Splines Are Universal Solutions Of Linear Inverse Problems With Generalized Tv Regularization∗

Keywords

Compressed sensing; Inverse problems; Sparsity; Splines; Total variation

Abstract

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This results in the definitions of a generalized total variation seminorm and its corresponding native space, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV), subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that nonuniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L1, or total-variation-like regularization constraints. Remarkably, the type of spline is fully determined by the choice of L and does not depend on the actual nature of the measurements.

Publication Date

1-1-2017

Publication Title

SIAM Review

Volume

59

Issue

4

Number of Pages

769-793

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1137/16M1061199

Socpus ID

85034234225 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85034234225

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