AUTHORS: Debarre T, Aziznejad S, Unser M

IEEE Transactions on Signal Processing, 67(22): 5824-5836, October 2019

ABSTRACT

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 $hybrid text{-} spline$ 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.

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