Primal-Dual Decomposition by Operator Splitting and Applications to Image Deblurring

We present primal-dual decomposition algorithms for convex optimization problems with cost functions f(x) + g(Ax), where f and g have inexpensive proximal operators and A can be decomposed as a sum of two structured matrices. The methods are based on the Douglas–Rachford splitting algorithm applied to various splittings of the primal-dual optimality conditions. We discuss applications to image deblurring problems with nonquadratic data fidelity terms, different types of convex regularization, and simple convex constraints. In these applications, the primal-dual splitting approach allows us to handle general boundary conditions for the blurring operator. Numerical results indicate that the primal-dual splitting methods compare favorably with the alternating direction method of multipliers, the Douglas–Rachford algorithm applied to a reformulated primal problem, and the Chambolle–Pock primal-dual algorithm.