Cosine Transform Based Preconditioners for Total Variation Deblurring

Image reconstruction is a mathematically ill posed problem and regularization methods are often used to obtain a reasonable solution Recently, the total variation (TV) regularization, proposed by Rudin, Osher and Fatemi (1992), has become very popular for this purpose In a typical iterative solution of the nonlinear regularization problem, such as the fixed point iteration of Vogel or New tons method, one has to invert linear operators consisting of the sum of two distinct parts One part corresponds to the blurring operator and is often a convolution the other part corresponds to the TV regularization and resembles an elliptic operator with highly varying coefficients. In this paper, we present a preconditioner for operators of this kind which can be used in conjunction with the conjugate gradient method It is derived from combining fast transform (e.g. cosine-transform based) preconditioners which the authors had earlier proposed for Toeplitz matrices and for elliptic operators separately. Some numerical results will be presented. In particular, we will compare our preconditioner with a variant of the product preconditioner proposed by Vogel and Oman.