## **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 log-likelihood  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, edge-preserving 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 region-based and the edge-based 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 region-based and edge-based 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.