Local regularization for n-dimensional integral equations with applications to image processing

Abstract

We examine the method of local regularization for the solution of linear first-kind integral equations on . We provide a theoretical analysis of the method and prove that the regularized solutions converge to the true solution as the level of error in perturbed data goes to zero. We also develop an iterative numerical algorithm based on this theory and describe its implementation. Our testing with a number of examples shows that local regularization tends to perform better than a classical method we call Tik-CG (a method based on a conjugate gradient algorithm with stopping criteria applied to standard Tikhonov regularization) when performance is measured in terms of relative error in solutions and/or in perceived sharpness of images. Unfortunately, this improvement can come at a cost as testing shows that the local regularization algorithm tends to be slower than the Tik-CG approach when applied to 2D images. As we illustrate with our numerical results, however, a compromise can be found by using the converged Tik-CG image as the starting value for the iterative local regularization method.