Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration

Abstract

Regularization is an effective method for obtaining satisfactory solutions to image restoration problems. The application of regularization necessitates a choice of the regularization parameter as well as the stabilizing functional. For most problems of interest, the best choices are not known a priori. We present a method for obtaining optimal estimates of the regularization parameter and stabilizing functional directly from the degraded image data. The method of generalized cross-validation (GCV) is used to generate the estimates. Implementation of GCV requires the computation of the system eigenvalues. Certain assumptions are made regarding the structure of the degradation so that the GCV criterion can be implemented efficiently. Furthermore, the assumptions on the matrix structure allow the regularization operator eigenvalues to be expressed as simple parametric functions. By choosing an appropriate structure for the regularization operator, we use the GCV criterion to estimate optimal parameters of the regularization operator and thus the stabilizing functional. Experimental results are presented that show the ability of GCV to give extremely reliable estimates for the regularization parameter and operator. By allowing both the degree and the manner of smoothing to be determined from the data, GCV-based regularization yields solutions that would otherwise be unattainable without a priori information.