Deconvolving Images with Unknown Boundaries Using the Alternating Direction Method of Multipliers

The alternating direction method of multipliers (ADMM) has sparked recent interest as an efficient optimization tool for solving imaging inverse problems, such as deconvolution and reconstruction. ADMM-based approaches achieve state-of-the-art speed, by adopting a divide and conquer strategy that splits a hard problem into simpler, efficiently solvable sub-problems (e.g., using fast Fourier or wavelet transforms, or proximity operators with low computational cost). In deconvolution problems, one of these sub-problems involves a matrix inversion (i.e., solving a linear system), which can be performed efficiently (in the discrete Fourier domain) if the observation operator is circulant, that is, under periodic boundary conditions. This paper proposes an ADMM approach for image deconvolution in the more realistic scenario of unknown boundary conditions. To estimate the image and its unknown boundary, we model the observation operator as a composition of a cyclic convolution with a spatial mask that excludes those pixels where the cyclic convolution is invalid, i.e., the unknown boundary. The proposed method can also handle, at no additional cost, problems that combine inpating (recovery of missing pixels) and deblurring. We show that the resulting algorithm inherits the convergence guarantees of ADMM and illustrate its state-of-the-art performance on non-cyclic deblurring (with and without inpainting of interior pixels) under total-variation (TV) regularization.