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 three-task 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 hand-traced contours. However the similarity values for other methods against human gold-standard were not reported.