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Automated segmentation of various anatomical structures of interest from medical images has been a well grounded field of research in medical imaging. One such problem is related to segmenting whole heart region from a sequence of magnetic resonance imaging (MRI), which is currently done manually, and is time consuming and tedious. Although many automated techniques exist for this, the task remains challenging due to the complex nature of the problem, partly because of low contrast between heart and nearby tissue. Moreover many of the methods are unable to incorporate prior information into the process. To this end, Pluempitiwiriyawej et al. proposed a version of active contour energy minimization based method to segment the whole heart region, including the epicardium, and the left and right ventricular endocardia. The proposed method follows the framework laid out by Chan and Vese\cite{Chan2001}. However Pluempitiwiriyawej et al. propose a modified energy function, which consists of four energy terms. The energy function is given below, where $C$ is the contour represented as a level set function $\phi(x,y)$: $J(C) = \lambda_1 J_1(C) + \lambda_2 J_2(C) + \lambda_3 J_3(C) + \lambda_4 J_4(C)$ The coefficients $\lambda_{1..4}$ determine the weight of terms $J_{1..4}$. The first term $J_1(C)$ is designed to add stochastic models $\mathcal{M}_1, \mathcal{M}_2$ corresponding to the regions inside and outside of the active contour $C$. The models dictate the probability distribution from which the image intensities making up the inside and outside region of the contour are sampled. The negative log of this term is minimized, which essentially maximizes the probability $p(u  C, \mathcal{M}_1, \mathcal{M}_2)$ given the active contour $C$, and the models $\mathcal{M}_1, \mathcal{M}_2$. The second term $J_2(C)$ is designed similar to the classical Snakes\cite{Kass1988} in the sense that it uses edges to guide the contour towards the structure of interest. For this term, a simple edge map is used after convolving with a Gaussian filter which smooths out the noise. The term $J_3(C)$ encodes an shape prior which constraints the contour to follow an elliptical shape, and guides it in conjunction with the region and edge information. The final term $J_4(C)$ which encodes the total Euclidean arc length of the contour. This forces the contour to be ``smooth", without rough edges. The process of minimizing the energy function follows a threetask approach. The first task is to estimate the stochastic model parameters $\mu_k, \sigma^2_k$, and is performed by fixing the position of initial contour $C$, taking derivatives of $J$ w.r.t stochastic model parameters, and solving by equating to zero. The second task estimates the parameters of the ellipse using least squares method. The third and final task involves the contour using the estimated parameters in task one and two, such that it minimizes the function $J$. The method also performs stochastic relaxation, by dynamically changing the values of parameters $\lambda_1, \lambda_2, \lambda_3, \lambda_4$ as the optimization process proceeds. The intuition is that when the optimization starts, the edge and region terms must guide the contour, and as the process proceeds to it's end, the shape prior and contour length term should carry more weight to regularize the effective shape of the contour. The study used 48 MRI studies acquired by imaging rat hearts, and compared the proposed method with two earlier methods, namely Xu and Prince's GVF \cite{ChenyangXu1998}, and Chan and Vese \cite{Chan2001}. The authors also design a new quantitative metric, which is a modification of the Chamfer matching \cite{Barrow} technique. The reported results are observed to be in excellent agreement with the gold standard handtraced contours. However the similarity values for other methods against human goldstandard were not reported.
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## **Introduction** This paper presents a stochastic active contour model for image segmentation from cardiac MR images. The proposed algorithms aims to minimize an energy functional by the level set method while incorporating stochastic region based and edge based information as well as shape priors of the heart and local contour properties. Moreover, the algorithm also uses a parameter annealing component to dynamically balance the weightage of the components of the energy functional. ## **Method** The paper locates a contour $C$ in a cardiac MR image that segments the image into two groups  the heart and the background. The corresponding objective energy functional can be represented by $$ J(C) = \lambda_1 J_1(C) + \lambda_2 J_2(C) + \lambda_3 J_3(C) + \lambda_4 J_4(C) $$ Looking at these individual components, we have * _Region Based Term: Model Matching_ **$J_1(\phi)$**: In order to segment the image into 2 regions, let the two regions inside and outside the contour C be represented by $\Omega_1$ and $\Omega_2$ respectively. For each region, consider a stochastic model to describe the pixel statistics of that region. Assuming that the pixel intensities of all the pixels in each region are statistically independent, the objective is to minimize the negative loglikelihood of pixels belonging to the correct regions. * _Edge Based Term_ **$J_2(\phi)$**: In order for the contour C to be aligned to the prominent edges in the image, the edge map of the image (which can be obtained by various image smoothing methods such as Gaussian kernel blurring, edgepreserving anisotropic diffusions, Min/Max flow algorithms, etc.) has to be minimized. * _Heart Shape Prior Term_ **$J_3(\phi)$**: In order to distinguish between similar looking tissues in the foreground and the background, an elliptical heart shaped prior is used. An ellipse can be described with 5 parameters with certain constraints in the conic equation. * _Contour Smoothing Term_ **$J_4(\phi)$**: In order to obtain a smooth contour of the segmented heart, the total Euclidean arc length of the contour C should be minimized. The parameters ($\lambda_1$, $\lambda_2$, and $\lambda_3$) of the energy functional $J(C)$ need to be dynamically updated during the energy minimization. For example, the regionbased and the edgebased terms should have a higher weightage in $J(C)$ during the initial steps of the segmentation, and at the later stages, their weightage should be reduced and that of the shape prior should be increased in order to keep the segmented output similar to the desired shape. ## **Results** The two metrics used for assessing the performance were Area Similarity and Shape Similarity. The algorithm was tested on 48 images covering 143 contours, including manually annotated contours by an expert on six rat cardiac sequences of eight frames each, and the results indicated excellent segmentation agreement with the manually traced contours. ## **Discussion and Shortcomings** Since STACS uses stochastic models instead of deterministic models, it can be applied to a large variety of images, and is especially helpful when distinguishing between visually similar adjacent regions. Since STACS incorporates both regionbased and edgebased information in its energy functional, this makes it more robust to noise as well as reduces the susceptibility to curve initialization. Perhaps the most highlighting feature of STACS that distinguishes it from other active contour based models is that it incorporates shape based priors into the energy functional. This helps segment the heart from the chest wall, which is especially difficult since the two regions share similar texture. Moreover, the scheduled parameter annealing adjusts the weights of the components of the energy functional, which helps dynamically vary the importance to different components at different stages of the segmentation process. 