Nonlocal Patch-based Image Inpainting through Minimization of a Sparsity Promoting Nonconvex Functional

We propose a convex model for nonlocal image inpainting that fills in missing or damaged areas with different convex combinations of available image patches. The visual quality of inpainted solutions is much better when patches are copied instead of averaged, and we show how this can be achieved by adding a nonconvex sparsity promoting penalty. To promote sparsity of a variable that is constrained to be nonnegative and sum to one, we add a concave quadratic function to an otherwise convex objective. Standard difference of convex programming can be used to find critical points of this nonconvex problem, but it requires solving a sequence of expensive convex problems. We argue that a faster and more practical approach is to use a modified primal dual hybrid gradient method, which is an efficient method for minimizing large non-differentiable convex functions. Although its convergence for nonconvex problems is only empirical, it shows promise as a practical method for minimizing large non-differentiable nonconvex functions for which the nonconvex component is smooth and can be made strongly convex by the addition of a quadratic. In particular, we show how to apply this strategy to a nonlocal patch-based image inpainting problem.