Hybrid-Spline Dictionaries for Continuous-Domain Inverse Problems

We study one-dimensional continuous-domain inverse problems with multiple generalized total-variation regularization, which involves the joint use of several regularization operators. Our starting point is a new representer theorem that states that such inverse problems have hybrid-spline solutions with a total sparsity bounded by the number of measurements. We show that such continuous-domain problems can be discretized in an exact way by using a union of B-spline dictionary bases matched to the regularization operators. We then propose a multiresolution algorithm that selects an appropriate grid size that depends on the problem. Finally, we demonstrate the computational feasibility of our algorithm for multiple-order derivative regularization operators.