MAP Estimation: When Does It Work?

Abstract

In this chapter we analyze the performance of the maximum a posteriori probability (MAP) method of estimating the image and the point spread function (PSF), which we term as joint MAP estimation for blind deconvolution since both the unknowns are estimated simultaneously. Many authors have reported the failure of direct application of the MAP estimator in blind deconvolution, the details of which we explain in this chapter. We show that joint MAP estimation fails only when the PSF regularizer is not chosen properly. We also demonstrate that the MAP estimation does produce good results with an appropriate choice of the PSF prior. The emphasis on the word appropriate is very important. We provide a theoretical justification to show when joint MAP estimation works and when it does not. The arguments are substantiated through experimental validation. We also show how the regularization factor is to be selected so that the PSF regularization is effective. Our analysis provides a feasible range for the regularization factor without using cross validation techniques. We give an exact lower bound and an approximate upper bound for the PSF regularization factor.