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.