Preconditioned Douglas–Rachford Algorithms for TV- and TGV-Regularized Variational Imaging Problems

Abstract

The recently introduced preconditioned Douglas---Rachford iteration (PDR) for convex---concave saddle-point problems is studied with respect to convergence rates and applied to variational imaging problems with total variation (TV) and total generalized variation (TGV) penalty. A rate of $${\mathcal {O}}(1/k)$$O(1/k) for restricted primal---dual gaps evaluated for ergodic sequences generated by the PDR iteration is established. Based on PDR, new fast iterative algorithms for TV-denoising, TV-deblurring, and TGV-denoising of second order with $$L^2$$L2 and $$L^1$$L1 discrepancy are proposed. While for denoising, symmetric (block) Red---Black Gauss---Seidel preconditioners are effective, fast Fourier transform-based preconditioners are employed for the deblurring problems. Finally, for the $$L^2$$L2-TGV-denoising problem, an effective modified primal---dual gap is developed which may serve as a stopping criterion. All algorithms are tested and compared in numerical experiments. In particular, for problems where strong convexity does not hold, it turns out that the proposed preconditioning techniques are beneficial and lead to competitive results.