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Continuous-Domain Formulation of Inverse Problems for Composite Sparse-Plus-Smooth Signals

T. Debarre, S. Aziznejad, M. Unser

IEEE Open Journal of Signal Processing, vol. 2, pp. 545–558, November 9, 2021.

We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction problem as a continuous-domain regularized inverse problem with multiple penalties. We prove that these penalties induce reconstructed signals that indeed take the desired form of the sum of a sparse and a smooth component. We then discretize this problem using Riesz bases, which yields a discrete problem that can be solved by standard algorithms. Our discretization is exact in the sense that we are solving the continuous-domain problem over the search space specified by our bases without any discretization error. We propose a complete algorithmic pipeline and demonstrate its feasibility on simulated data. 

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AUTHOR="Debarre, T. and Aziznejad, S. and Unser, M.",
TITLE="Continuous-Domain Formulation of Inverse Problems for Composite
	Sparse-Plus-Smooth Signals",
JOURNAL="{IEEE} Open Journal of Signal Processing",
YEAR="2021",
volume="2",
number="",
pages="545--558",
month="November 9,",
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