Extremal points of total generalized variation balls in 1D: characterization and applications

TL;DR

This paper characterizes the extremal points of the 1D total generalized variation balls, showing that TGV-regularized inverse problems favor piecewise affine solutions and providing tools for optimality conditions and algorithms.

Contribution

It formalizes the extremal points of 1D TGV balls, enabling analysis of TGV-regularized problems and deriving optimality conditions and solution methods.

Findings

1D TGV-regularized problems admit piecewise affine minimizers.

Characterization of extremal points aids in deriving optimality conditions.

Provides a simple algorithm for TGV-regularized minimization.

Abstract

The total generalized variation (TGV) is a popular regularizer in inverse problems and imaging combining discontinuous solutions and higher order smoothing. In particular, empirical observations suggest that its order two version strongly favors piecewise affine functions. In the present manuscript, we formalize this statement for the one-dimensional TGV-functional by characterizing the extremal points of its sublevel sets with respect to a suitable quotient space topology. These results imply that 1D TGV-regularized linear inverse problems with finite dimensional observations admit piecewise affine minimizers. As further applications of this characterization we include precise first-order necessary optimality conditions without requiring convexity of the fidelity term, and a simple solution algorithm for TGV-regularized minimization problems.