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SSM with NURBS Surface Representation

SSMs with explicit representation of the surfaces of target organs are also widely employed for the segmentation of given medical images. The surfaces are explicitly represented with some parameters, and the prior probabilities of those values are represented by the SSMs. For image segmentation, the values of those parameters are estimated so that the resultant surfaces fit to the boundaries of target organs in given images. Different from the segmentation methods with implicit organ representation, the likelihoods of the parameter values are defined based on the distances between the model surfaces and the boundary candidate points extracted from the given images. In the followings, SSMs that represent the surfaces with NURBS surfaces are described.

NURBS surfaces are determined by a set of control points and a set of basis functions. Given the set of basis functions, one can vary the shape of the NURBS surfaces by changing the locations of the control points: The surfaces are parametrized by the coordinates of the control points. It is not difficult to compute the normal direction at each point on the surface, and this helps the computation of the distances between the surface and the boundary candidates extracted from given images along the normal directions.

NURBS represent surfaces thus:

where Vl(s) denote B-spline basis functions of order n that are periodic over the range 0 < s < L, wuv denote weights of control points, and the three-vectors, Puv, denote the coordinates of the control points. Let

Then, the Eq. (2.194) can be rewritten as follows: where B(s, t) is a 3 x 3NsNt matrix such that

Here, Buv = Buv(s, t) are the products of the B-spline basis functions as shown in (2.195), and в denotes the coordinates of all control points: в = [Pl0,PTm,... ,PTNsNt]T. The matrix B(s, t) can be determined in advance, and the shape of the surfaces can be varied by changing the values of the components of the vector в. Figure 2.21 An example of a surface (kidney) represented by NURBS.

An example of closed surface represented by NURBS appeared in [129]

Fig. 2.21 An example of closed surface represented by NURBS appeared in [129]

When the model in (2.197) is employed, the model can be registered to given images by estimating the values of в by means of a MAP estimation or Bayesian estimation.

A NURBS-based SSM represents a prior probability distribution of в, which is constructed from a set of training images. Assume the boundaries of target organs in the training images are manually extracted by some experts. Letting the boundary surface in the i-th image be denoted by S (i = 1,2, and letting в' denote

the parameter of NURBS describing Si, referring to the set of the parameters, {ei|i = 1,2,...,M}, allows estimation of the prior probability distribution, р(в). Once the prior, р(в), is determined, the simultaneous prior probability distribution of the points, x(s , t), of the NURBS surfaces can be determined.

For statistically registering the NURBS surfaces, statistical models of the residuals between the NURBS surfaces and the corresponding candidates of organ boundaries detected by, e.g., edge detectors, are also needed. Letting Xj = x(sj, tj) (j = 1,2, ... ,L) be a point of a NURBS surface and letting Xj denote a detected candidate point corresponding to Xj, the conditional probability distributions of the residuals, p.Xj — Xj|xj) = p(Xj|xj), supply the likelihood distributions when Xj for each j is given. Using р(в) and p(Xjx) allows computation of a posterior probability of в as follows:

where L denotes the number of the samples and it is assumed that the residuals are determined independently. The MAP estimates of в can be obtained by maximizing р(в |X1; X2,..., XL) in (2.198).

For example, a Gaussian distribution can be employed for representing р(в): The mean and the covariance matrix, в and E, can be estimated from the set of the training parameters, {в j i = 1,2 , ••• , M}:

where Z is a constant for the normalization, в is a 3NsNt vector, and E is a 3NsNt x 3NsNt matrix. The likelihood can also be modeled by a Gaussian. For example:

where ^ denotes the distance between Xj and Xj measured along the direction of the unit normal vector, nj, perpendicular to the NURBS surface at Xj such that щ = nj (Xj — Xj).Then, the cost function, Е(в), which is derived from the negative logarithm of the posterior probability distribution, is obtained:

Minimizing E(0), the estimates, в, that maximize the posterior probability of в can be obtained. E{6) has a quadratic form of в, and the computation of the minimizer is straightforward if Xj (j = 1,2,N) are fixed. Once в is updated to в, then the locations of xj move, and the corresponding candidate points, Xj, change. The registration methods, hence, iteratively minimize E{6) and update Xj until they are converged.

As mentioned, a NURBS-based SSMs are constructed from a set of training data, which are obtained by manually fitting a NURBS surface to the boundary of a target organ in each of the training images. The number of control points should be identical among all fitted NURBS surfaces, and each of the control points of one NURBS surface should be corresponded to one of the control points of each of the other surfaces. Making this correspondence is not straightforward.

The prior distribution, р.в), is estimated based on the control points of the training data. Different from the SSMs with implicit representation, it is guaranteed that the surfaces represented by the SSMs are single and closed if the surfaces in the training images are all single and closed. It is not guaranteed, though, that the surfaces represented by the NURBS surfaces are simple: The surfaces would have self-intersections even if all of the training surfaces are simple and have no selfintersections. It is difficult to find the global maximum of the posterior probability, and the algorithm described above can find only the local maximum. The estimated parameters, в, hence vary, depending on the initial values of the parameters, в.

 
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