Continuous-Domain Signal Reconstruction Using [Formula Omitted]-Norm Regularization

• Published in: IEEE transactions on signal processing 2020-01, Vol.68, p.4543
• Main Authors: Bohra, Pakshal, Unser, Michael
• Format: Article
• Language: English
• Subjects: Domains; Ill posed problems; Interpolation; Mathematical analysis; Operators (mathematics); Polynomials; Regularization; Signal reconstruction; Spline functions; Well posed problems
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2020.3013781

We focus on the generalized-interpolation problem. There, one reconstructs continuous-domain signals that honor discrete data constraints. This problem is infinite-dimensional and ill-posed. We make it well-posed by imposing that the solution balances data fidelity and some [Formula Omitted]-norm regularization. More specifically, we consider [Formula Omitted] and the multi-order derivative regularization operator [Formula Omitted]. We reformulate the regularized problem exactly as a finite-dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid. Our splines are represented in a B-spline basis, which results in a well-conditioned discretization. For a sufficiently fine grid, our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted. This remarkable property is due to the approximation power of splines. We use the alternating-direction method of multipliers along with a multiresolution strategy to compute our solution. We present numerical results for spatial and Fourier interpolation. Through our experiments, we investigate features induced by the [Formula Omitted]-norm regularization, namely, sparsity, regularity, and oscillatory behavior.