Analysis and Application of a Nonlocal Hessian

Abstract

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.

A Nonlocal Laplacian-Based Model for Bituminous Surfacing Crack Recovery and its MPI Implementation

“…The original local functional intertwining decomposition and fine structure detection is first introduced. Inspired then by the works of Bourgain et al ( [11]), Dávila ([17]) or Ponce ([35]) devoted to the design of nonlocal counterparts of Sobolev and BV semi-norms using radial mollifiers and integral operators, we extend their results to the case of second order operators (also inspired by the work of Lellmann et al ( [30])) and rephrase our original model in the obtained nonlocal setting. The main theoretical result -Γ -convergence result -connecting the nonlocal minimization problem to the local one is provided.…”

“…ρ n (r) dr = 0, and extend the previous results to second order operators. Our nonlocal extension is also deeply inspired by the formulation of a nonlocal Hessian and the novel characterization of higher Sobolev and BV-spaces introduced in [30]. This new operator is derivative free and only requires the considered function u to be in an L p space as in our model.…”

“…dx is approximated by an integral operator involving a differential quotient and the radial mollifiers. The resulting model inherits fine analytical properties, has the advantage of being numerically more tractable than [30] and of being straightforwardly connected to our imaging problem. We would like to point out that the qualifying term "nonlocal" might sound inadequate in the sense that the "nonlocal counterpart" includes a parameter n, via the mollifier ρ n , that is destined to tend to infinity, concentrating the measure around the point of interest and removing in some way the nonlocal nature of the component.…”

Space–time adaptive finite elements for nonlocal parabolic variational inequalities

“…For nonlocal material laws, they are intensely studied in peridynamics [18,47,55], but even for local material laws boundary integral formulations give rise to nonlocal problems [29,36,39]. Friction also plays a role in nonlocal evolution equations in image processing [17,33,46], and obstacle problems arise in the study of nonlocal interaction energies in kinetic equations [21].…”

Exact solutions for the denoising problem of piecewise constant images in dimension one

“…Today's literature on the study of problem (3) is extensive, and here we limit ourselves to recall that properties of the solutions have been studied, for instance, in [2,3,4,9,10,14,15,20,23,26,28,31,35,41,45,46], the analysis of variants of (3) that use the generalized total variation have been performed in [11,12,38,40,39], anisotropic models are undertaken in [24,27,29,34], while the effects of considering high-order models have been investigated in [19,21,25,32,39]. Finally, other variants of (2) have been addressed in [6,7,37], and algorithmic considerations may be found in [13,16,18,36].…”