Computation of "Best" Interpolants in the Lp Sense

Abstract

We study a variant of the interpolation problem where the continuously defined solution is regularized by minimizing the L p -norm of its second-order derivative. For this continuous-domain problem, we propose an exact discretization scheme that restricts the search space to quadratic splines with knots on an uniform grid. This leads to a discrete finitedimensional problem that is computationally tractable. Another benefit of our spline search space is that, when the grid is sufficiently fine, it contains functions that are arbitrarily close to the solutions of the underlying unrestricted problem. We implement an iteratively reweighted algorithm with a grid-refinement strategy that computes the solution within a prescribed accuracy. Finally, we present experimental results that illustrate characteristics, such as sparsity, of the L p -regularized interpolants.

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

“…• the basis functions h m are typically increasing at infinity, which contradicts the Riesz-basis requirement and leads to severely ill-conditioned optimization tasks [24]; • depending on the measurements operator ν, h m may lack a closed-form expression. We therefore focus on these criteria, in a spirit similar to [55]. The ϕ 2,k are chosen to be regular shifts of a generating function ϕ 2 , with ϕ 2,k = ϕ 2 (• − k) such that {L 2 {ϕ 2,k }} k∈Z forms a Riesz basis in the sense of Definition 3.…”

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

“…We therefore focus on these criteria, in a spirit similar to [54]. The ϕ 2,k are chosen to be regular shifts of a generating function ϕ 2 , with ϕ 2,k = ϕ 2 (• − k) such that {L 2 {ϕ 2,k }} k∈Z forms a Riesz basis in the sense of Definition 3.…”