Letting the regions of target organs in given images be denoted by Rt .t = 1,2,...,N/, where t denotes an index number or a label of each organ and N is the number of the organs to be detected, allows determination of the regions, Rt, in a given image, I(x) : R3 ! R, by using a classifier that judges if each voxel in the image is inside of the t-th organ or not. The optimal classifier, which minimizes the expected error ratio, decides that a voxel is inside the region, Rt, only when the following condition is satisfied (Sect. 2.2.3):
where p.x e RtI(x)) and p.x ? RtI(x)) denote the posterior probability distributions. These posterior probability distributions can be computed by multiplying the prior distributions by the likelihood distributions:
The prior probability distributions, p.x e Rt/, are called the probabilistic atlas of target organs [22, 121-123]. A probabilistic atlas of a target organ represents the probability of each voxel being inside of Rt and is constructed from a set of training images in which the target organs are manually labeled. Let M denote the number of the training images and let I‘(x) (i = 1,2, ...,M) denote the training images in which the location, the size, and the shape of the body are normalized. Let blt(x) denote labeled images corresponding to I‘(x) such that
where t = 1,2 , ... denotes the ID of a target organ and Rt denotes the region of the target organ in I‘(x). When the number of the training images, M, is large enough, the probabilistic atlas can be constructed as follows:
When M is not large enough, though, a probabilistic atlas estimated by (2.175) overfits to the training images. For example, it often happens that the estimated prior probability distributions have many zero values at inappropriate locations. One of the techniques for avoiding this over-fitting blurs b‘(x) and estimates p(x 2 Rt) as follows:
where v 2 R3 and w(v) > 0 is a unimodal weighting function for the spatial blurring that satisfies
For example, a Gaussian function can be used for w(v). Once p((x,y,z) 2 Rt) is estimated, the prior of the voxel being outside of Rt can be estimated as p((x, y, z) ? Rt) = 1 — p((x, y, z) 2 Rt). Figure 2.19 shows an example of a probabilistic atlas of the liver.
For computing the posterior probabilities in (2.172), not only the prior probability distribution but also the likelihood distributions are needed, p(I(x)|x 2 Rt) and p(I(x)|x ? Rt). Using these likelihoods, one can rewrite Eq. (2.172) as follows:
The models of the conditional probability distributions, p(I(x)|x 2 Rt) and p(I(x)|x ? Rt), are estimated from the set of training images. For example, let ht(Ijx 2 Rt) (ht(Ijx ? Rt)) denote the histogram of pixel values observed in (out
Fig. 2.19 An example of an input image and of the probabilistic atlas of the liver appeared in 
of) Rt .i = 1,2,... , M). Then, the conditional distributions can be represented using, e.g., Gaussian mixture models, of which parameters are estimated by fitting the models to the histograms by means of an expectation/maximization (EM) algorithm. Once the representations of the conditional distributions are obtained, the likelihoods in (2.178) can be computed straightforwardly.
The segmentation method described above determines whether each voxel is included in the region of a target organ or not based only on the location of each voxel and on its voxel value. No image features that are obtained by observing multiple voxels, e.g., the continuity of the regions or the shapes of the region boundaries, are referred to for the segmentation. As a result, the boundaries of the segmented regions can have shapes different from those of target organs. Hence, SSMs of organs that represent the shapes of target organs are often employed for obtaining regions consistent with the targets.