Deconvolving images with unknown boundaries using the alternating direction method of multipliers

The alternating direction method of multipliers (ADMM) has recently sparked interest as a flexible and efficient optimization tool for inverse problems, namely, image deconvolution and reconstruction under non-smooth convex regularization. ADMM achieves state-of-the-art speed by adopting a divide and conquer strategy, wherein a hard problem is split into simpler, efficiently solvable sub-problems (e.g., using fast Fourier or wavelet transforms, or simple proximity operators). In deconvolution, one of these sub-problems involves a matrix inversion (i.e., solving a linear system), which can be done efficiently (in the discrete Fourier domain) if the observation operator is circulant, i.e., under periodic boundary conditions. This paper extends ADMM-based image deconvolution to the more realistic scenario of unknown boundary, where the observation operator is modeled as the composition of a convolution (with arbitrary boundary conditions) with a spatial mask that keeps only pixels that do not depend on the unknown boundary. The proposed approach also handles, at no extra cost, problems that combine the recovery of missing pixels (i.e., inpainting) with deconvolution. We show that the resulting algorithms inherit the convergence guarantees of ADMM and illustrate its performance on non-periodic deblurring (with and without inpainting of interior pixels) under total-variation and frame-based regularization.