Deflated Preconditioned Conjugate Gradients for Nonlinear Diffusion Image Enhancement

Abstract

Nonlinear diffusion equations have been successfully used for image enhancement by reducing the noise in the image while protecting the edges. In discretized form, the denoising requires the solution of a sequence of linear systems. The underlying system matrices stem from a discrete diffusion operator with large jumps in the diffusion coefficients. As a result these matrices can be very ill-conditioned, which leads to slow convergence for iterative methods such as the Conjugate Gradient method. To speed-up the convergence we use deflation and preconditioning. The deflation vectors are defined by a decomposition of the image. The resulting numerical method is easy to implement and matrix-free. We evaluate the performance of the method on a simulated image and on a measured low-field MR image for various types of deflation vectors.