TV-based Spline Reconstruction with Fourier Measurements: Uniqueness and Convergence of Grid-Based Methods

TL;DR

This paper proves the uniqueness and convergence of grid-based methods for reconstructing piecewise-polynomial periodic functions from low-frequency Fourier data using TV regularization, introducing a B-spline algorithm for practical computation.

Contribution

It establishes the first proof of uniqueness for TV-based spline reconstruction and demonstrates convergence of grid-based solutions to the continuous problem.

Findings

Solution of the TV-based problem is always unique.

Reconstructed spline has a bounded number of knots, at most twice the cutoff frequency.

Proposed B-spline algorithm effectively solves the discretized problem.

Abstract

We study the problem of recovering piecewise-polynomial periodic functions from their low-frequency information. This means that we only have access to possibly corrupted versions of the Fourier samples of the ground truth up to a maximum cutoff frequency $K_c$. The reconstruction task is specified as an optimization problem with total-variation (TV) regularization (in the sense of measures) involving the $M$-th order derivative regularization operator $\mathrm{L} = \mathrm{D}^M$. The order $M \geq 1$ determines the degree of the reconstructed piecewise polynomial spline, whereas the TV regularization norm, which is known to promote sparsity, guarantees a small number of pieces. We show that the solution of our optimization problem is always unique, which, to the best of our knowledge, is a first for TV-based problems. Moreover, we show that this solution is a periodic spline matched to the regularization operator $\mathrm{L}$ whose number of knots is upper-bounded by $2 K_c$. We then consider the grid-based discretization of our optimization problem in the space of uniform $\mathrm{L}$-splines. On the theoretical side, we show that any sequence of solutions of the discretized problem converges uniformly to the unique solution of the gridless problem as the grid size vanishes. Finally, on the algorithmic side, we propose a B-spline-based algorithm to solve the grid-based problem, and we demonstrate its numerical feasibility experimentally. On both of these aspects, we leverage the uniqueness of the solution of the original problem.