Expectation-maximization algorithms, null spaces, and MAP image restoration

A computationally efficient, easily implementable algorithm for MAP restoration of images degraded by blur and additive correlated Gaussian noise using Gibbs prior density functions is derived. This algorithm is valid for a variety of complete data spaces. The constraints upon the complete data space arising from the Gaussian image formation model are analyzed and a motivation is provided for the choice of the complete data, based upon the ease of computation of the resulting EM algorithms. The overlooked role of the null space of the blur operator in image restoration is introduced. An examination of this role reveals an important drawback to the use of the simulated annealing algorithm in maximizing a specific class of functionals. An alternative iterative method for computing the nullspace component of a vector is given. The ability of a simple Gibbs prior density function to enable partial recovery of the component of an image within the nullspace of the blur operator is demonstrated.