Sparse Dictionaries for Continuous-Domain Inverse Problems

We study 1D continuous-domain inverse problems for multicomponent signals. The prior assumption on these signals is that each component is sparse in a different dictionary specified by a regularization operators. We introduce a hybrid regularization functional matched to such signals, and prove that corresponding continuous-domain inverse problems have hybrid spline solutions, i.e. they are sums of splines matched to the regularization operators. We then propose a B-spline-based exact discretization method to solve such problems algorithmically.