Continuous-Domain Signal Reconstruction Using $L_{p}$-Norm Regularization

Abstract

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 $L_{p}$ -norm regularization. More specifically, we consider $p\geq 1$ and the multi-order derivative regularization operator ${\mathrm{L}}={\mathrm{D}}^{N_{0}}$ . 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 $L_{p}$ -norm regularization, namely, sparsity, regularity, and oscillatory behavior.