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Graph cut segmentation with a statistical shape model in cardiac MRI

Graph cut segmentation with a statistical shape model in cardiac MRI
Graph cut segmentation with a statistical shape model in cardiac MRI

Graph cut segmentation with a statistical shape model in cardiac

MRI

D.Grosgeorge a ,C.Petitjean a ,?,J.-N.Dacher b ,S.Ruan a

a LITIS EA 4108,Universitéde Rouen,22bd Gambetta,76183Rouen Cedex,France b

INSERM U1096,Universitéde Rouen,1rue de Germont,76031Rouen Cedex,France

a r t i c l e i n f o Article history:

Received 5February 2012Accepted 8January 2013

Available online 30April 2013Keywords:

Image segmentation Graph cut Shape prior MRI

Cardiac ventricle

a b s t r a c t

Segmenting the right ventricle (RV)in magnetic resonance (MR)images is required for cardiac function assessment.The segmentation of the RV is a dif?cult task due to low contrast with surrounding tissues and high shape variability.To overcome these problems,we introduce a segmentation method based on a statistical shape model obtained with a principal component analysis (PCA)on a set of representative shapes of the RV.Shapes are not represented by a set of points,but by distance maps to their contour,relaxing the need for a costly landmark detection and matching process.A shape model is thus obtained by computing a PCA on the shape variations.This prior is registered onto the image via a very simple user interaction and then incorporated into the well-known graph cut framework in order to guide the seg-mentation.Our semi-automatic segmentation method has been applied on 248MR images of a publicly available dataset (from MICCAI’12Right Ventricle Segmentation Challenge).We show that encouraging results can be obtained for this challenging application.

ó2013Elsevier Inc.All rights reserved.

1.Introduction

Magnetic resonance imaging (MRI)is increasingly used as a standard tool in the evaluation of the right ventricle (RV)function [14].The segmentation of the RV cavity is a prerequisite to the computation of clinical parameters.Although some relatively ef?-cacious methods are commercially available for segmenting the left ventricle (LV)on MR images,such as MASS (Medis,Leiden,The Netherlands)[40]and Argus (Siemens Medical Systems,Ger-many)[24],the segmentation of the RV is currently performed manually in clinical routine.This lengthy and tedious task requires about 20min by a clinician and is also prone to intra and inter-ex-pert variability.An automatic segmentation method would avoid these drawbacks,but has to deal with the following issues:(i)fuzz-iness of the cavity borders due to blood ?ow and partial volume ef-fect,(ii)the presence of trabeculations (wall irregularities)in the RV,which have the same grey level as the surrounding myocar-dium,(iii)the complex crescent shape of the RV,which varies according to the imaging slice level (see Fig.1).This is probably one of the reasons why RV functional assessment has long been considered secondary compared to that of the LV,leaving the prob-lem of RV segmentation wide open [27].

The literature of RV segmentation is thus much less abundant than the one of LV segmentation.Most of RV segmentation meth-ods are based on a joint segmentation of both ventricles.These methods take bene?t from the relative positions of the ventricles and the similarity of the gray levels in their respective blood cavi-ties.For example,this information is used within active contours [29],or within a framework combining thresholding,clustering and morphological operations [4].Another possibility is to use higher-level prior anatomical information to guide the segmenta-tion process,such as a biomechanical model [33]or statistical shape models.In this latter case,the statistical shape model maybe a Point Distribution Model (PDM),obtained by a principal compo-nent analysis (PCA)on the set of aligned shapes,and integrated into the well-known active shape and appearance modeling frame-work [6].This technique ensures to have a realistic solution since only shapes similar to the training set are allowed,but at the ex-pense of building a training data set with manually generated seg-mentations.It has been applied to the segmentation of both LV and RV in [23,25].Statistical prior information may also take the form of an heart atlas.An atlas describes the different structures present in a given type of image.The segmentation of the ventricles is ob-tained by registering a single [19]or multiple atlases [15]onto the image to be segmented.

Shape prior based segmentation:More generally,automatic organ segmentation can bene?t from the use of information regarding shape and/or gray levels,to increase its robustness and accuracy [8].This especially applies if the nature of the shape does not change much from an example/individual to another.Making use a shape prior for object segmentation involves two steps:(i)The de?nition of a mathematical shape representation.In

the literature,shapes are usually represented by an explicit

1077-3142/$-see front matter ó2013Elsevier Inc.All rights reserved.https://www.sodocs.net/doc/a910905784.html,/10.1016/j.cviu.2013.01.014

?Corresponding author.

E-mail addresses:damien.grosgeorge@univ-rouen.fr (D.Grosgeorge),caroline.petitjean@univ-rouen.fr (C.Petitjean),su.ruan@univ-rouen.fr (S.Ruan).

model such as a point distribution model (PDM)[6]or an implicit one e.g.signed distance function [16].We discuss both representations,the PDM vs.the SDF in Section 2;(ii)The de?nition of the segmentation framework.Initially pro-posed in the deform models framework,the PDM represen-tation has been mostly used in the active shape and appearance modeling approaches [6].The SDF representa-tion allowed to easily extend the use of a shape prior into variational frameworks [16,7],and more recently,the graph cut framework [35,11].In this paper,our aim is to take advantage of the versatility and the low computational cost of the graph cut,in order to propose an ef?cient prior-based segmentation approach for the RV in cardiac MRI as described in Section 3.Graph cut with shape prior:The graph cut approach is based on the global optimization of a cost function and is very computa-tionally ef?cient in 2D [1].It is also ?exible enough to easily take into account shape information.As of today,its use in medical im-age segmentation has been restricted to a few applications.In [12],a ‘‘blob’’is created to constraint the segmentation of the entire heart in Computer Tomography (CT)cardiac scans.The shape mod-el may be either an arbitrary ?xed template [11],or a parametric curve:an elliptical mask is used for segmenting circular structures (blood vessel)in MRI [35],and circles are used for the LV segmen-tation in cardiac MRI [46].But the object to be segmented might not be easily described with a parametric shape.Statistical shape models take into account the variabilities of the shape,at the ex-pense of building a set of manually segmented shapes [36,45].The use of a speci?c shape model often imposes an iterative pro-cess,with alternates between shape model registration and graph-cut based segmentation,which can be computationally expensive.For example,in [45],the method consists in alterna-tively searching the PCA,GMM and the pose parameters using gra-dient descent (maximization step)and segmenting by graph cut using the current shape from the PCA (estimation step),as in an EM framework.Our idea is to design a statistical shape model and to design a graph cut based segmentation framework without the need for an iterative process.

In the remainder of the article,we introduce in Section 2statis-tical shape models and two possible representations:point-wise based and signed distance function-based.We propose a shape representation,that will be used for the shape prior.We then recall graph cut basics for image segmentation and introduce our method in Section 3.Experiments provided in Section 4include a compar-ison to a state of the art,namely Freedman and Zhang’s approach [11].Conclusion and future works are drawn in Section 5.2.Representation of statistical shape models

There has been a lot of work on the building and the use of statistical shape models for image segmentation

[13,3,31,34,44,17,10].The use of such models can indeed be of great help when the boundaries of the objet are ill-de?ned with occlusions.Point-distribution model (PDM)is one of the most widely used representation,used in the ASM framework.The PDM is an explicit shape representation:objects are represented by a ?nite number of landmarks [5,43,20].The PDM requires to have point correspondences before performing an analysis.A prob-lem arises in practice when dealing with PDM.The number of available shapes in the training set is often insuf?cient,where a manual labeling of medical data can be very tedious while estab-lishing correspondences can also be very challenging.This process is prone to variability if manual,and prone to detection error if automatic.

To address this problem,another representation consists in considering the shape as a zero-level set and the values of the remaining voxels by their shortest (usually Euclidian)distance to the boundary [16,7].This shape representation does not re-quire point correspondences,the shape variability being implic-itly represented by the distance variability.Eigenshape decomposition using SDF implicit representation provides toler-ance to slight misalignment of object features,since slightly mis-aligned pixels in a SDF are generally highly correlated.But using a linear (Euclidean)vector space to compute and combine eigen-shapes might be an invalid space [30].However,as point out in [16],the surfaces generally still have advantageous properties of smoothness,local dependence,and zero level sets consistent with the combination of original curves with the use of SDF representation.

In the following section,a statistical shape model based on dis-tance map is proposed,derived from SDF as previously introduced in [16,32].A comparison with the PDM is carried out to show the similarity of both models.

2.1.Shape representation based on distance function

PDM is an ef?cient model to represent shape variability.Our idea is to build a statistical shape model which does not need to match corresponding points,but can represent the shape in a sim-ilar way as PDM does,as initially proposed in [16].Consider a set of n two-dimensional binary aligned images of size H ?W composed of shapes of the same class of object.The SDF to each curve is clas-sically de?ned as H ?W samples encoding the distance to the nearest point on the curve,with for example negative values inside the object.Let Z be the matrix of SDF of each curve,where each col-umn vector are the H ?W distance samples to the corresponding curve and where each row are the distances for the same location point for each curve of the training set.The objective is to extract shape variations of this matrix Z .The average distance map is con-sidered as the reference by averaging each row of Z :

q ?1n X n i ?1

Z :;i

e1

T

images.The RV is outlined in red (hand drawn by an expert).(For interpretation of the references to colour in this of this article.)

q is a vector of length H.W.To capture the variability,we propose a different approach for distance function representation from the one used in the literature[16,37].Usually,a mean signed distance map is chosen as the reference,which implies that the PCA is per-formed in the distance function space.As the mean of signed dis-tances is not a signed distance,we reset this map to the distance values.We thus propose to calculate?rst the mean shape from the mean signed distance maps and then compute its signed dis-tance map.Let p be a point of the image domain included in R2. The binary mean shape is de?ned as:

v q epT?

à1if qepTP0

t1if qepT<0

8

><

>:e2T

Let C be the set of the curve points of the binary mean shape v q(p). The reference l is computed by determining distances to the contour:

lepT?v

q epT?inf

q2C

j pàq je3T

Individual shapes are centered:

M?eZ:;1àlTáááeZ:;nàlTe4TIn this way,PCA is performed in the space of shape variations.The variability in shape is captured using a https://www.sodocs.net/doc/a910905784.html,ing Singular Value

Decomposition(SVD),the covariance matrix,de?ned by1

n MM T,is

decomposed to?nd orthogonal modes of shape variation and corre-sponding singular values:

1

n

MM T?U R U Te5Twhere U is a matrix whose column vectors represent the set of orthogonal modes of shape variation,namely eigenvectors,and R is a n?n diagonal matrix of corresponding singular values,or eigenvalues.Note that dimension of1MM T is large and belong num-ber of pixels,H.W?e H.W.Eigenvectors and eigenvalues computa-tion of this matrix is computationally expensive.A solution exists to deal with large sample size in PCA[38].

In the following,we consider eigenvalues and eigenvectors have been sorted according to size of eigenvalues.Let k6n be the num-ber of modes to consider,which de?nes the amount of retained shape variation.Let z be a novel shape of the same class of object,where a i are obtained with:

a?U T

k

ezàlTe7T

with U T

k

being a matrix consisting of the?rst k columns of U that is used to project z into the subspace.

2.2.Shape model comparison

An empirical comparison to illustrate the difference between both representations has been carried out on a dataset of12jet images of size114?114[37](Fig.2).All images are?rst aligned to the one of them.Both models are then constructed from the aligned images,the PDM being computed on‘landmarks.The superposition of both mean shapes and their two principal modes of variation is shown in Fig.3,that illustrates a similarity of the both mean shapes.Both models do not represent the shape vari-ability in the same space but axes of variation look similar.

In the following,we try to quantify the difference between both models,when used to describe a novel shape instance.Each shape of the jet dataset is being reconstructed with both a SDF and a PDM representation,on a leave-one-out basis:nà1shapes are used to build both shape models and the last one being used for estima-tion.Note that the PDM consists of‘corresponding landmarks on easily identi?able points.Two standard metrics are then com-puted to compare the estimated shape and the real one:(i)the Dice coef?cient DM(A,B),an overlap measure between two shapes A and B de?ned by:

DMeA;BT?

2j A\B j

e8T

and(ii)the average point-to-curve(P2C)error between two con-tours,de?ned by:

P2CeA;BT?

1

j A j

X

a2A

min

b2B

dea;bTe9T

where j A j denotes the number of points of contour A.Results,shown in Table1,provide the overlap rates and P2C error in pixels with a varying number of points‘and a varying number of eigenvectors k used for the reconstruction.Not surprisingly,reconstruction errors decrease as k increases.Note as well that the PDM seems to capture details slightly better than the SDF representation,whatever the

Fig.2.Binary database of12aligned jet images(from[37]).

?ghter jet when using the SDF and the PDM.(a)Overlap image,(b)±2r variations of the?rst principal mode,(c)

common to both the SDF and the PDM representation.The red part is part of the PDM only(‘=37).The green part

(For interpretation of the references to colour in this?gure legend,the reader is referred to the web version of this

D.Grosgeorge et al./Computer Vision and Image Understanding117(2013)1027–10351029

correct reconstruction of shape can be obtained,without the need for a costly landmark detection and matching process.In summary,the SDF representation has the following advantages over the PDM representation:

It does not require the positioning of several landmark points,a process prone to variability if manual,and prone to detection error if automatic;

It does not require landmark matching,a prerequisite before shape analysis,that is dif?cult to establish;

It it more robust than the PDM to initial misalignment of the shapes [16].

This SDF representation is thus retained in our segmentation method,described in the next section.3.Graph cut segmentation with shape prior

In this section,we ?rst outline the graph cut segmentation framework as described in [1].Then,we introduce the construction of the shape model based on a PCA and show how it can be inte-grated into the graph cut cost function.3.1.Graph cut segmentation

Let us consider the image I as a graph G ?hV ;Ei ,where V is the set of nodes (pixels)and E the set of edges.Each pair of nodes ep ;q T2E in a neighborhood N is connected by a segment called n-link and weighted by B p ,q ,a regularization or boundary term,de-signed to provide spatial coherence in a neighborhood of pixels.B p ,q is typically de?ned as:

B p ;q /exp à

eI p àI q T2

r

!:

1e10T

where I p and I q are the gray levels of pixels p and q ,dist (p ,q )the Euclidean distance between p and q ,and r a constant usually

related to acquisition noise.Consider two additional nodes,called terminal nodes:the source S representing the object O (in our case,the RV cavity),and the sink T representing the background B .Each node p 2V is connected to the terminal nodes S and T by two respective segments called t-links and weighted by the so-called re-gion term,denoted R p and de?ned by:

R p ex T?àln Pr eI p j x T

e11T

where Pr (I p j x )is the likelihood of observing I p given that pixel p be-longs to class x .

A cut C in the graph consists in cutting t-links and n-links to attribute a label O or

B to each pixel p of the image,which boils down to segmenting the image.The energy of a cut

C is de?ned by:

E eCT?

X p 2V

R p ex p Ttk X p ;q 2N

B p ;q :d ex p –x q T

e12T

where d (x p –x q )is 0if p and q have the same label,1otherwise.

The optimal segmentation is obtained by searching for the cut of minimal energy.This global search can be very ef?ciently per-formed due to mincut-max?ow algorithms,in polynomial time [2].3.2.Adding the shape prior into the graph cost function

To guide the segmentation process,constraints or models for the object can be introduced through an additional term in the en-ergy formulation of Eq.(12).How to incorporate this prior informa-tion depends on the available information:either the constraints are weak and are simple assumptions about the general shape of the object (e.g.convex)or the constraints are strong and concern a precise shape to ?nd in the image.

Weak constraints.In the literature,the constraints on rough shapes are speci?ed through the n-links ,changing the labeling of the neighboring pixels under the assumption made.In [9],the val-ues of the boundary energy B p ,q are modi?ed by prohibiting certain relative positions of p and q ,thus promoting compact shapes.The same methodology is used in [41]for more general shapes than convex shapes:it requires that if C is the center of the shape and p a point in the shape,any point q on the line (C ,p )after p be also inside the shape.Note that this method has an interesting effect countering shrinkage typically seen in the graph cut segmentation,but implies signi?cant discretization problems.Requiring that the result of segmentation is convex can also be done through an addi-tional energy term as 1àcos (a ),where a is the angle between (p ,q )and (p ,C )where C is the center of the object designated by a user click [12].This shows how important angles are penalized by high values of energy,encouraging roughly convex cuts in the graph (Fig.5).

Strong constraints.When a model of the object is available,it Table 1

Mean Dice metric (DM)and P2C (P2C)errors (in pixels)between reconstructed shape and real shape for both the PDM and the SDF representations.DM varies from 0(total mismatch)to 1(perfect match).‘is the number of points of the PDM.k is the number of modes considered for reconstruction.

k =3

k =6k =10DM

SDF

0.89±0.040.91±0.020.93±0.02PDM ‘=160.92±0.030.94±0.020.96±0.01constraint imposed on the graph allowing the contour (red)to interpretation of the references referred to the web version of this Two sample shapes reconstructed with a PDM (red curve)and a SDF This ?gure is best viewed in color.(For interpretation of the references in this ?gure legend,the reader is referred to the web version of this 1030 D.Grosgeorge et al./Computer Vision and Image Understanding 117(2013)1027–1035

object by labels of the shape prior [35]or a map of probability [36,21],in which case the energy associated with the prior takes the form:E s ex p T?àln ePr A ex p TT

where Pr A (x p )is the probability of a pixel p belongs to the class x p depending on the model.Note that the use of a shape prior implies a dif?cult problem of matching the model with the image.The regis-tration can be done iteratively and therefore computationaly expen-sive:the pose parameters estimation and the segmentation calculation are alternated [45,46,42,21].When registration is a prequisite for the segmentation,process is based on a user interac-tion [11,36].The model is de?ned in this case by a distance map or an atlas.Note that these models are limited to represent judiciously For each binary shape,a signed distance map /i to the RV contour is computed.Shapes are rigidly aligned on an arbitrary reference

shape and averaged into a mean shape U

(Fig.7a): U ?

1N X N

i ?1/i

e13T

Since averaging does not ensure to obtain a distance function,

we propose to reset U

to a SDF to the RV contour.A PCA is then performed on the set of centered shapes and yields eigenshapes denoted by U i ,with i =1...N ,and their associated eigenvalues,de-noted k i [37].A number k 6N of eigenshapes is retained,with k chosen large enough to account for the most important shape vari-ations present in the training set.

steps of the prior map.(a)The binary mean shape,(b)and (c)extremal areas of variation of the mean shape for the second axis,axis,(e)–(g)distances to the mean shape for the 3?rst axis,(h)?nal prior map (the darker the higher the distance).This ?gure Fig.6.Overview of the proposed method with shape prior.For the prior map,the darker the higher the distance.

D.Grosgeorge et al./Computer Vision and Image Understanding 117(2013)1027–10351031

where HeáTis the Heaviside function.C i is a binary map that con-tains areas of variation of the mean shape,for eigenmode i (Fig.7d).This map is superimposed to the distance values of the

mean shape U

(Fig.7e and g):PM i ep T?C i ep Tá U

;for all i ?1...k e16T

The k distance maps are averaged into a single distance map

(Fig.7h):

P S ep T?1X

k i ?1

PM i ep T

e17T

P S includes a distance based region where the contour is ex-pected to be and the complementary region ?lled with null values.Now how can this prior map be integrated into the graph cut framework?In the literature,additional energy terms on the t-links [36,21]or the n-links [11]are added to the graph cost function.In any case,the shape prior must be rigidly registered onto the image to be segmented (see Section 4).We suggest that the shape prior contribute to weighting both t-links and n-links .The region term R p can straightforwardly be de?ned with:

R S p eOT?àln ePr eOj I p TTif P S ep T–0

t1if P S ep T?0and He U

ep TT?1eBackground T0if P S ep T?0and He U

ep TT?0eObject T8><>:e18T

R S p eBT?àln e1àPr eOj I p TTif P S ep T–00if P S ep T?0and He U

ep TT?1eBackground Tt1if P S ep T?0and He U

ep TT?0eObject T8><>:e19T

with Pr eOj I p Ta posterior probability model computed from the im-age gray level pixels p such as P S ep T?0and He U

ep TT?0.Our shape prior is contour-based and may be added as a prior term weighting n àlinks .We thus propose to add a new frontier term denoted by B S p ;q and de?ned by:

B S p ;q ?

P S ep TtP S eq T

e20T

The ?nal energy of a cut C for a graph integrating a shape prior is then:

E eCT?k

X p ;q 2N

B p ;q tc B S p ;q

:d ex p –x q TtX p 2V

R S

p ex p Te21T

where B p ,q is de?ned with Eq.(10),k weights the relative contribu-tions of the n àlink and t àlink terms and c weights the frontier

shape prior term B S p ;q and the image frontier term B p ,q .

4.Experimental results 4.1.Cardiac MR images

Our method has been applied to the datasets of the MICCAI

2012Cardiac MR Right Ventricle Segmentation Challenge (RVSC).1Images have been collected at Rouen University Hospi-tal,as part of the clinical routine.The image database includes 491short-axis MR images acquired on 32patients with diag-nosed pathologies,who gave written informed consent.For each patient,two volumes a total of 16images (in average)are avail-able at two time points:9images (or slices)at End Diastole (ED,time of maximum ?lling)and 7images at End Systole (ES,time of greatest contraction).Training set and Test1set include both 16patients.Cardiac images have been zoomed and cropped to a 256?216(or 216?256)pixel ROI,leaving the LV visible for joint ventricle segmentation,if necessary.

Manual segmentation of the endocardium was performed by a cardiac radiologist,with the convention that trabeculae and papil-lary muscles were included in the RV cavity.For more information on the data,please refer to [28].

4.2.Shape model construction and method parameterization The shape models are built based exclusively on the Training

Set.There is one model per time point (ED or ES),and several models describing the slice level from base to apex:6for ED,5for ES.Each of the 11PCA is performed using between 16and 32images.Since size of each eigenvalue indicates the amount of importance its corresponding eigenvector has in determining the shape,we have empirically chosen k to keep 99%of mean shape variations.This corresponds to 7–10eigenvectors depend-ing on the slice level.Preliminary registration is performed by manually positioning two anatomical landmarks on the inter-ventricular septum (Fig.6).We have chosen our landmarks based on two criteria,(i)a minimal user interaction,(ii)easily identi?able anatomical landmarks by expert.Based on these cri-teria,the choice of two points on the junction of the septum to register the model on the 2D images seems to be consistent.Parameters are derived empirically from the training set of 16patients:r =10and k =100for both ED and ES,c =0.001for ED and c =0.005for ES.The implementation of Boykov and Kol-mogorov of the mincut-max?ow algorithm 2is used to compute the cut of minimal cost in the graph [2]

.

the user for object (green)and background (red),used for original graph cut method and Freedman and Zhang’s method.?gure legend,the reader is referred to the web version of this article.)

1http://www.litislab.eu/rvsc .

2

Available online at http://pub.ist.ac.at/$vnk/software.html .

4.3.Segmentation results

Our segmentation algorithm is run on the Test1Set which con-tains16unseen patients.Our method is compared to one of the ?rst methods for graph cut segmentation with shape prior,as pro-posed by Freedman and Zhang[11].Their method consists of using a single shape template described by an unsigned distance map /. We have also compared our method to the original graph cut[1].

Table2

Mean(±standard deviation)Dice Metric(DM)and Point to Curve Distance(P2C,in mm)between automatic and manual delineation of the contour at ED and ES from base(B)to Apex(A)for endocardium.

Our method Freedman’s method[11]Original graph cut[1]

Dice P2C(mm)Dice P2C(mm)Dice P2C(mm)

B0.91±0.09 2.25±1.860.87±0.12 4.37±4.120.86±0.1210.99±6.78

0.90±0.10 2.31±1.760.90±0.09 3.80±3.490.88±0.0910.34±8.12

0.88±0.12 2.11±1.800.80±0.187.55±7.340.77±0.1915.29±10.61

E0.83±0.10 2.55±1.280.75±0.199.87±9.480.71±0.2020.01±12.40 D0.81±0.12 2.39±1.390.66±0.2312.73±11.260.61±0.2423.62±13.54 A0.70±0.18 2.27±1.350.56±0.2112.24±11.140.48±0.2229.62±14.29

Mean0.83±0.15 2.32±1.570.74±0.228.77±9.370.70±0.2419.22±13.67

B0.84±0.14 2.89±2.460.83±0.127.12±6.680.83±0.0816.97±9.84

0.82±0.15 2.85±1.670.75±0.179.06±8.240.74±0.2116.47±9.77

0.73±0.19 3.59±1.980.65±0.249.67±9.600.64±0.2318.07±11.62

E0.66±0.19 2.96±1.280.56±0.2111.62±10.010.50±0.2123.59±13.20 S A0.52±0.21 2.86±1.570.39±0.2219.30±18.790.40±0.1632.30±16.43 Mean0.70±0.22 3.05±1.820.61±0.2512.00±12.980.60±0.2422.26±14.28

Bold values indicate best values for each column.

obtained with automatic algorithm(green)and manual ground truth(red)for Patient20of Test1at ED(from base

Zhang’s method[11],bottom:original Graph Cut[1].(For interpretation of the references to colour in this?gure legend,

D.Grosgeorge et al./Computer Vision and Image Understanding117(2013)1027–10351033

Note that both methods require object and background seeds for shape model matching(only in[11]),object and background gray level modeling,and are used as hard constraints in the graph.In average,5landmarks have been used for the RV cavity and10land-marks for the background(see Fig.8).Note that parameters have been set empirically from the training set.

For each image of each patient,the user is required to point out two landmarks in order to register the shape prior for our method. Segmentation results for each method are compared to manual ground truth through the Dice Metric(DM)(see Eq.(8))and the Point to Curve Distance(P2C)(see Eq.(9)).Results are provided in Table2and examples of segmentation are shown in Fig.9.

Accurate segmentation results would be expected to reach in-tra and inter-observer variability of manual segmentation,that is in the range1–2mm[22,26,20,39].3Our method yields encourag-ing results which compare favorably to a state of the art method and outperform the original graph cut approach.Our method pro-vides better results than Freedman and Zhang’s,especially in api-cal images,which are much more dif?cult to segment(see Fig.9): they exhibit small structures and are often fuzzy(due to partial volume effect).Freedman and Zhang’s prior model might not be speci?c enough for this kind of images.Regarding basal slices,re-sults are comparable between our method and Freedman and Zhang’s,at the cost of substantial user interaction for Freedman and Zhang’s method.

Not surprisingly,segmentation results are better for ED images than for ES images,for the three methods:ED images are easier to process,as the heart is then the most dilated.Results are also bet-ter for basal and mid-ventricular slices than apical ones.A poor segmentation result in apical slices has little in?uence on the vol-ume computation but can be a limiting factor in other?elds such as studies on the?ber structure.To a lesser extent,this is also true for ES images.Regarding computational cost,our algorithm is implemented in C++without any particular optimization and re-quires about45s by patient(including both ED and ES volumes) on a Dell E6510laptop with4Go RAM and Intel(R)Core(TM)i7 CPU,M460@ 2.80GHz.This time is compatible with clinical practice.

5.Conclusion and perspectives

In this paper,we have presented a graph cut based method to segment the RV,using a shape prior.The shape model is con-structed via a PCA from a set of representative shapes of the RV ob-tained by manual segmentation.An original prior term is then integrated into the graphcut cost function.Our segmentation method has been validated on248images acquired on16patients. It is shown to outperform the native graph cut approach and to compare favorably to the state-of-the-art method.Note however that if results are satisfying for basal and mid-ventricular slices, room for improvement is left in apical slices.

Further investigation concerns a possible3D extension of our algorithm.Even if our method is scalable to3D from a theoretical and a memory consuming point of views,a problem arises from the MR clinical data:the space between slices is quite large,typically 8.4mm,as compared to a spatial resolution of0.7mm per pixel in the short-axis view.This raises the general question of the cre-ation of a3D neighborhood when resolution is anisotropic.One possible solution could be to use a hierarchical approach to de-crease time and memory consumption,such as in[18]where a multilevel banded graph cut method is presented.Further investi-gation will also concern making the registration step automatic.Acknowledgments

The authors thank Dr.Jér?me Caudron for providing the images and for fruitful discussions.They also would the anonymous reviewers whose insightful comments helped to deeply improve the manuscript.

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