The probabilistic atlases and the SSMs are constructed from training images, in which target organs are manually labeled. Here, a multi-atlas-based method that uses training images more directly for the segmentation of given images is described [134-137, 137, 138, 138-141].

A multi-atlas-based method segments target organs from given images as follows: First, each of the training images is nonrigidly registered to a given target image. Let P(x) (j = 1,2, ... ,M) denote the training images and R denote the region of a target organ in I‘(x) and b(x) denote binary label images where

and letting the given target image be denoted by Ig(x), the nonrigid registration deforms Г(x) so that it is aligned to the given image. Letting a two-vector, u^{1}(x), denote the image coordinates obtained by the registration where Г (u^{1} (x)) and Ig(x) are aligned, the deformed image is represented by Г(u^{1}(x)), and b^{1}(u(x)) represents the deformed binary labeled image. Each of the deformed binary labeled images, b^{1}(u(x)), is an estimate of the region of a target organ in /_{tgt}(x).

The multi-atlas-based method, then, computes the absolute difference between Г(u^{1}(x)) and hgt(x): D^{1}(x) = Г(u(x)) — /_{tgt}(x), and a weight, X(x), that is a

decreasing function of D^{1} (x), such that

where g(xa^{2}) = N(x, E^{2}) is a Gaussian filter of which variance is a^{2} and e > 0. The regions indicated by the deformed binary labeled images would be more reliable and would have larger values of Xx) if the difference, D^{1} (x), were smaller.

Finally, the method computes a weighted average of the deformed labeled images, b^{1} (u^{1} (x)) ( i = 1,2,..., M), as follows and classifies voxels as the inside of the organ if the average is larger than 1/2:

i

where Z(x) = P^{M}=1 X^{1} (x).

The algorithm described here can be interpreted as a MAP estimation of the region, as will be described below. It should be remembered that the singleatlas-based method also determines the region of a target organ by means of MAP estimation as shown in (2.172). The difference between the multi-atlas-based method and the single-atlas method is the model used for computing the posterior probability distribution.

For the interpretation of the multi-atlas-based method, let us introduce a latent variable, в, that represents the shape of the target organ in an image. For example, the shape of the organ in b^{i}(x) is explicitly indicated as Ъ^{1}(хв^{i}). As described in (2.175), a single probabilistic atlas can be estimated by averaging all of the binary labeled images, b'{x). That computation of the average is nothing but the marginalization of the binary labeled images, Ь.хв), over the latent shape parameters as follows:

In the single-atlas-based method, the posterior probability distribution of each voxel being included in the target region is evaluated by multiplying the prior in (2.217) by the likelihood which is defined based only on the voxel value; no information on the organ shape is used for computing the likelihood. In the multi-atlas-based method, the differences between the shapes are more explicitly considered in the computation.

For a simple example, let again b_{t}(x) denote a binary image, where b_{t}(x) = 1 if x e R_{t} and otherwise b_{t}(x) = 0. Using the latent shape parameter, the posterior probability distribution can be represented thus:

Because p(bt(x)I_{XgX}, в) / p(I_{XgX}t(x), в)p(bt(x)e), р.в I_{tg}t), p(b_{x}(x)e), and p(hgt_{t}(x), в) need to be estimated for computing the posterior probability in (2.218). The multi-atlas-based method estimates р.в I_{tgt}) and p(b_{t}(x)e) by the nonrigid registration between I^{i}(x) and I_{tgt}.

The nonrigid registration computes the transformation u(x) so that the two input images align. Once the transformation is obtained, the deformed binary labeled image, b^{i}(u(x)), should be an estimate of the region of the target organ in the target image, I_{tgt}(x). Let the true and unknown shape parameters of the organ region in

the given image be denoted by в_{tgt} and let в^{i} and в [_{gt} denote the shape parameters representing the organs in b^{i}(x) and in b^{i}(u(x)), respectively, в^{i} (i = 1,2, ... ,M) obey p^), and в[_{gt}, which are the estimates of в_{tgt}, obey the posterior probability

distribution, p^ I_{tgt}). In addition, from the definition of в [_{gt}, the following equation can result:

The last factor to be estimated is p(I_{tgt}(x)_{t}(x), в). As defined above, the label, b_{t}(x), is uniquely determined by the shape parameter, в, and hence p(I_{tgt}(x)_{t}(x), в) = p(I_{tgt}(x)e). When в = вt_{gt}, it is natural to assume that the probability of the voxel value I(x) being equal to I_{tgt}(x) decreases as the absolute

difference, D(x) = |T(u(x)) — I_{tgt}(x), increases because I'(u(x)) = r(u(x)(°t_{gt}). Following (2.215), assume that the following equation holds:

From the discussion above, the following equation results:

where Z is a normalization term. As shown in (2.216), S(x) represents the posterior probability distribution, and MAP estimates can be obtained by binarizing S (x) with a threshold 1/2.