a

N



_e

z|V |δ

_{(1 + δ)}

|γ|

_{− 1 − |γ|δ − z|V |δ}



_{= O(δ + δz + δz}

2

_).

(55) Hereweusedthat

(1 + δ)

|γ|

= e

|γ| ln(1+δ)

= e

δ|γ|

+ O(δ

2

)

forsmall

δ > 0.

Finally,from(51),(52),(53),(54),(55)itfollowsthat forany

f (γ) ∈ B(Γ

d

(V ))

sup

γ

|∆

δ

f (γ)| → 0

as

δ → 0,

and onsequently,

kL

β,δ

f − L

β

f k

B(Γ

d

(V ))

→ 0

as

δ → 0.

Theorem5is ompletelyproved.

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