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SSM with PDM (Subspace Representation)

PDMs represent surfaces of target organs using sets of points distributed on them. Letting the coordinates of the j-th point on a surface be denoted by Xj e R3 (j = 1,2, ... ,N), a PDM represents a surface with a 3N-vector,x = [x[x ...xtl]t, and a corresponding SSM represents the prior probability distribution of x. Combining the prior probability distribution with the likelihood distribution of X allows computation of the posterior probability distribution of X and registration of the models to given images by estimating the coordinates of each point, X;, by maximizing the posterior probability.

A prior probability distribution of x is estimated from a set of training data. Again letting S; (i = 1,2,..., M) denote surfaces manually extracted by experts from the i-th image, letting xj (j = 1,2 , ..., N) denote a set of N corresponding points generated on S;, and letting 3N-vectors x; = [(x1-^(x2)t ... (x;n)t]t (i = 1,2 , ..., M) denote a set of the generated corresponding points, the prior distribution of X can be estimated from the set, {x‘|i = 1,2 , ..., Mg. Letting the prior be denoted by

p(x), many PDM-based SSMs including the ASMs represent p(x) by degenerated Gaussian functions that constrain represented surfaces to subspaces. Letting the sample mean of the training data be denoted by X = (J2N=1 x')/M and letting the empirical covariance matrix of them be denoted by ?emp such that

a subspace for the representation is spanned by the eigenvectors of the empirical covariance matrix. Letting the eigenvalues of the covariance matrix, ^emp, be denoted by Xb X2,..., X3L, where they are in decreasing order, X1 > X2 > ..., and letting the corresponding eigenvectors be denoted by u1, u2,..., u3L, using the eigenvectors and the mean vector, the surfaces can be linearly represented as

where T denotes the number of the eigenvectors used for the representation and вк denotes a weight for each eigenvector. Equation (2.203) can be rewritten as follows:

where a 3N x T matrix, U, is composed of the eigenvectors as follows:

and a T-vector, в, denotes the shape parameters that controls the shapes, where в = [в12,... 6T ]T. The model can be registered by estimating the values of the shape parameters, в, in (2.204).

Assuming that x obeys a Gaussian and is constrained to the subspace shown in (2.204):

Then, the mean of the Gaussian satisfies X, д = X, and the covariance matrix, X'sub, in (2.206) is obtained as follows:


where ^eigen = diag (Xi, X2,..., XT/. It is often assumed that the shape parameters в in (2.204) obey a Gaussian distribution:

Visualization of the shape model appeared in [130]. Top row

Fig. 2.22 Visualization of the shape model appeared in [130]. Top row: Axial, sagittal, and coronal views of the mean shape, a vertebra. Second row: First eigenmode of the generated model visualized by an overlay of the mean shape and two deformed shapes according to the first eigenvector. Third row: Mean shape and deformed shapes according to the second mode

where I denotes a unit matrix. Figure 2.22 shows an example of an SSM of a lumbar vertebra represented by a PDM that appeared in [130]. The PDM-based SSMs are registered to given images by estimating the values of the shape parameters, в, so that every point on the surface locates on the boundary of a target organ. In the registration process, candidate points of the boundaries are first extracted from given images, and then a corresponding point, Xj, is selected for each of a point of the PDM, xj. The likelihood of x is computed based on the distances between the corresponding points, Xj (j = 1,2 ,..., L), and the model points, xj. Assuming that the residuals between Xj and the corresponding candidate points Xj obey an isotropic Gaussian such that

then the posterior probability distribution, p(xx1, xxN) can be computed as

Substituting (2.206) and (2.209)-(2.210) and computing the negative logarithm of the resultant posterior distribution result in the following quadratic cost function,


where x = [xT, xT,... , xN]r. The MAP estimate that maximizes the posterior probability in (2.210) can be obtained by minimizing the cost function, E(0), shown in (2.211). If the candidate points, x, are fixed, then the optimal parameter, 0, that minimizes E(0) can be computed analytically. Once the values of the shape parameters are updated to 0, then the surface represented by the PDM varies and the corresponding points, xj, should also be updated. The final estimates of the shape parameters, 0, are obtained by iteratively updating the shape parameters and the corresponding points until they converge.

The statistical models shown above are constructed from the corresponding points generated on the training surfaces, S (j = 1,2 , M), and it is not easy

to generate these corresponding points. It should be noted that many methods for generating the corresponding points on surfaces have been proposed and that different SSMs can be constructed from an identical set of training data if different methods for the corresponding generation are employed.

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