SSM with Point Distribution Model (MRF Representation)

MRF [54, 131] can be employed for representing PDM-based SSMs [66, 70, 132, 133]. With MRF, multivariable probability distributions are represented with products of single-variable and two-variable probabilities. Let xj (j = 1,2 , ..., N) denote the coordinates of the N points that represent a target surface and let Ij = Ij(x) denote an appearance observed in a local small region around xj in a given image. The following equation represents an MRF model of the SSMs:

where C is a set of pairs of indexes: (s, t) e [1,2, In (2.212), p(xj)

represents the prior probability distribution of xj, p(Ijjxj) represents the likelihood of xj when a target image is given, and p(x_{s}, x_{t}) represents the simultaneous probability distribution of two of the points. MRF models can be represented by using undirected graphical models, in which a node represents one of the variables and an edge connecting two nodes represents the conditional dependencies of the two variables. The set of the pairs, C, corresponds to a set of the edges in the graphical model.

p(xj), p(Ijjxj), andp(x_{s}, x_{t}) can be estimated using a set of training data, {xj|i =

1,2 , ..., N, j = 1,2 , ..., M}, where xj denotes the j-th point on the surface of a target organ in the i-th training image. The prior distribution,p(xj), can be estimated directly from the set, {xj|i = 1,2 , ..., N}. For example, employing a Gaussian distribution to representp(xj), then the mean and the covariance matrix are estimated from the set. The simultaneous distribution, p(x_{s}, x_{t}), can also be estimated from the training set. Letting x'_{st} = [(xS)^{r} , (xj)^{T}]^{T} allows estimation of p(x_{s}, x_{t}) from the set, {xl = 1,2 , ..., N}. Estimating p(Ijjxj) requires a training set of local appearances in addition to the training set of the points. For example, letting an L^{3}-vector, Ij, denote an appearance observed in a L x L x L local cube in I' (x) whose center is located at xj, the conditional probability of the appearance, p(Ij|xj), can be estimated from a set of the appearances, {Ijji = 1,2 , ..., N}. The prior probability and the likelihood do not contain shape information and can be computed without using any information of other points. In (2.212), the statistical variety of the shape is represented by the simultaneous probability distributions, Пp(x_{s}, x_{t}).

Using the model shown in (2.212) allows segmentation of target organs in given images by estimating the marginal posterior probability distribution of each point, p(xj|I), as follows: first, the posterior probability is temporally evaluated without using the simultaneous probability distributions, as p'(xj) / p(xj)p(Ij|xj). Employing this temporal estimate as the initial state, the posterior probability of each point is then estimated based on the model shown in (2.212). The shape model that is represented by the simultaneous probabilities is now explicitly used. For this estimation, several techniques developed for inferring on undirected graphical models can be employed, e.g., belief propagation or MCMC.

Here, it should be remembered that multivariable Gaussian functions can be represented by products of single-variable and two-variable Gaussian functions. The statistical model shown in (2.206), hence, can be written as follows:

i

where C = {(s , t) | (X“b)_{st}Ф 0} and (X“b)_{st} denotes the (s,t) component of a matrix X“b. An inverse matrix of a covariance matrix is called a precision matrix, and its zero components correspond to the pairs of variables that are conditionally dependent. When a multivariable Gaussian distribution is employed for the prior probability distribution of the points of a PDM, the structure of the corresponding undirected graphical model, or the set of the edges, C in (2.212), is determined by the precision matrix.