SSMs

SSMs represent the prior distributions of the boundary shapes of target organs. An SSM is constructed from a set of training data, which are the descriptions of the target organs in the training images. Depending on the representation employed for the descriptions, different SSMs may result from identical training data. In this section, MAP estimation approaches of the registration are described for some of the representations.

SSM with Level-Set Representation

A level-set function, ft(x), represents organ regions in an implicit fashion [124126], and SSMs based on the level-set functions [126, 127] represent the statistical varieties of the shapes of the target organs with probability distributions of the values of ft(x). The probability distributions are estimated from sets of the training data. Let Rt denote the region of an organ, where t (t = 1,2,...,N) denotes the label of organs. The region, Rt, can be represented by a level-set function, ft(x), as follows:

Given a region Rt in an image, such a level-set function (2.179) can be constructed by computing a signed distance function as described in [58]. For the construction of the statistical models, the level-set function defined in an image space is converted to a vector. Letting the number of voxels in an image be denoted by W, and letting the coordinates of the k-th voxel (k = 1,2 , ..., W) be denoted by xk, then the level-set function, ft(x), can be represented using a W-vector, ft such that ft = [ft(x1) ,ft(x2), ..., ft(xW)]r. As described above, an SSM of a target organ is constructed from a set of training data. Letting a region of the target organ in the i-th training image, T(x), be denoted by R‘t (i = 1,2 , ..., M) and letting the corresponding level-set representation be denoted by fti, applying a PCA to the training set, Dt = fftji = 1,2 , ..., M} allows construction of a linear SSM as follows (see also Fig. 2.20):

An example of a SSM for level-set representation of the spline

Fig. 2.20 An example of a SSM for level-set representation of the spline

Let the eigenvalues obtained by applying PCA to Dt be denoted by A(1) > A(2) > ... > At(W), which are in decreasing order, and let the corresponding eigenvectors be denoted by u(tx) (s = 1,2,..., W). The following linear model is widely employed for an SSM:

where ft is a mean vector of ft (i = 1,2,..., M), Tt < W is a positive integer, and в t = [91,9j,..., 9f‘ ]T is a Tt-vector of the shape parameters.

As described in Sect. 2.2.3, the prior distribution of the shape parameters can be approximated using a multivariable Gaussian function such that

The covariance matrix of the Gaussian distribution in (2.181) is diagonal, and the diagonal components consist of the eigenvalues, A(t). The number of eigenvectors, Tt, in (2.180) can be determined as the minimum integer that satisfies ETLi Astl sWDi AS > yt, where yt is a threshold less than one (1), e.g., yt =

0.95. The linear model (2.180) constrains the level-set representation, ft, to an Tt- dimensional subspace.

Given an image, the region of the target organ can be segmented by estimating the values of вt in (2.180), and the estimates can be obtained by maximizing the posterior probability distribution. Letting the MAP estimates be denoted by в, then,

where I(x) denotes a given image. It is difficult, though, to solve the problem (2.182) directly.

To make the problem tractable, a latent variable, bt(x), is introduced, where bt(x) = 1 if x e Rt and bt(x) = 0 if x ? Rt. If not only an image, I, but also bt(x) are measured, then the problem of the estimation of вt based on p(et, bt(x)|I(x)) is tractable. An EM algorithm is useful for the estimation.

In the EM algorithm, the parameters, в, are estimated by maximizing the following MAP estimation problem [128]:

where b 2 {0,1}. The function to be maximized is the marginal distribution of logр.вt, bt(xk)I(x)) with respect to bt(xk). It should be noted that

logp(01, bt(x) = bI(x)) / logp(It(x) = b, вt)Clogp(bt(x) = b/)Clogр(в/),

(2.185)

and that p(I(x)t(x) = b, вt) = p(I(x)t(x) = b). Here, p(I(x)t(x) = 0)

and p(I(x)t(x) = 1) can be evaluated based on the histograms of the voxel values observed outside of Rt and inside of Rt in training images, respectively. p(bt(xk) = be) in (2.185) can be approximated by using ft(в). Let p(bt(xk) = 1) and p(bt(xk) = 0) be abbreviated by pin and pout, respectively. Then, pin C pout = 1 holds. For the evaluation of the values of the probabilities, pin and pout, an inverse of a logit function [128], whose definition is described below, is often introduced. A logit function of pin is defined as follows:

The inverse of the logit function is given as follows: where

It should be noted that 0 < pin, pout < 1 and pin C pout = 1 hold and that pin = pout = 1/2 when gin = 0. Now the values of pin and pout can be evaluated by using the level-set function, ft(e), as

At the boundary of the region where ft(xe) = 0, pin = pout = 1/2 holds and pin monotonically increases as ft increases.

The MAP estimates shown in (2.182) can be obtained by an EM algorithm as follows:

1. Set the initial value of the parameters: в t = в °LD

2. E-STEP: Compute wk(b) = p(bt(xk) = bI, в°LD) as follows:

3. M-STEP: Solve the following problem and obtain вNEW:

4. в OLD -- вNEW and back to E-STEP if it is not converged.

The target function to be maximized shown in (2.192) can be rewritten as follows:

The second term in the right side of (2.193), р.вt), is the prior probability distribution of в t and is given as (2.181).

In the process of the estimation of в , no candidate points of the boundaries of target organs are explicitly extracted from given images; hence the distances between the region boundary represented by the model and the image points extracted as the candidates of the region boundary are not explicitly measured at all. The fit between the region represented by the implicit function and the given images is evaluated based on the differences in voxel values between the interior and the exterior of each target organ. This methodology is quite different from that employed for SSMs with explicit representation of region boundaries, which will be described below.

 
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