Video Object Segmentation Using Eulerian Region-Based Active Contours

Video Object Segmentation Using Eulerian Region-Based Active Contours (International Conference in Computer Vision,Vancouver,Canada,juillet2001)

St′e phanie Jehan-Besson Michel Barlaud Laboratoire I3S,CNRS-UNSA

2000,route des Lucioles

06903Sophia Antipolis,France

jehan,barlaud@i3s.unice.fr

Gilles Aubert Laboratoire J.A.Dieudonn′e,CNRS-UNSA

Parc Valrose

06108Nice Cedex2,France

gaubert@math.unice.fr

Abstract

We address the problem of moving object segmentation using active contours.As far as segmentation of moving objects is concerned,region-based terms must be incorpo-rated in the evolution equation of the active contour,in ad-dition to classical boundary-based terms.In this paper,we propose a general framework for region-based active con-tours.Novel aspects of the segmentation method include a new Eulerian proof to compute the evolution equation of the active contour from the minimization of a criterion,and the introduction of functions named“descriptors”of the re-gions.In this proof,the dynamical scheme is directly intro-duced in the criterion before differentiation.With such a method,the case of descriptors depending on the evolution of the curve,i.e.depending upon features globally attached to the region,can readily be taken into account.The vari-ation of these descriptors upon the evolution of the curve induces additional terms in the evolution equation of the active contour.The proof ensures the fastest decrease of the active contour towards a minimum of the criterion.Inside this theoretical framework,a set of descriptors is evaluated on real sequences for the detection of moving objects.

1.Introduction

Detection and localization of moving objects in a se-quence of images become a crucial issue for the develop-ment of the new standard of video compression MPEG-4 [8].Generally speaking,two classes of approaches can be considered for segmentation:region-based approaches such as Markov Random Fields[17]and boundary-based ap-proaches.Originally,active contours were boundary-based methods:snakes[13],balloons[6]and geodesic active con-tours[2]are driven by the minimization of an energy to-wards the edges of an image.The information used is lo-cated strictly along the boundary.Nevertheless,active con-tours are powerful tools for the segmentation of moving ob-jects since they enclose the whole object by a continuous curve,which can be?tted to any shape.

In order to use active contours for the segmentation of mov-ing objects,region-based information must be incorporated in the evolution equation of an active contour.Previous works[3,4,5,18,19,22,23]proposed a way of integrating region-based terms in the evolution equation of an active contour for many different applications.A short survey of these methods is proposed in section2.

Goals and Contributions:Our goal is to elaborate a gen-eral framework for region-based active contours.Novel as-pects of the segmentation method include:

The introduction of functions describing each region, named“descriptors”of the regions.Thanks to these func-tions,the algorithm can easily be adapted to various appli-cations by only changing the descriptors.In this paper,a set of descriptors is evaluated for the segmentation of moving objects in a video sequence.

A new Eulerian proof to compute the evolution equa-tion of an active contour from a criterion incorporating both region-based and boundary-based terms.In this proof,the dynamical scheme is introduced immediately in the crite-rion before derivation.This method allows us to readily take into account a possible variation of the descriptors dur-ing the evolution of the curve.We consider both kinds of descriptors:time-dependent ones(dependent on the evolu-tion of the curve)and time-independent ones.The evolution equation is computed for time-independent descriptors and for some examples of time-dependent descriptors.We show that the velocity vector is not the same in the two cases. There are some additional terms in the evolution equation of time-dependent descriptors that have to be considered.

The paper is organized as follows.In section2,we present a survey of the region-based active contours found in the literature.In section3,we introduce a general frame-work for region-based active contours and a new Eulerian proof to compute the evolution equation of the active con-tour.In section4,a set of descriptors is proposed for the detection of moving objects.Finally,our method is evalu-

ated on real sequences in section5.

2.Problem statement and overview of existing

region-based active contours

We present a state-of-the-art on region-based methods

using active contours.We focus on differences and similar-

ities between all these works.

2.1.Problem statement

First,we describe the equations related to the segmen-

tation of an image in two regions.Let us de?ne the

region containing the objects to segment and the back-

ground region.Their common boundary is noted.

Let us consider an active contour

,where can be the arc length of

the contour and is an evolution parameter.It classically

evolves according to the following partial differential

equation(PDE):

where is the curvature of.

Recently,Paragios and Deriche[18]have proposed an ex-

tension of the work of Zhu and Yuille by changing the de-

scriptor of the contour in order to incorporate the image gra-

dient as in geodesic active contours[2].In their study,some

descriptors are evaluated for texture or moving objects seg-

mentation.

Chakraborty et al.[3]have introduced a similar region-

based approach for the segmentation of medical images.As

in Zhu and Yuille[23],the whole problem is reduced to

computing line integrals with the Green-Riemann theorem,

rather than both line and area integrals.The criterion is in-

troduced and numerically evaluated to maximize it.In the

same way,Chesnaud et al.[5]search to maximize a likeli-

hood function by choosing at each step the better displace-

ment of the active contour.

A later approach has been introduced by Samson et al.

[20]for classi?cation and by Chan and Vese[4]for segmen-

tation of still images.Here the level set method is directly

used by representing the curve as the zero level set of

a Lipschitz continuous function.The criterion is

then minimized with respect to and the evolution equa-

tion is directly expressed with the level set function.

What emerges from this survey is that these pioneer works are complementary.However,all proofs leading to the evolution equation of the active contour are based on the derivation of the criterion using Euler-Lagrange equations and the dynamical scheme is introduced after the compu-tation of the derivative[4,18,23].With such a method, the case of descriptors depending upon features globally attached to the region(time-dependent descriptors)cannot readily be taken into account.The major contribution of our work is to compute the evolution equation of the active contour for descriptors depending upon features globally at-tached to the region without computing the Euler-Lagrange equations.We propose the powerful idea of introducing a dynamical scheme on the criterion itself,and a therorem of ?uid dynamic to avoid computing the Euler-Lagrange equa-tions.For time-dependent descriptors,we show that addi-tional terms appear in the evolution equation of the curve. With these additional terms,we obtain the fastest decrease of the active contour towards a minimum of the criterion.

3.Setting a general framework for region-

based active contours

The main idea lies in the elaboration of a general frame-work for the segmentation of an image in two different re-gions using region-based active contours.

Each region is described using a function that we name “descriptor of the region”.The introduction of such descrip-tors is interesting for two reasons.Firstly,for a given appli-cation like detection of moving objects,various descriptors can be easily tested inside the same theoretical framework. Secondly,this framework can be applied to other applica-tions[1,7].

Once the criterion is introduced,we propose a new Eule-rian proof for the computation of the evolution equation of the contour.It fundamentally differs from other approaches: we now directly introduce a dynamical scheme in the cri-terion before differentiation.The criterion becomes a functional of the evolving parameter.In this dynamical scheme,the evolving contour appears directly in the criterion.Moreover,we can readily take into account

a possible variation of the descriptors with the evolving pa-rameter.Two cases are considered:time-dependent de-scriptors(descriptors that can vary with the evolution of the curve)and time-independent descriptors.In the case of time-dependent descriptors,the evolution equation of the active contour is not the same as the one computed for time-indepedent descriptors.Some additional terms appear and so it is of prime necessity to consider the two kinds of de-scriptors,time-dependent or not,separately.3.1.Introduction of a general criterion

An image is a function de?ned for

,where is the image domain.The image domain is considered to be made up of two parts(?g.1):the fore-ground part,containing the objects to segment,noted, and the background part noted.The discontinuities set is a curve noted that de?nes the boundary between the two domains.

Figure1.The two regions of an image

We then search for the two domains and which minimize the criterion given by(3).In criterion(3), the?rst two terms are region-based while the third term is boundary-based.The functions,and are respectively the descriptors of the background part,the ob-jects to segment and the contour.

To compute an optimal solution,a dynamical scheme is introduced where the unknown regions become a function of an evolving parameter.The triplet

acts as a minimizing sequence for

as evolves.So we introduce the func-tional,noted,which has to converge towards the minimum value of as.

(5)

In contrast with other active contours approaches,the dy-namical scheme is directly introduced in the criterion.With such a scheme,the introduction of an active contour evolv-ing with is straightforward.Hence is modeled as an active contour that converges towards the?nal expected segmentation.

Let be the initial curve de?ned by the user.We recall that we search for as a curve evolving according to the following PDE:

where is the velocity vector of the active contour.The main problem lies in?nding the velocity from the crite-rion(3)in order to get the fastest curve evolution towards the?nal segmentation.

https://www.360docs.net/doc/2d18068722.html,putation of the derivative of the criterion

In order to obtain the evolution equation,the criterion

must be differentiated with respect to.The inte-gral bounds depend on and the descriptors and

may also depend on.Let us de?ne the functional

such that:

if

if(7)

The functional and are de?ned on the whole image domain.The criterion writes as:

(8) where is the image domain.We de?ne:

In order to compute the derivative of the criterion,the discontinuities must explicitly be taken into account.For such a purpose,we?rst recall a general theorem concerning the derivative of a time-dependent criterion[10].Let us de?ne as a region included into.

Theorem1Let be a smooth function on where represents the jump of across:

,and the unit normal of directed from to.

Proof:We can apply Theorem1to the domain and to the domain.By adding the two equations,we obtain the corollary.

Figure2.The domains and the vectors involved in

the derivation

It is now straightforward to get the derivative of the criterion :we take the image domain(?g.2).Then, by taking explicitly into account the discontinuities thanks to the corollary,the derivative is given by:

(9)

Obviously,the second term of the derivative(9)is null since the external boundary of the image is?xed.By replac-ing by its expression,we?nd:

(10)

The derivative of is classical[2]:

(11)

where is the curvature of.Thus for the derivative of the whole criterion,we?nd:

(12)

3.3.Evolution equation with either time-dependent

or time-independent descriptors

The goal of this part is to compute the expression of the velocity vector which makes the curve evolve the fastest as possible towards a minimum of the criterion.For such a purpose,two main cases have to be considered:

a)the descriptors do not depend on(time-independent de-scriptors);

b)the descriptors depend on(time-dependent descrip-tors).

3.3.1.Evolution equation with time-independent

descriptors

In this case,we can compute the theoretical expression of the velocity vector for any descriptors.In fact,as the de-scriptors do not depend on,the derivative of the criterion reduces to:

(13)

Therefore,according to the inequality of Cauchy-Schwartz, the fastest decrease of is obtained by choosing ,where:

(14) Thus,the curve evolves according to the following PDE:

with

Region-dependent descriptors with the mean:

does not depend on

(16)

with an even positive function of class. Thus,we have:

The derivative of with respect to is computed using the-orem1:

The expression of the whole derivative is then:

(17) with:

So,for this?rst set of descriptors,the fastest decrease of is obtained by choosing,where:

(18)

In this example,the computed force is not the same as the one computed for time-independent descriptors.We re-mark that there is an additional term in the expression of the velocity.We also notice that,in this case,the force depends on the evolution parameter.

Let us consider some examples of functions.In the par-ticular case of,we have and thus,the additional term is null.However,in most cases,this addi-tional term is not null.For example,if

And if,we have:

Using theorem1,we get:

(20) And so for the derivative of,we obtain:

(22) As far as this second set of descriptors is concerned,there is also an additional term in the evolution equation of the active contour.

(23)with a positive constant.

In this case,we obtain for the derivative of:

(24)

So,for this third set of descriptors,the fastest decrease of is obtained by choosing,where:

(25) As far as this third set of descriptors is concerned,we can remark that the computed force does not depend on.

The additional terms found in equations(18),(22)and (25)cannot readily be obtained when the dynamical scheme is introduced after the computation of the derivative using Euler-Lagrange equations.Thus,it is of prime necessity to introduce the dynamical scheme on the criterion itself in order to easily get these additional terms in the case of time-dependent descriptors.Thanks to these additional terms,we ensure the fastest evolution of the curve towards a minimum of the criterion.

3.3.3.Remarks on the evolution equation

Considering equations(14),(18),(22)or(25),we no-tice that the velocity can be positive or negative.Con-sequently,region-based methods allow some parts of the curve to shrink while others expand.Thus the initialization is less constrained.

3.4.Implementation with the level set method

In a video sequence,several objects may appear in a scene.Therefore segmentation of video objects requires a method where topological changes are well handled in or-der to detect several objects from the same initial curve.On that account we use the level set method approach proposed by Osher and Sethian[21].The key idea is to introduce an auxiliary image such that is the zero level set of:.The function is often chosen to be the signed distance function of,and is then characterized by:.This Eulerian formu-lation presents several advantages[21].Firstly,the curve may break or merge as the function evolves,and topological changes are thus easily handled.Secondly,the function always remains a function allowing ef?-cient numerical schemes.Thirdly,the geometric properties of the curve,like the curvature and the normal vector?eld ,can be estimated directly from the level set function:

(26)

The evolution equation(6)then becomes:

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