Maximum a Posterior Probability

Given an observed image Y, for any pixel at location s in Y, we assume it is with intensity ys and with label xs in segmentation label matrix. Then the a posteriori probability of segmentation result can be expressed as

where P(Y | X) is the conditional probability of the observed image given the scene segmentation. The goal of maximum a posterior probability (MAP) criterion is to find an optimal estimate of X, Xopt, given the observed image Y. Since P(Y) is not a function of X, the maximization process only applies over the upper portion of Eq. (8.10), P(Y | X)P(X). More accurately, given the observed image, the target of solving an MRF is to find the optimal state Xopt that maximizes the a posterior probability and take that state as the optimal image segmentation solution.

In this study, the conditional density is modeled as a Gaussian process, with mean ms and variance a2 for the region that s belongs to in the image domain. Thus, the intensity of each region can be regarded as a signal ¡xs plus additive zero mean Gaussian noise with variance a2, and the conditional density can be expressed as

By substituting Eqs. (8.9) and (8.11) into (8.10), the general form of the a posterior probability can be written as

0 0

Post a comment