Fast deconvolution by a two-step method

A method for solving a discretized Fredholm integral equation of the first kind with convolution kernel on the d-dimensional torus (d greater than or equal to 1) is presented. A two-step regularization method is used, based on the minimization of two objective functions, made up of a convex coherence-to-data term and a term which enforces a roughness penalty. This is convex in the first initialization step and nonconvex in the second one. The computational aspect is the main point addressed in the paper. A deterministic, as opposed to stochastic, approach is followed. Auxiliary variables are used. The separability of the original and the auxiliary variables is proven and exploited to reduce the computational burden. The first step provides an initial estimate by computing the unique minimum of the convex functional. The second step refines this initial estimate by computing a stationary point of the nonconvex functional. Experiments are reported on simulated data and in magnetic resonance (MR) imaging. A significant reduction in computer time and similar restoration quality is observed when the method is compared with dynamic Monte Carlo methods applied to global optimization of the nonconvex functional.