Continuous-Domain Solutions of Linear Inverse Problems with Tikhonov vs. Generalized TV Regularization

TL;DR

This paper derives the form of solutions for continuous-domain linear inverse problems using Tikhonov and generalized TV regularizations, revealing smooth and sparse solutions respectively, with implications for measurement models and solution uniqueness.

Contribution

It provides the first parametric form (representer theorems) of solutions for continuous-domain inverse problems with these regularizations, highlighting their structural differences.

Findings

Tikhonov solutions are smooth and confined to a fixed subspace.

gTV solutions are sparse, composed of few dictionary elements.

An algorithm is proposed to find sparse extreme solutions in the gTV case.

Abstract

We consider linear inverse problems that are formulated in the continuous domain. The object of recovery is a function that is assumed to minimize a convex objective functional. The solutions are constrained by imposing a continuous-domain regularization. We derive the parametric form of the solution (representer theorems) for Tikhonov (quadratic) and generalized total-variation (gTV) regularizations. We show that, in both cases, the solutions are splines that are intimately related to the regularization operator. In the Tikhonov case, the solution is smooth and constrained to live in a fixed subspace that depends on the measurement operator. By contrast, the gTV regularization results in a sparse solution composed of only a few dictionary elements that are upper-bounded by the number of measurements and independent of the measurement operator. Our findings for the gTV regularization resonates with the minimization of the $l_1$ norm, which is its discrete counterpart and also produces sparse solutions. Finally, we find the experimental solutions for some measurement models in one dimension. We discuss the special case when the gTV regularization results in multiple solutions and devise an algorithm to find an extreme point of the solution set which is guaranteed to be sparse.