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Level Set with CA

This section describes a combination of level-set-based segmentation and an SSM of an organ, or a statistical model of a signed distance function, called a level-set distribution model (LSDM) in Sect. 2.3.2.

An important finding was made by Leventon [157], who introduced an LSDM into a level-set-based segmentation algorithm consisting of following steps: Letting a statistical shape variation of an organ be modeled by a principal component vector a under the assumption that the distribution is a Gaussian distribution:

the segmentation algorithm finds a set of shape parameter vector a and pose parameter vectorp using the following equation:

where P(a,pu, VI/ is a posterior probability of parameters of shape and pose given a boundary u and gradient image VI. In practice, the posterior probability is transformed using the formula for Bayes’ theorem and several assumptions on probabilistic distributions of parameters. The maximization is performed using a gradient ascent algorithm. Subsequently the estimated shape and pose parameters are incorporated into the following update equation of a geodesic active contour [158]:

where g is a stopping function based on image gradient, c is a constant value, and к is the curvature of a boundary u. The updated shape of the boundary u at time t + 1 can be computed from u(t) by

where u*(t) is the estimated boundary with parameter aMAP, pMAP and two parameters Ab A2 balance the influence of the gradient-curvature model and the shape model. The above process is repeated until convergence of deformation.

Figure 2.25 shows the segmentation results of corpora callosa from MR brain images. The algorithm was tested on unknown sections that were not used for the training. It is found from parts (a), (b), and (c) that the MAP estimator of shape and pose guides the model to the true boundary. Part (d) presents the result without the shape model, which failed in segmentation. In addition to this example, Leventon et al. showed results of applying the algorithm to 2D slices of MR images of femur

Shape-based segmentation that combines level-set method with a statistical shape model

Fig. 2.25 Shape-based segmentation that combines level-set method with a statistical shape model. The red curve is the zero level set of the evolving surface. The green curve is the next step in the curve evolution. The yellow curve is the MAP estimate of the position and shape of the curve. The cyan contour is the standard evolution without the shape influence

and corpus callosum as well as 3D CT volumes to segment vertebrae, all of which showed successful segmentation results.

An alternative approach was presented in the paper [125], in which LSDMs for multiple objects with neighbor constraints were employed and an energy functional including Mahalanobis distances computed in eigenshape spaces of the models was minimized. The proposed method was employed to extract left and right ventricles from a 2D cardiac MR image as well as eight subcortical structures in an MRI brain image.

In the paper [126], an implicit representation of the shape was combined with the Chan Vese region-based energy functionals [159]. After the training phase of a statistical shape model, the segmentation phase was carried out, in which the region- based functional is minimized. In practice, shape parameters for the eigenshapes and pose parameters to handle pose variations are iteratively updated to generate a new level set that determines the segmenting curve implicitly. The image statistics inside and outside the curve are used to compute the update function for the next iteration, and the iterative scheme is continued until convergence is reached for segmentation. The experimental results of left ventricle segmentation on cardiac MRI and prostate segmentation on pelvic MRI acquired with an endorectal coil proved that the proposed segmentation algorithm was computationally efficient and robust to noise.

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