Abstract

We consider the problem of reconstructing a finite image x from a blurred observation y, where y = A_kx + n. In this equation A_k is an unknown blurring operator from a known class {A_k; k = 1, K} of such operators and n is an observation noise. The operator A_k may be continuous or discrete and n is modeled as a 2-D white Gaussian noise process. The reconstruction procedure is based on the observation that A_k is singular (or highly ill-conditioned and thus singular for all practical purposes). It consists in projecting the received image y onto the null space of each of the possible blurring operators A_k. The projection onto the null space of the actual blurring operator will consist of noise only whereas the projections onto the other null spaces will have a signal component in addition to the noise component. Thus, we can estimate the actual blurring operator by picking the projection that has minimum energy. It is shown that this is equivalent to performing a simultaneous maximum likelihood estimation of A_k and x. To minimize the probability that the wrong projection is chosen a projection dependent constant is subtracted from the energy of each projection before the comparison is done. The constant depends on the variance of the noise process and the actual null space being considered. This modification allows us also to deal efficiently with the case where the null space of a blurring operator is entirely included in that of another operator. Once the blurring operator is identified, the image x is reconstructed using the pseudoinverse of the identified blurring operator.

© 1989 Optical Society of America