Biomedical Engineering Reference
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duced into variational segmentation frameworks [9, 10, 11, 12, 13]. The authors
of [9] propose an energy functional that is a piecewise constant Mumford-Shah
functional. This is equivalent to representing a region by its mean intensity value.
A voxel is assigned to a region if its intensity is close to the region's mean intensity
value. Without a curve length constraint, the method reduces to K-means cluster-
ing [14]. Another more general variational segmentation model is obtained from
observed data by the Maximum Likelihood principle [11, 12, 13]. According to the
assumptions that regions are independent, voxels follow identical an independent
distribution in a region, and given that the prior probability is uniform, it becomes
the Maximum Likelihood of observed data. In addition, edge information can be
incorporated into the model. The models employed in [9, 11, 12, 13] behave very
well when the noise level is not too high. However, when the image is corrupted
with prominent noise, the results are not satisfactory. In [13], the authors suggest
multiscale preprocessing to filter the noise and accelerate evolution of the hyper-
surface. This is equivalent to using a voxel's neighborhood weighted mean instead
of its own intensity. This method improves the results. However, it is ineffective
if regions have similar means but different variances. To solve this problem, more
information should be considered, such as prior shape knowledge. However, shape
prior data are not guaranteed to be easily available in all cases. Incorporation of
variance information is another natural way to deal with this issue [15, 16]. The
authors of [16] use average probability in the neighborhood of a voxel so that
second-order statistics are considered. Besson and Barlaud [15] propose a general
region-based variational segmentation framework where any information from the
region and boundary can be generalized as descriptors.
In this chapter, we propose an energy functional using voxels' neighborhood
information. The variational framework can be derived from the Minimum De-
scription Length (MDL) criterion, as in [16], or as a special case of region-based
segmentation using active contours [15]. The major contribution of the present
work is the formulation of an energy measuring the distribution discrepancy be-
tween local and global regions in the sense of J-divergence. With the Gaussian
distribution hypothesis, the proposed energy is formulated with local and global
region mean and variance, respectively.
It is worth noticing that in the work of [17], where segmentation is based on
region competition and implemented within a level set framework, it is deduced
that the average probability distribution function (PDF) is equivalent to Kullback-
Leibler (KL) divergence between the PDF of the local and global regions. Recently,
[18, 19] proposed a variational segmentation framework incorporating mean and
variance in the same manner as we do. However, as their application is to extract
objects from natural images, the distribution of regions is assumed to be a Gen-
eralized Laplacian. We make the assumption that each region follows a Gaussian
distribution. In addition, the dissimilarity between distributions in their work is
KL divergence, which is a directional but not symmetric form like ours; therefore,
their derived evolution equation is completely different. In this chapter we apply
