On the Uniqueness of Solutions for the Basis Pursuit in the Continuum

Debarre, Thomas; Denoyelle, Quentin; Fageot, Julien

doi:10.48550/arxiv.2009.11855

Cited by 1 publication

(3 citation statements)

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“…Moreover, we slightly improve the upper bound on the number of knots to K ≤ 2K c in (6). Our proof relies on a result of our previous paper [13,Corollary 1]. To the best of our knowledge, Theorem 1 is the first systematic uniqueness result for the analysis of TV-based variational problems such as (2).…”

Section: Contributionsmentioning

confidence: 92%

“…The operator L specifies the transform domain in which sparsity is enforced together with the regularity properties of the recovery. In the absence of a regularization operator L, Problem (2) leads to the recovery of periodic Dirac masses; this scenario is the subject of our previous paper [13]. In the latter, we thoroughly analyze the cases of uniqueness of the solution of the resulting optimization problem.…”

Section: Reconstruction Via Tv-based Optimizationmentioning

confidence: 98%

“…Optimization over Radon measures: There exists a vast literature concerned with Dirac recovery using the TV norm as a regularizer, that is, problems of the form (2) in the absence of a regularization operator L. Some of the more recent works concerning this topic include [1,2,7,17,18,19,20,21,22]. However, our focus in this paper is on generalized TV regularization with a nontrivial operator L. We refer to the introduction of our previous paper [13] for a more detailed coverage of the Dirac-recovery literature.…”

Section: Related Workmentioning

confidence: 99%

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

Debarre¹,

Denoyelle²,

Fageot³

2022

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We study the problem of recovering piecewise-polynomial periodic functions from their lowfrequency 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 L = D M . The order M ≥ 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 L whose number of knots is upperbounded by 2K c . We then consider the grid-based discretization of our optimization problem in the space of uniform 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.

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Section: Contributionsmentioning

confidence: 92%

Section: Reconstruction Via Tv-based Optimizationmentioning

confidence: 98%