Abstract
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.
Highlights
In the traditional discrete formalism of linear inverse problems, the goal is to recover a signal c0 ∈ RN based on some measurement vector y ∈ RM
The recovery is often achieved by solving an optimization problem that aims at minimizing the discrepancy between the measurements Hc of the reconstructed signal c and the acquired data y
We have introduced a continuous-domain framework for the reconstruction of multicomponent signals
Summary
In the traditional discrete formalism of linear inverse problems, the goal is to recover a signal c0 ∈ RN based on some measurement vector y ∈ RM These measurements are typically acquired via a linear operator H ∈ RM×N that models the physics of our acquisition system (forward model), so that Hc0 ≈ y. The recovery is often achieved by solving an optimization problem that aims at minimizing the discrepancy between the measurements Hc of the reconstructed signal c and the acquired data y. This data fidelity is measured with a suitable convex loss function E : RM × RM → R, the prototypical example being the quadratic error E(x, y) =. Despite its non-differentiability, numerous efficient proximal algorithms based on the proximity operator of the 1 norm have emerged to solve 1-regularized problems [10]–[13]
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