Computational Anatomical Model Hidekata Hontani, Yasushi Hirano, Xiao Dong, Akinobu Shimiz, and Shohei Hanaoka

Models for Segmentation

One of the main objectives in medical image analysis is to segment given medical images into organs. One can segment the images by determining boundaries of the organs in the images. Organ boundaries mainly consist of edges. It is pointed out in [50] that the concept of determining edges in given images is an ill-posed problem. A problem is well posed if the following three conditions are satisfied:

  • 1. The solution exists,
  • 2. The solution is unique, and
  • 3. The solution depends continuously on the initial data.

One can convert ill-posed problems to well-posed ones by introducing some prior knowledge of the problems in order to restrict the class of admissible solutions. A major technique for the restriction is regularization [50, 51].

Assuming that the objective is to detect the boundary of a target organ in a given image, let D denote a set of edge points detected from the given image as the candidates for points on the boundary, and let the parameter of a model representing the surface be denoted by в. LData(e |D) describes a cost function that defines the error between the edge points, D, and the model surface в. In general, edge detection often fails to detect some portions of the boundary and detects many false candidates. It is hence very difficult to design a cost function, LData(e |D), such that the boundary of the target organ is delineated correctly only by minimizing it.

In the regularization technique, prior knowledge introduces some regularity of the sizes and shapes of surfaces by adding new terms, regularization terms, to the cost function, allowing detection of the boundary by minimizing the new cost function, Ь(в; D) = LData(e; D) + ALreg), where A controls the compromise between the strength of the regularization and the closeness of the model to the data. For example, an Active Contour Model (ACM) detects contours in given images by minimizing a cost function with regularization terms that restrict the curves represented by the model to short and smooth ones [52]. The regularization terms are manually determined based on geometrical features of the boundaries of target regions, and the regularization techniques work well only if all of the boundaries have common geometric features that can be represented by the regularization terms. It is difficult in general to find such features and to determine appropriately the regularization terms.

More powerful strategies convert ill-posed problems to well-posed ones by introducing a framework of the Bayes’ estimation or of the maximum a posteriori

(MAP) estimation [53, 54], in which the class of admissible solutions is restricted by prior probability distributions of targets. In medical image analysis, the regions or their boundaries are determined by maximizing the posterior probability distributions which consist of the prior distributions and the likelihood distributions. Assuming again that the objective is to detect the boundary surface of a target organ in a given image and letting the posterior probability distribution be denoted by р.в D), in the framework of the MAP estimation, the boundary can be detected by maximizing p(e D). As described in ***, p(e D) a p.O)p(De), where p.O) is the prior probability distribution of the model parameters and p(De) denotes the likelihood of the parameters with respect to the given data, D. The prior distribution, p.e), is introduced to restrict the admissible solutions and can be constructed by learning from sets of training data. Assuming that these probability distributions can be represented using exponential functions such as p{6) = exp(Fpri(Aв)) and p(D) = exp(Flike(e; D)), the posterior probability distribution can be maximized by maximizing logp(e D) = Flike(в; D) + AFpri(e), the form of which is analogous to the cost function with the regularization terms.

There are many models that represent regions or boundaries in medical images: For example, regions may be represented by labeling voxels in the images, and boundaries may be represented by using parametric surface models. The statistical models of organs represent the varieties observed among patients. Even when a single set of training data is given for constructing an SSM, the resultant SSM would vary with respect to the representation of the regions or of the boundaries employed, and the performance of the image segmentation would largely depend on the SSM. In the following section, the representation and the corresponding segmentation methods will be described.

 
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