REGION-BASED ACTIVE CONTOUR WITH NOISE AND SHAPE PRIORS
F.Lecellier a,S.Jehan-Besson a,J.Fadili a,
G.Aubert b,M.Revenu a,E.Saloux c
a GREYC UMR6072Caen France,
b Laboratoire J.A.Dieudonn´e Nice France,
c CHU Caen France
ABSTRACT
In this paper,we propose to combine formally noise and shape priors in region-based active contours.On the one hand,we use the general framework of exponential family as a prior model for noise.On the other hand,translation and scale in-variant Legendre moments are considered to incorporate the shape fidelity to a reference shape).The combi-nation of the two prior terms in the active contour functional yields thefinal evolution equation whose evolution speed is rigorously derived using shape derivative tools.Experimental results on both synthetic images and real life cardiac echog-raphy data cle
arly demonstrate the robustness to initialization and noise,flexibility and large potential applicability of our segmentation algorithm.
Index Terms—Image segmentation
1.INTRODUCTION
The current work is devoted to the segmentation of regions of a priori known shape in noisy images using region-based active contours[1,2,3].This method allows the use of pho-tometric image properties,such as texture and noise,as well as geometric properties such as the shape of the object to be segmented.The shape prior can prove very useful in cases where the object is occluded or partially missing.Further-more,by including an a priori on the shape,sensitivity of the active contour model to initialization will also be alleviated.
On the one hand,attempts to incorporate shape priors have been proposed by some authors using various methods, such as diffusion snakes[1]or distance function[2,3,4]. On the other hand,there are only few proposals in the lit-erature that tried to take benefit of a noise prior[5,6,7] within region-based active contours.In these works,image intensity)are considered as random variables whose distribution belongs to some parametric family which is chosen according to the physical acquisition m
odel of the considered images.However,to the best of our knowledge, shape and noise priors have never been combined,at least formally,in active contour models.This would enable to per-form the segmentation of poor noisy images,beyond the sim-ple classical white Gaussian noise model,with a strong shape constraint.Example of such data is encountered in echocar-diographic images.For instance,in echographic data,it is notably well known that under appropriate conditions(large number of randomly located scatters),the Rayleigh distribu-tion is well suited to model the noise[8].
The main contribution of this paper is to combine formally noise and shape priors in region-based active contours for seg-mentation purposes.In order tofix ideas,let us consider a region of interestΩin the image.We propose tofind the partition of the image that minimizes the following generic criterion which is able to handle both noise and shape priors:
J(Ω)= Ωf n(x,Ω)d x+αd(Ω,Ωref)
whereΩref represents the reference region shape,and x= [x,y]T stands for the location of the pixel.
Thefirst term corresponds to the noise prior term.This term takes benefit of statistical properties of the image inten-sity.It is based on functions of parametric probability density functions(pdf)belonging to the exponential family.Proba-bility models with these common features include Normal, Bernoulli,Binomial,
Poisson,Gamma,Beta,Rayleigh,etc. These models are the most commonly encountered in imag-ing acquisition systems.This term is detailed in Section3. The second term d(Ω,Ωref)corresponds to the shape prior. Shapes are here described using scale and translation invari-ant Legendre moments as in[9].With such a shape descrip-tor,the registration step is avoided.This term is discussed in Section4.
The evolution equation of the deformable curve is de-duced from the functional to minimize using shape derivative tools[10]and the framework set in[11,12].
This paper is organized as follows:we briefly remind the shape derivation tools in Section2.The noise model term is presented in Section3.In Section4,we introduce the shape prior model and the invariances that were added.The seg-mentation algorithm is presented in Section5.Experimental results are discussed in Section6.Wefinally conclude and give some perspectives.
2.SHAPE DERIV ATIVE TOOLS
In order to be comprehensive,we here give a brief summary of the shape derivation theory.The interested reader may refer to[10,11,12]for further details.
Let U be a class of domains(open,regular bounded sets, i.e.C2)of R n,andΩan element of U.The boundary∂Ωof
Ωis sometimes denoted byΓ.
The region-based term is expressed as a domain integral of a function f named descriptor of the region:
J r(Ω)= Ωf(x,Ω)d x(1)
In the general case,this descriptor may depend on the do-main such as the descriptors introduced thereafter for noise and shape priors.The derivation of this term is performed us-ing domain derivation tools.We apply a fundamental theorem [10]which establishes a relation between the Eulerian deriva-tive of J r(Ω)in the direction V,and the domain derivative of f denoted f s(x,Ω,V):
<J r(Ω),V>= Ωf s(x,Ω,V)d x
− ∂Ωf(x,Ω)(V·N)d a(x)(2)
where N is the unit inward normal to∂Ω,d a its area element. Thefirst integral comes from the dependence of the descriptor f(x,Ω)upon the region while the second term comes from the evolution of the region itself.
From the shape derivative,we can deduce the evolution equation that will drive the active contour towards a minimum of the criterion.
Let us suppose that the shape derivative of the regionΩmay be written as follows:
<J r(Ω),V>=− ∂Ωv(x,Ω)(V(x)·N(x))d a(x)(3) We can then deduce the following evolution equation:
∂Γ(p,τ)
∂τ
=v(x,Ω)N(x)
withΓ(τ=0)=Γ0,x=Γ(p,τ).
3.THE NOISE MODEL
In this section we focus our attention on the noise model.The chosen descriptor for this part is:
f n(x,Ω)=Φ(p(y(x),η)(4) where p is the pdf of some image features y(x)∈R d whose associated parameters are denoted byη,andΦis at least C1.
In our study,we consider that p belongs to the exponen-tial family.This family is comprehensive enough to cover noise models in most image acquisition systems encountered in Gaussian,Exponential,Poisson,Rayleigh to cite a few.The multi-parameter exponential families are natu-rally indexed by a k-dimensional real natural parameter vec-torη=(η1,...,ηk)T and a k-dimensional natural sufficient statistic vector T=(T1,...,T k)T.A simple example is the normal family when both the location and the scale parame-ters are unknown(k=2).Formally,the pdf of a vector of random variables Y belonging to the k-parameter canonical exponential family is:
p(y,η)=h(y)exp[ η,T(y) −A(η)](5) where η,T denotes the scalar product.
This statistical criterion is now derived according to the domain in order to deduce the evolution equation of the ac-tive contour.For the sake of simplicity,we denoteηfor the natural parameter of a pdf of the exponential family and itsfi-nite sample estimate over the domain(without a slight abuse of notation,this should beˆη).
Theorem1The Gˆa teaux derivative,in the direction of V, of the functional J n(Ω)= ΩΦ(p(y(x),η(Ω)))d a(x)where p(.)belongs to the multi-parameter exponential family with natural hyperparameter vectorη,is:
<J n(Ω),V>=− ∂ΩΦ(p(y))(V·N)d a(x)
+ Ωp(y)Φ (p(y)) ∇Vη,T(y)−∇A(η) d x(6)
with∇Vηthe Gˆa teaux derivative ofηin the direction of V, and x,y the scalar product of vectors x and y.
In afinite sample setting,when using the ML estimator, we can replace∇A(η)by T(Y)(the1st order sample mo-ment of T(Y)).Thus,when using the-log-likelihood func-tion,the second term becomes equal to Ω ∇Vη,T(y)−T(Y)) d x,and hence vanishes.The following corollary fol-lows:
Corollary1The Gˆa teaux derivative,in the direction of V, of the functional J n(Ω)=− Ωlog(p(y(x),ηML(Ω))d a(x) whenˆηML is the ML estimate,is the following:
<J n(Ω),V>= ∂Ω(log(p(y(x),ηML(Ω)))(V·N)d a(x)
These general results can be easily specialized to some pdf of Gaussian,Rayleigh,etc).We let the reader refer to[6]for more details.
4.THE SHAPE PRIOR MODEL
The shape prior is used as an additionalfidelity to a reference shape),designed to make the behaviour of the segmentation algorithm more robust to occlusion and missing data and to alleviate initialization issues.Here,orthogonal Legendre moments with scale and translation invariance were used as shape descriptors[9].Indeed,moments[13]give a region-based compact representation of shapes through the projection of their characteristic functions on an orthogonal basis such as Legendre polynomials.
The shape prior is then defined as the Euclidean distance between the moments of the evolving region and ones of the reference shape,
d(Ω,Ωref)= λ(Ω)−λ(Ωref) 22(7) whereλ(Ω)are the moments of the regionΩ.In practice, infinite moment expansion is generally limited to a sufficient finite number resulting in a good approximation of the origi-nal shape.The criterion then reduces to:
d(Ω,Ωref)=p+q≤N
p,q
(λpq(Ω)−λpq(Ωref))2(8)
where theλpq are defined as follows,using the geometric mo-ments M pq and the coefficients a pq of the Legendre polyno-mials[13]:
λpq=C pq
p
u=0
q
v=0
a pu a qv M uv(9)
where C pq=(2p+1)(2q+1)
4,M pq(Ω)= Ωx p y q dxdy,and
the Legendre polynomials are defined as:
P p(x)=
p
k=0
a pk x k=
1
2p!
d p
dx
(x2−1)p.
In general,the reference shape can have different orien-tation and size compared to the shape to be segmented.This will then necessitate an explicit registration step in order to re-align the two shapes.In order to avoid this generally problem-atic registration step,we here use scale and translation invari-ant Legendre moments as in[9].In the geometric moments definition,the scale invariance is embodied as a
normaliza-tion term:1.As far as translation invariance is con-cerned,we replace x and y in the geometric moments M pq by x−¯x and y−¯y,(¯x,¯y)are the shape barycenter coordinates.
The derivation of the criterion(8)is relatively complex. We here give the main formula.
<d (Ω,Ωref),V>=u+v≤N
u,v
A uv(H uv+L uv)N
where
A uv=2p+q≤N
X
p,q
(λpq−λref pq)C pq a pu a qv H uv=
(x−¯x)u(y−¯y)v
Ω(u+v+2)/2
L uv=u¯x M u−1,v
Ω3/2
(1−x)+
v¯y M u,v−1
Ω3/2
(1−y)−
(u+v+2)M u,v
2Ω
The reader may refer to[9]for further details.
5.SEGMENTATION WITH JOINT NOISE AND
SHAPE PRIORS
The region-based active contour functional to be minimized isfinally written as:
J(Ωin,Ωout)=ZΩin f n(x,Ωin)d x+αd(Ωin,Ωref)
+ZΩout f n(x,Ωout)d x+βE b(Γ)(10)
where we assign a specific noise model to the background (outside)region,possibly different from the noise model of the object(inside)region.The energy term E b is a regular-ization term balanced with a positive real parameterβ.It can be chosen as the curve length and classically derived using calculus of variation or shape derivation tools.
To drive this functional towards its minimum,the geomet-rical PDE corresponding to(10)is iteratively run without the shape prior,then the shape prior term is updated,and the ac-tive contour evolves again by running the PDE with the shape prior.This procedure is repeated until convergence.This it-erative optimization scheme has aflavour of coordinate re-laxation.At this stage,it is worth pointing out some major differences between our algorithm and the one developed in [9].Thefirst one is that we here consider both photometric (noise)and geometrical(shape)priors,while[9]focused on the shape prior and
did not considered noisy data.This dif-ference has a clear impact on the evolution algorithm since those authors propose to run the evolution equation only once without the shape prior and then incorporate the shape prior in the evolution.This is fundamentally different from our al-ternating scheme.
Algorithm1Evolution algorithm of the active contour
1:repeat
2:Evolution using noise prior for n iterations
3:repeat
4:Evolution using shape prior for1iteration.
5:until Maximum shape speed<threshold
6:until Convergence
6.EXPERIMENTAL RESULTS
The above evolution scheme was applied on some synthetic data with y(x)=I(x)the image intensity.Fig.1.(a)depicts a shape corrupted by an additive white Gaussian noise with SNR=1,with the initial curve.To bring to the fore the con-tribution of the shape prior term,parts of the objects are de-liberately missing.Fig.1.(b)(resp.(c))shows the segmenta-tion result with the noise model(Gaussian)but without(resp. with)the shape prior.As expected,one can clearly see that: (i)without a shape prior,thefinal curve sticks to the appar-ent boundaries of object,(ii)owing to the shape prior,the algorithm managed to recover properly the missing parts of the object.Furthermore,in addition to its robustness to miss-ing data,we have also observed that the shape prior allows to mitigate initialization issues.As far as the noise prior is con-cerned,choosing the appropriate model has a clear impact on the quality of the results as it has been shown in[6].
We then tested our approach on real echocardiographic images.As the Rayleigh distribution is well suited to model the noise in these data[8],this noise model was used in Corol-lary1.The original image(Fig.2.(a))is shown with the ini-
(a)(b)(c)
Fig.1.a.Noisy image with initial contour,b.Final contour without
shape prior,c.Final contour with shape
prior.
(a)
(b)
(c)
(d)
(e)(f)
Fig.2.a.Echocardiographic image with initial contour,b.Contour
draw by an expert,c.Final contour without shape prior,d.Final
contour with shape prior,e.Final contour using AAMM method,
f.Hamming distance for one echocardiographic sequence of14im-
ages.
tial contour position.We compared the result of our method
(fig.2),with(d)and without(c)the shape prior,to an expert
manual segmentation(b),and a segmentation provided by the
Active Appearance and Motion Model(AAMM)method(e)
designed for echocardiography[14,15].Again,the saliency
of our method is obvious.Our method gives the closest seg-
mentation to the expert manual delineation.This is quanti-
tavely by the Hamming distance plots(f),showing that our
method outperformes AAMM.
7.CONCLUSION AND PERSPECTIVES
This paper concerns the incorporation of both noise and shape
priors in region-based active contours.The evolution of the
active contour is derived from a global criterion that combines
statistical image properties and geometrical information.Sta-
tistical image properties take benefit of a prespecified noise
model defined using parametric pdfs belonging to the expo-
nential family.The geometrical information consists in mini-
mizing the distance between Legendre moments of the shape
and those of a reference.The Legendre moments are designed
to be scale and translation invariant in order to avoid the reg-
istration step.The combination of these terms gives accurate
results on both synthetic noisy images and real echocardio-
graphic data.Our ongoing research is now directed towards
the integration of a complete shape learning step.
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