Bayesian Image Restoration

We can use MRF models together with fundamental principles from statistical decision theory to formulate optimal image-processing algorithms. One such optimality criterion is based on the MAP estimate. Let us consider the uncorrupted image ${\bf x}$ as a realization of a MRF ${\bf X}$, and the observed degraded image ${\bf\tilde x}$ as a realization of a MRF ${\bf\tilde X}$. Given the true image ${\bf x}$, let us assume, for simplicity, that the RVs in the MRF ${\bf\tilde X}$ are conditionally independent. This is equivalent to saying that the noise affects each image location independently of any other location. Given the stochastic model $P (\tilde x_t \vert x_t)$ for the degradation process, conditional independence implies that the conditional probability of the observed image given the true image is

$$P ({\bf\tilde x} \vert {\bf x}) = \prod_{t \in \mathcal{T}} P (\tilde x_t \vert x_t).$$ (86)

Our goal is to find the MAP estimate ${\bf\hat x}^*$ of the true image ${\bf x}$

$${\bf\hat x}^* = \mathop{\mbox{argmax }}_{{\bf x}} P ({\bf x} \vert {\bf\tilde x})$$ (87)

This MAP-estimation problem is an optimization problem that, like many other optimization problems, suffers from the existence of many local maxima. Two classes of optimization algorithms exist to solve this problem: (a) methods that guarantee to find the unique global maximum and (b) methods that converge only to local maxima. Typically, the former class of methods are significantly slower. Here we face a trade-off between finding the global maximum at a great expense and finding local maxima with significantly less cost.