Analysis and Application of a non-local Hessian

TL;DR

This paper introduces a non-local Hessian formulation that enhances image restoration by combining higher-order and non-local regularization, improving upon existing methods like Total Generalized Variation.

Contribution

It presents a novel non-local Hessian model that extends non-local gradients to higher-order derivatives and demonstrates convergence to classical regularizers.

Findings

Improves image restoration quality over TGV

Converges to classical second-order regularizers

Provides new characterization of Sobolev and BV spaces

Abstract

In this work we introduce a formulation for a non-local Hessian that combines the ideas of higher-order and non-local regularization for image restoration, extending the idea of non-local gradients to higher-order derivatives. By carefully choosing the weights, the model allows to improve on the current state of the art higher-order method, Total Generalized Variation, with respect to overall quality and particularly close to jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights, it can be shown to converge to classical second-order regularizers, and in fact allows a novel characterization of higher-order Sobolev and BV spaces