Keywords: least squares, regularization

Image deblurring or signal restoration can be formulated as a data fitting least squares problem, but the problem is severely ill-posed and regularization is needed, hence introducing the need to find a regularization parameter. I will review the background on finding the regularization parameter dependent on the properties of the regularized least squares functional

for the solution of discretely ill-posed systems of equations. It was recently shown to follow a χ² distribution when the a priori information x₀ on the solution is assumed to represent the mean of the solution x. But of course for image deblurring, we don’t wish to assume knowledge of a prior image to obtain the image. On the other hand, it is possible to obtain statistical properties of the given image, hence given the mean value of the right hand side, b, the functional is again a χ² distribution, but one that is non-central. These results can be used to design a Newton method, using a hybrid LSQR approach, for the determination of the optimal regularization parameter λ when the weight matrix is W_x=λ² I. Numerical results using test problems demonstrate the efficiency of the method, particularly for the hybrid LSQR implementation. Results are compared to another statistical method, the unbiased predictive risk (UPRE) algorithm. The method has potential for efficient image deblurring, and current work is aimed at extending the method for determining local regularization parameters. Results are illustrated for image deblurring.

This document was translated from L^AT_EX by H^EV^EA.