Hessian Schatten-Norm Regularization for Linear Inverse Problems

TL;DR

This paper introduces a new family of regularizers based on Schatten norms of the Hessian for linear inverse imaging problems, improving over total variation by reducing artifacts and enhancing performance.

Contribution

The paper proposes a novel convex regularizer using Hessian Schatten norms, along with an efficient primal-dual algorithm and matrix projection techniques for inverse imaging tasks.

Findings

Reduces staircase artifacts compared to TV regularization

Effective on real and simulated inverse imaging problems

Provides a computationally efficient projection method onto Schatten norm balls

Abstract

We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, computed at every pixel of the image. They can be viewed as second-order extensions of the popular total-variation (TV) semi-norm since they satisfy the same invariance properties. Meanwhile, by taking advantage of second-order derivatives, they avoid the staircase effect, a common artifact of TV-based reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primal-dual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto $\ell_q$ norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data.