Multiphase Evolution and Variational Image Classiﬁcationfrey/papers/applications... ·...

ISS

N 0

249-

6399

appor t de r echerche

THÈME 3

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Multiphase Evolution and Variational ImageClassification

Christophe Samson — Laure Blanc-Féraud — Gilles Aubert — Josiane Zerubia

N° 3662

Avril 1999

Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles, B.P. 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : 04 92 38 77 77 - International : +33 4 92 38 77 77 — Fax : 04 92 38 77 65 - International : +33 4 92 38 77 65

Multiphase Evolution and Variational Image

Classication

Christophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia

Thème 3 Interaction homme-machine,images, données, connaissances

Projet Ariana

Rapport de recherche n° 3662 Avril 1999 42 pages

Abstract: This report presents a supervised classication model based on a vari-ational approach. This model is devoted to nd an optimal partition compound ofhomogeneous classes with regular interfaces. We represent the regions of the imagedened by the classes and their interfaces by level set functions, and we dene afunctional whose minimum is an optimal partition. The coupled Partial Dieren-tial Equations (PDE) related to the minimization of the functional are consideredthrough a dynamical scheme. Given an initial interface set (zero level set), thedierent terms of the PDE's are governing the motion of interfaces such that, atconvergence, we get an optimal partition as dened above. Each interface is guidedby internal forces (regularity of the interface), and external ones (data term, no vac-uum, no regions overlapping). We conducted several experiments on both synthetican real images.

Key-words: Variational model, classication, labelling, level set formulation, ac-tive regions, active contours, multiphase, satellite images.

Laboratoire J.A. dieudonné UMR 6621 CNRS, Université de Nice-Sophia Antipolis, 06108 Nice

Cedex 2, FRANCE

Evolution de Phases Multiples et Classication d'Images

par modèle variationnel

Résumé : Dans ce rapport, nous présentons un modèle de classication super-visée basé sur une approche variationnelle. Nous souhaitons obtenir une partitionoptimale de l'image constituée de classes homogènes séparées par des interfaces régu-lières. Pour cela, nous représentons les régions dénies par les classes ainsi que leursinterfaces par des fonctions d'ensembles de niveaux. Nous dénissons une fonction-nelle sur ces ensembles de niveaux dont le minimum est une partition optimale. LesEquations aux Dérivées Partielles (EDP) relatives à la minimisation de la fonction-nelle sont couplées et plongées dans une schéma dynamique. En xant un ensemblede niveaux initial, les diérents termes des EDP guident l'évolution des interfaces(ensembles de niveaux zéro) vers les frontières de la partition optimale, par le biaisde forces internes (régularité de l'interface) et externes (attache aux données et pasde chevauchement des régions ni de vide dans la partition). Nous avons eectué denombreux tests sur des images synthétiques ainsi que sur des images réelles.

Mots-clés : Modèle variationnel, classication d'images, formulation par ensemblesde niveaux, régions actives, contours actifs, phases multiples, images satellitaires.

Multiphase Evolution and Variational Image Classication 3

Contents

1 Introduction 4

2 Image classication as a partitioning problem 5

3 Multiphase model : image classication in terms of level set 73.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Multiphase functional . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Remark about length minimization . . . . . . . . . . . . . . . . . . . 12

4 Multiphase evolution scheme 134.1 System of coupled PDE's . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 A way to include a restoration process 15

6 Experimental results 166.1 Synthetic images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6.1.1 Non noisy data . . . . . . . . . . . . . . . . . . . . . . . . . . 176.1.2 Noisy data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.2 Real images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Conclusion 37

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4 Christophe Samson, Laure Blanc-Féraud, Gilles Aubert, Josiane Zerubia

1 Introduction

Image classication, which consists of assigning a label to each pixel of an observedimage, is one of the basic problems in image processing. This concerns many ap-plications as, for instance, land use management in teledetection. The classicationproblem is closely related to the segmentation one, in the sense that we want to get apartition compound of homogeneous regions. Nevertheless, within the classicationprocedure, each partition represents a class, i.e. a set of pixels with the same label.In the following, the feature criterion we are interested in is the spatial distributionof intensity (or grey level). This work takes place in the general framework of super-vised classication which means that the number and the parameters of the classesare known. The proposed method could be extended to other discriminant featuresthan grey-level such as texture for instance. The unsupervised case, including a pa-rameter estimation capability, will be studied in the future.

Many classication models have been developed with structural notions as regiongrowing methods for example [20], or by stochastic approach as in [2, 3, 7, 8, 13,15, 17], but rarely in the eld of variational approach. In [22, 23] we proposed asupervised variational classication model based on Cahn-Hilliard models, such thatthe solution we get is compound of homogeneous regions separated by regularizedboundaries. The classes are considered as phases separated by interfaces bound-aries. The model was developed through considerations of regularity on the phasesby dening a set of functionals whose expected minimum at convergence is an imagewith expected properties of regularity.Herein, the approach is dierent, mainly because the proposed model is based onactive contours, and the functional of interest is dened over the regions with asso-ciated interfaces through a level set model. The resulting dynamical Partial Dier-ential Equations (PDE's), governing the evolution of the set of interfaces, consist ofa moving front converging to a regularized partition. This model is inspired by thework of Zhao et al. about multiphase evolution [25], and takes place in the generalframework of active contours [4, 5, 12, 16] for region segmentation [19, 21, 26]. Weuse a level set formulation [18] which is convenient to write functional depending onregions and contours, and allows a change of topology of the evolving fronts. Eachactive interface is coupled to the other ones through a term which penalizes overlap-ping regions (i.e. pixels with two labels) and the formation of vacuum (i.e. pixelswithout any label). The evolution of each interface is guided by forces that imposethe following constraints : the interface exhibits a minimal perimeter (internal force)and it encloses one and only one homogeneous class (external force).

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First, we state the problem of classication as a partitioning problem. We clearlyset the framework and dene the properties we expect on the classication. Sec-ond, we expose the classication statement through a level set formulation. TheEuler-Lagrange derivative of the proposed functional leads to a dynamical schemewe propose to implement. We then propose to introduce a restoration process inthe model which permits to deal with degradated data. We nally present someexperimental results on both synthetic and real images, and also on noisy data.

2 Image classication as a partitioning problem

This section is devoted to present the properties we want the classication model tosatisfy. In the following, we consider a classication problem in which a partitionof the observed data u0, with respect to the predened classes, is searched. Thispartition is compound of homogeneous regions, say the classes, separated by regular-ized interfaces. Herein, we suppose that the classes have a Gaussian distribution ofintensity, therefore a class is characterized by its mean i and its standard deviationi. The number K of classes and the parameters (i;i)i=1:::K are supposed to begiven from a previous estimation. We choose to assign the label value i to eachelement of the ith class. All indexes i or j are going from 1 to K.

Let be an open domain subset of R2 with smooth boundary, and let u0 : ! R

represent the observed data function. Let i be the region dened as

i = fx 2 =x belongs to the ith classg: (1)

A partitioning of consists of nding a set figi=1:::K such that (see Fig. 1)

=

K[i=1

i and i

\i6=j

j = ?: (2)

We note i = @i \ the boundary of region i (excepted the common points with@), and let the interface between i and j be

ij = ji = i \ j \ ; 8i 6= j: (3)

We have

i =[j 6=i

ij : (4)

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1ΩΩ 3

Γ

Γ

Γ13

Ω 4

Γ12 14

24

23ΓΓ13

Ω 2Γ121Ω

Figure 1: A partition of .

Let remark that in (3) and (4) we eventually have ij = ?. We note jij the one-dimensional Hausdor measure of i verifying

jij =Xj 6=i

jij j and j?j = 0: (5)

The classication model we consider for an image uo dened over , is a set figidened by (1) and satisfying :

Condition a : figi is a partition of :

=[i

i and i

\i6=j

j = ?:

Condition b : The partition figi is a classication of the observed data u0 and

takes into account the Gaussian distribution property of the classes (data term) :

minimizeXi

Zi

u0 ii

2with respect to i:

Condition c : The partition is regular in the sense that the sum of the length of

interfaces ij is minimum :

minimizeXi;j

ij jijj with respect to ij (ij 2 R are xed):

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The solution of the classication model proposed in the next section has to take intoaccount the three conditions. This is done by associating a functional to the set ofinterfaces such that minimizers will respect conditions a, b and c.

3 Multiphase model : image classication in terms of

level set

The classication model developed further is based on coupled active interfaces, andthe approach we adopt is inspired from Zhao et al. [25]. The evolution of eachinterface is guided by forces constraining the solution to respect conditions a, band c exposed in the previous section. We use a level set formulation to representeach interface and also each region i element of the partition figi.

3.1 Preliminaries

Let i : R+ ! R be a Lipschitz function associated to region i (we assume the

existence of such a i) such that8><>:i(x; t) > 0 if x 2 i

i(x; t) = 0 if x 2 i

i(x; t) < 0 otherwise :

(6)

Thus, the region i is entirely described by the function i (see Fig. 3). In thefollowing, for a sake of clarity, we will sometimes omit spatial parameter x and timeparameter t in i(x; t).Let dene the approximations and H of Dirac and Heaviside distributions (cf.Fig. 2) with 2 R+

(s) =

(12

1 + cos(s

)

if jsj

0 if jsj > (7)

H(s) =

8>><>>:

12

1 + s

+ 1

sin(s

)

if jsj

1 if s >

0 if s <

(8)

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α−α

1

δ α

0.5

1/α

Η α

x

Figure 2: Approximations and H of Dirac and Heaviside distributions.

and we have 8><>:

D0()! as ! 0+

HD

0()! H as ! 0+

where D0() is the space of distributions dened over . From (6),(7) and (8) we

can write (see Fig. 3)

fx 2 = lim!0+

H(i(x; t)) = 1g = i (9)

fx 2 = lim!0+

(i(x; t)) 6= 0g = i: (10)

3.2 Multiphase functional

Let u0 : ! R be the observed data (grey level for instance).

Thanks to the level set i's dened in (6) and by the use of (9) and (10), a partitionfigi respecting conditions a, b and c stated in section 2 can be found through theminimization of a global functional depending on the i's. This functional containsthree terms, each one being related to one of the three conditions. In the following,we express each condition in term of functional minimization. Minimizers of the

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Ω

φ

δα

>2α<

i

i

φ )i( Hα φ i )(

>α<

2 01 -1 -2 -3

Figure 3: Illustration of the signed distance function i (top right handside) associated to

region i (top left handside). Filled regions on both gures on bottom handside represent

non-zero regions for (i) (on the left) and for H(i) (on the right).

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following functionals are supposed to exist.

Functional related to condition a (partition condition) :

Let dene the following functional :

F a

(1; :::;K) =

2

Z

KXi=1

H(i) 12dx with 2 R+ : (11)

The minimization of F a

, as ! 0+, penalizes the formation of vacuum (pixels withno label) and regions overlapping (pixels with more than one label).

Functional related to condition b (data term) :Taking into account the observed data and the Gaussian distribution property of theclasses, we consider :

F b

(1; :::;K) =

KXi=1

ei

ZH(i)

(uo i)2

2idx with ei 2 R;8i: (12)

The family fgi minimizing F b

as ! 0+ leads to a partition figi satisfyingcondition b.

Functional related to condition c (length shortening of interface

set) :

The last functional we want to introduce is related to condition c about the min-imization of the interfaces length. We would like to minimize

1

2

Xi;j

ijjij j with ij being real constants: (13)

The factor 12 expresses the symmetry ij = ji and will be introduced in the weight-

ing parameters ij . We turn the minimization of interfaces length into the minimiza-tion of boundaries length :

KXi=1

ijij with i being real constants: (14)

From (13) and (14) we obtain the constraint ij = i+ j which permits to select theweighting parameters i in the problem of boundaries length minimization to retrieve

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the interfaces length minimization one1. According to Lemma 1 exposed below, theminimization of (14) is operated by minimizing the functional (as ! 0+) :

F c

(1; :::;K) =KXi=1

i

Z(i)jrijdx: (15)

Let recall the Coarea formula :

Theorem 1 (Coarea formula [9]) Let f : R2 ! R be a Lipschitz function andg 2 L1(R2), then : Z

R2

g(x)jrf(x)jdx =

ZR

hZf=

g(x)dsid

Then, from the Coarea formula we get :

Lemma 1 According to the previous denitions, let dene

L(i) =

Z(i(x; t))jri(x; t)jdx;

then we have

lim!0

L(i) =

Zi=0

ds = jij:

Proof : in the Coarea formula, we choose g(x) = (i(x; t)) and f(x) = i(x; t),then we have

L(i) =

ZR

hZi=

(i(x; t))dsid

=

ZR

h()

Zi=

dsid

By setting h() =Ri=

ds we obtain

L(i) =

ZR

()h()d

(7)=

1

2

Z

1 + cos(

)h()d

1P

i;j 6=iij jij j =

Pi ijij

(4)=P

i ij [j 6=i ij j =

Pi i

Pj 6=i

jij j =P

i;j 6=i( i + j)jij j

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If we take = we have

L(i) =1

2

Z 1

1

1 + cos()

h()d

Thus, when ! 0 we obtain

lim!0

L(i) =1

2h(0)

Z 1

1

1 + cos()

d

= h(0) =

Zi=0

ds = jij

Global functional :

The sum F a

+ F b

+ F c

leads to the global functional :

F(1; :::;K) =KXi=1

ei

ZH(i)

(uo i)2

2idx+

KXi=1

i

Z(i)jrijdx

+

2

Z

KXi=1

H(i) 12dx (16)

As ! 0+, the solution set figi minimizing F(1; :::;K), if it exists2 and accord-ing to (6), denes a classication compound of homogeneous classes (the so-calledi phases) separated by regularized interfaces.

3.3 Remark about length minimization

Consider the length functional :

L(t) =

Z 1

0j@C(p; t)

@tjdt (17)

where fC(p; t)gt is a set of closed parametrized (p 2 [0; 1]) curves over such that

C(0; t) = C(1; t) and @C(0;t)@t

= @C(1;t)@t

. Then, L(t) is decreasing most rapidly if

@C(p; t)

@t= ~N (18)

2If they exist, minimizers figi should be found in the space fi : R+ ! R=jri j 2 L1()g

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being the local curvature of C(p; t) and ~N the inward normal. Curve evolutionthrough PDE (18) is known as mean curvature motion (see [14] for instance). Activecontours guided by (18) tends to regular curves in the sense that the length is mini-mized. PDE (18) can be written through a level set formulation [18] which is moreconvenient to manage curves breaking and merging. Assume that d : R+ ! R isa smooth continuous function such that, from the value of d(x; t), we can determineif x is interior, exterior or belongs to C(p; t). Let suppose that : C(p; t) = fx 2=d(x; t) = ag (i.e. the contour is represented by level set a of function d). PDE(18) formulated by the use of level set becomes

@d(x; t)

@t= div(

rd

jrdj)jrdj; (19)

with div( rdjrdj) being the local curvature of level set a. Equation (19) was studied for

instance in [1, 10]. Evolution of level sets of function d (and so evolution of contourC(p; t) through level set a) from (19) is the level set formulation of mean curvaturemotion. The level set formulation allows breaking and merging fronts which is notpossible from formulation (18). Since contour C(p; t) is represented by level set a, weonly need to update PDE (19) in a narrow band around a. In this case, the level setformulation (19) comes from a reformulation of (18) to track the motion of contoursC(p; t). In our case, we directly dene a length functional F c

over contours i's bythe use of level set i's. The associated Euler-Lagrange equations lead to K PDE'sof the form

@i(x; t)

@t= div(

ri

jrij)(i): (20)

Compared to PDE (19), we get from (20) a "natural" narrow band from the Diracoperator whose width depends on the value of (for i's dened as signed distancein (6)). Fig. 4 shows the zero level set evolution of the i's through PDE (20).

4 Multiphase evolution scheme

Herein describe the way we minimize F in (16) with respect to 1; :::;K . Weobtain a dynamical system of K coupled PDE's.

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4.1 System of coupled PDE's

If (1; :::;K) is solution of the minimization of F, then necessarily3

@F

@i= 0; 8i = 1:::K: (21)

With Neumann conditions (@i

@~n(x; t) = 0;8x 2 @), the Euler Lagrange equations

associated to F give (see Appendix for more details about the computation of deriva-tives) the K following coupled PDE's

@F

@i= (i)

hei(u0 i)

2

2i idiv

ri

jrij

+

KXi=1

H(i) 1i

= 0; i = 1:::K

(22)

with div denoting the divergence operator, and divri

jrij

being the (mean) cur-

vature of level set i at point x. We note that the term (i) in (22) delimitsa "natural" band in which the ith PDE is non zero valued (for i's being signeddistance functions) : Bi

= fx 2 =ji(x; t)j g.

4.2 Algorithm

We embed (22) into a dynamical scheme, we get a system of K coupled equations(i = 1:::K) :

t+1i = t

i dt(i)

hei(u0 i)

2

2i idiv

ri

jrij

+

KXi=1

H(i) 1i

;

(23)

where dt is the step in time.Let be a grid with N lines and M columns. The horizontal and vertical steps ofdiscretization are set to 1. We select a small value for . According to the chosenstep of discretization and for i's such that jrij = 1, as it is for signed distancefunctions, we can see from (7) that the band Bi

contains 2 1 (if 2 N) pixels.We do not make decrease to 0, but we directly set to a small value (not too smallin order to have enough pixels in Bi

for the calculus of the dierent terms of (23)),and in our experiments = 3:0. Let remark that we initially set the i's to signed

3if the 's are regular enough

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distance functions which is commonly used for level set schemes. But as for (19),PDE's (20) and (23) do not maintain the the constraint jrij = 1, and we regularlyneed to regularize the level sets i to be sure they remain signed distance functions.This can be done for instance by the use of PDE [24]

@i(x; t)

@t= sign(i)(1 jrij); (24)

sign being the function returning the sign of the argument.Here is the algorithm resulting from (23)

0 - Fix 0i for i = 1:::K

1 - t t+ 12 - For i = 1:::K solve the K coupled PDE's (23)

3 - each n iteration of (23), regularize the i's with PDE (24)

4 - go to 1

Step 0 is the initialization of the i's. This initialization is very important. A badinitialization leads to a bad classication. The 0

i 's can be manually chosen, or canbe automatically computed from an initial pre-segmentation process, or automati-cally selected from computed "seeds" (see next section). The algorithm is stoppedwhen the evolution of the solution between two iterations with respect to t is notsignicant, or when a xed maximum number of iterations is reached.

5 A way to include a restoration process

The observed data u0 can be corrupted by dierent sources of degradation, usuallymodeled by the following linear degradation equation :

u0(x) = Ru(x) + (25)

with u : ! R being the original non noisy data, R is the impulse response of thesystem, and is an additive Gaussian noise. This noisy information can interferewith the classication process and may induce misclassied pixels. So, in additionto the partitioning capability, we want the model to remove degraded pixels withoutdamaging the set of edges of the image which represent important features of data

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u0. Based on edge-preserving restoration [6], we introduce the following functional :

G(u;1; :::;K) =

KXi=1

i

Z(i)jrijdx+

12

Z

KXi=1

H(i) 12dx

+KXi=1

ei

ZH(i)

(u i)2

2idx

+hZ

(Ru uo)

2 + 2

Z'(jruj)

i(26)

Function ' : R ! R+ is a regularizing function with at least the two following

properties : '(t) has a quadratic behavior near t = 0 (smoothing eect), and islinear or sub-linear for t 1 (edge-preserving eect). We can nd some examplesof regularizing functions in [6]. The parameter 2 R+ is weighting the restorationpart of functional G. The minimization of G with respect to u leads to the edge-preserved restoration of u taking into account the informations given by the partitionterm (third term on the right handside of (26) that attracts the values of u towardsthe mean of the classes), and the minimizations of G with respect to the i's givesa partition of u0. The Euler-Lagrange equations associated to the minimization ofG are :

@G

@u=

KXi=1

eiH(i)(u i)

i+

hR(Ru uo) 2div

'0(jruj)

2jrujrui

= 0; (27)

with R being the conjugate of R, and for i = 1:::K we have

@G

@i= (i)

hei(u i)

2

2i idiv

ri

jrij

+ 1

KXi=1

H(i) 1i

= 0: (28)

We alternate resolutions of (27) and (28) until the solution set (u;1; :::;K) doesnot evolve anymore. The resolution of (27) is done thanks to the half-quadratic reg-ularization method [6, 11] based on the introduction of an auxiliary variable relatedto the discontinuity set. System (28) is embeded in a dynamical scheme as for theresolution of (22) with the dynamical system (23).

6 Experimental results

We made some experiments on both synthetic and real images. Some of the noisydata have been corrupted by an additive Gaussian noise. The Gaussian parameters

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of the classes i and i (i = 1:::K) are supposed to be known from a previous esti-mation for instance. For all the results shown hereafter, we x = 3:0. All initiali's are set to signed distance functions.

In the following, ZLS will stand for "zero level set" of functions i's, and Phaseimage represents the map of the K regions surrounded by the ZLS's, whose greylevel values are set to the mean values i.

6.1 Synthetic images

6.1.1 Non noisy data

On Fig. 4 we show the motion of the ZLS's of functions i's through PDE (20),i.e. parameters and ei are set to zero for all i in (23). This evolution reects theinuence of length shortening term F c

dened in (15). We have initialized threecircular and overlapping signed distance functions. Fig. 4 presents the motion ofthe ZLS's : three circles with initially same radius. We remark that the evolutionof circular level sets through (20) leads to results of same type than classical meancurvature motion by level set approach dened in (19) : the ZLS's remain smoothcircles whose radii are decreasing, and they nally shrink after iteration 800 for ourexperiment.

From Fig. 5 to Fig. 20, the initial i's are circular signed distance functions, andwe present both evolution of i ZLS's and what we call "phase evolution", i.e. themotion of regions surrounded by the ZLS's. When not mentioned, the values of iare equal to 1:0.

On Fig. 5 and Fig. 6, we give a result for three classes whose boundaries con-stitute a "T-junction". The initial overlapping and vacuum regions are vanishingthanks to the term F a

and the regularity of the level sets i's is obtained from F c

.We see that the interface between class 2 and class 1 is found later than the onebetween class 3 and class 1, because of the data term : the transition value betweenclass 3 (3 = 3:0) and class 1 (1 = 1:0) is higher than the one between class 2(2 = 2:0) and class 1.We expose on Fig. 7 and Fig. 8 results for three classes of dierent shape and dif-ferent curvature. We choose a small value for the i's in order to retrieve the nonsmooth boundary of the class 3 object on the right handside.Results on Fig. 9 and Fig. 10 reveal the behavior of the model when we start with

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a "bad" initialization. Initial ZLS's for class 2 and class 3 are not judiciously set(see initial ZLS's on Fig. 9). We remark that from iteration 50 to iteration 150, theZLS of 2 is cut in two curves, the left one being constituted with only few pixelsover the left part of class 2. This tiny seed is sucient to detect the boundary of theleft part of class 2. This phenomena shows that, even if the evolution of the ZLS'sdepends on the initialization set f0

i gi, we have a certain tolerance for the choice ofinitial sets.

6.1.2 Noisy data

The data on Fig. 11 and Fig. 12 are a noisy version of data treated on Fig. 5.A Gaussian noise has been added to the orignal data and the signal to noise ratio(SNR) in variances reaches -1.6 dB ( Fig. 11) and -6.4 dB (Fig. 12). The varianceparameters 2i are set to the value of the Gaussian noise. The initialization is thesame than for the non noisy case, but parameters are dierent. Due to noisy pixels,the data term is low valued. To avoid the detection of isolated pixels, the valueof the length shortening term has to be much more important than the data term(this leads to a quite wide vaccum region around the T-junction on Fig. 12, but weprefered to avoid the detection of noisy pixels, which are false alarm, than recoveringnon classied pixels). If we do not choose a high value for , the evolution of the zerolevel sets can prematurely stop. We present on Fig. 13 a comparison between themodel without restoration terms (i.e the minimization of (16)) and with restorationterm (i.e the minimization of (26) for R = Id) for noisy data exposed on (Fig. 12).We remark that the restoration capability of the model expressed through (26) leadsto a classication result containing less vacuum, and less perturbated boundaries.Data on Fig. 14 and Fig. 17 are a noisy version of data treated on Fig. 7. AGaussian noise has been added to the orignal data and the SNR is equal to 3.2 dB.We choose the same initialization as for the non noisy case of Fig. 7. We note thatthe classication contains a lot of unclassied pixels on interfaces set (black pixelson false color image on Fig. 14), and also on homogeneous regions (see the two smallisolated regions on the ZLS's nal result). This kind of initialization, in the case ofnoisy data, is quite sensitive to noisy pixels because each ZLS sweeps over a wideregion of the noisy data. We see on these results that we can not choose a set ofparameters leading to smaller vacuum regions at the risk of being more sensitive toisolated pixels. That is the reason why we performed another initialization proce-dure, which is automatic and less sensitive to noise in the sense that the initial i'sare sweeping over a wideless region of the noisy data.

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For the following results, we use an automatic initialization we will call seed ini-tialization. This method consists of cutting the data image of u0 into N windowsWl; l=1::N of predened size. We compute the average ml of u0 on each window Wl.We then select the index k such that k = argminj(ml j)

2. And we initialize thecorresponding circular signed distance function k on each Wl. Windows are notoverlapping and each of them is supporting one and only one function k, thereforewe avoid overlapping of initial k's. The size of the windows is related to the smallestdetails we expect to detect. The major avantages of this simple initialization methodare : it is automatic (only the size of the windows has to be xed), it acceleratesthe speed of convergence (the smaller the windows, the faster the convergence), andit is less sensitive to noise (in the sense that we compute the average ml of u0 overeach window before selecting the function k whose mean k is the closest one toml).

The data on Fig. 16 and Fig. 17 are the same noisy data treated as the onesshown on Fig. 14, but we use an automatic seed initialization whose windows are ofsize 55. The size of the windows are small in order to avoid a lost of many detailson the boundary of the perturbated shape on the right handside. The initial seedsare growing until they merge. Results are better than the ones obtained with theinitialization set of Fig. 14.

6.2 Real images

Orignal data on Fig. 18 are MRI medical brain data containing 4 classes of pre-estimated parameters i and i. We present a color classication result and threesteps of the ZLS and phase evolution obtained with an automatic seed initalization(on windows of size 55).

Data on Fig 19 and Fig 20 are SPOT satellite images provided by the French SpaceAgency CNES. Fig 19 contains 4 classes whose parameters were estimated in [2], andwe have 10 classes for Fig 20 whose parameters were given by an expert (see [13]).

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initialization iteration 100

iteration 300 iteration 500


Figure 4: Evolution of the three ZLS's. Parameters and ei in (23) are set to zero forall i = 1:::K. Only the length shortening term is taken into account : i = 10:0 8i,and dt = 0:2

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class 3

class 2

class 1

class 3

class 2

class 1

original data initial ZLS's classication

initialization iteration 30 iteration 70

iteration 100 iteration 230 iteration 360

Figure 5: ZLS evolution for three classes (1 = 1:0, 2 = 2:0 and 3 = 3:0).Parameters are : = 5:0, dt = 0:2, and for all i we have i = 1:0 and ei = 2:0. Theclassication result on top right handside is a false color image whose black colorrepresents unclassied pixels (pixels of vacuum).

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Figure 6: Phase evolution associated to Fig. 5.

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class 1

class 2

class 3class 3

class 2class 1

class 3

original data initial ZLS





Figure 7: ZLS evolution for three classes (1 = 100:0, 2 = 128:0 and 3 = 160:0).Parameters are : = 5:0, dt = 0:2, and for all i we have i = 0:1 and ei = 0:01.

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classication

Figure 8: Phase evolution associated to Fig. 7. Last gure is the false color classi-cation result whose black color represents unclassied pixels (pixels of vacuum).

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class 3

class 2

class 2class 1

class 3

class 2

class 3

class 1

original data initial ZLS Classication




Figure 9: ZLS evolution for three classes (1 = 0:0, 2 = 1:0 and 3 = 2:0).Parameters are : = 2:0, dt = 0:2, and for all i we have i = 0:4 and ei = 2:0. Thefalse color image on top right handside is the classication result whose black colorrepresents unclassied pixels (pixels of vacuum).

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Figure 10: Phase evolution associated to Fig 9.

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class 3

class 2

class 1

class 3

class 2

class 1

noisy data initial ZLS classication

ZLS evolution

phase evolutioninitialization iteration 100 iteration 800

Figure 11: Noisy version (-1.6 dB) of data presented on Fig. 5; ZLS and phaseevolution of the i's for three classes (1 = 1:0, 2 = 2:0 and 3 = 3:0) . Parametersare : = 7:0, dt = 0:2, and for all i we have i = 4:0 and ei = 1:0. The false colorimage on top right handside is the classication result whose black color representsunclassied pixels (pixels of vacuum).

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class 2

class 3

class 1

class 3

class 2

class 1

noisy data initial ZLS classication

ZLS evolution

phase evolutioninitialization iteration 100 iteration 1300

Figure 12: Noisy version (-6.4 dB) of data presented on Fig. 5; ZLS and phaseevolution of the i's for three classes (1 = 1:0, 2 = 2:0 and 3 = 3:0). Parametersare : = 10:0, dt = 0:2, and for all i we have i = 15:0 and ei = 1:0. The false colorimage on top right handside is the classication result whose black color representsunclassied pixels (pixels of vacuum).

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class 2

class 3

class 1

class 3

class 2

class 1

noisy data initial ZLS

without restoration

with restorationiteration 100 nal ZLS's classication

Figure 13: Noisy version (-6.4 dB) of data presented on Fig. 5. Comparison ofthe results given from the minimization of (16), i.e. without restoration, on the tophandside, and from the minimization of (26), i.e with restoration coupled to theclassication, on bottom handside. We present two steps of the ZLS's evolution, andthe classication results are false color images.

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noisy data (SNR=3.2 dB) classication




Figure 14: Noisy data : noisy version of data presented on Fig. 7 (SNR=3.2 dB).ZLS evolution for three classes with manual initialization. Parameters are : = 5:0,dt = 0:2, and for all i we have i = 0:2 and ei = 0:001. The false color image on topright handside is the classication result whose black color represents unclassiedpixels (pixels of vacuum).

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noisy data (SNR=3.2 dB) classication




Figure 16: Noisy data : noisy version of data presented on Fig. 7 (SNR=3.2 dB).ZLS evolution for three classes with seed initialization (on windows of size 55).Parameters are : = 5:0, dt = 0:2, and for all i we have i = 0:2 and ei = 0:001.The false color image on top right handside is the classication result whose blackcolor represents unclassied pixels (pixels of vacuum).

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data Classication


Figure 18: MRI brain data containing 4 classes with seed initialization. We showthree steps of the ZLS (top) and phase (bottom) evolutions. The classication re-sult is a false color image whose black color represents unclassied pixels (pixels ofvacuum).

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SPOT data classication


Figure 19: SPOT satellite image containing 4 classes with seed initialization (onwindows of size 99) : We show three steps of the ZLS (top) and phase (bottom)evolutions. The classication result is a false color image.

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SPOT data

ZLS result classication result

Figure 20: SPOT satellite image containing 10 classes with seed initialization (win-dows of size 55).

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7 Conclusion

We have presented a variational model based on level set formulation for image clas-sication. The level set formulation is a way to represent regions and set of interfaceswith a continuous function dened over the whole support of the image. The min-imization of the functional leads to a set of coupled PDE's which are consideredthrough a dynamical scheme. Each PDE is guiding a level set function according tointernal forces (length minimization), and external ones (data term, no vacuum andno region overlapping). Results on both synthetic and satellite images are given. Wealso propose a way of introducing a restoration capability in the model. First resultsare promising, and we will study more precisely this model in future work. Furtherwork will also be conducted to deal with the estimation of the class parameters (unsu-pervised classication). We also envisage to extend this model to multispectral data(with applications to multiband satellite data and applications to color imaging).

Acknowledgements

We thank Pierre Kornprobst for the synthetic data shown on Fig. 9 and for helpfullcomments. We also thank the French Space Agency CNES for providing SPOTsatellite images, the Gdr ISIS for original data on Fig. 7, and Dr. Dormont from LaPitié Salpétrière Hospital for providing MRI data.

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Appendix

We describe the intermediate calculus to obtain (22) from (16).From (11), (12) and (15) we can write

F(1; :::;K) = F a

(1; :::;K) + F b

(1; :::;K) + F c

(1; :::;K)

=

ZLa(1; :::;K ;r1; :::;rk; x)dx

+

ZLb(1; :::;K ;r1; :::;rk; x)dx

+

ZLc(1; :::;K ;r1; :::;rk; x)dx

If (1; :::;K) is solution of the minimization of F, then necessarily

@F

@i= 0;8i = 1:::K

and the Euler Lagrange equations for every i = 1:::K (with Neumann conditions on@) leads to

@F a

@i=

@La@i

div @La@ri

= (i) KXi=1

H(i) 1

@F b

@i=

@Lb@i

div @Lb@ri

=

@H

@i(i)

(u0 i)2

2i

= (i)(u0 i)

2

2i

@F c

@i=

@Lc@i

div @Lc@ri

= i

h0

(i)jrij div(i)

ri

jrij

i

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and if x = (x1; x2) 2 we have

div(i)

ri

jrij

=

@

@x1

(i)

ri

jrij

+

@

@x2

(i)

ri

jrij

=

@@i

(i)@i

@x1

ri

jrij+@@i

(i)@i

@x2

ri

jrij+ (i)div

ri

jrij

=

0

(i)jrij+ (i)div ri

jrij

thus,

@F c

@i= (i)div

ri

jrij

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References

[1] G. Barles, H. M. Soner, and P.E. Souganidis. Front propagation and phaseeld theory. SIAM J. Control and Optimization, 31:439479, 1993.

[2] M. Berthod, Z. Kato, S. Yu, and J. Zerubia. Bayesian image classication usingMarkov random elds. Image and Vision Computing, 14(4):285293, 1996.

[3] C.A. Bouman and M. Shapiro. A multiscale random eld model for Bayesianimage segmentation. IEEE Trans. on Image Processing, 3:162177, March 1994.

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[19] N. Paragios and R. Deriche. Geodesic active regions for texture segmentation.INRIA Research Report RR-3440 (http://www.inria.fr/RRRT/publications-eng.html), June 1998.

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