Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

A second order free discontinuity model for bituminous surfacing crack recovery and analysis of a nonlocal version of it

Pages: 49 – 88

DOI: https://dx.doi.org/10.4310/AMSA.2018.v3.n1.a3

Authors

Noémie Debroux (Laboratoire de Mathématiques, Normandie Université, INSA de Rouen, Saint-Etienne-du-Rouvray, France)

Carole Le Guyader (Laboratoire de Mathématiques, Normandie Université, INSA de Rouen, Saint-Etienne-du-Rouvray, France)

Luminita Vese (Department of Mathematics. University of California at Los Angeles)

Abstract

We consider a second order variational model dedicated to crack detection on bituminous surfacing. It is based on a variant of the weak formulation of the Blake–Zisserman functional that involves the discontinuity set of the gradient of the unknown, set that encodes the geometrical thin structures we aim to recover, as suggested by Drogoul et al. Following Ambrosio, Faina and March, an approximation of this cost function by elliptic functionals is provided. Theoretical results including existence of minimizers, existence of a unique viscosity solution to the derived evolution problem, and a $\Gamma$-convergence result relating the elliptic functionals to the initial weak formulation are given. Extending then the ideas developed in the case of first order nonlocal regularization to higher order derivatives, we provide and analyze a nonlocal version of the model.

Keywords

Blake–Zisserman functional, space of generalized special functions of bounded variation (GSBV), elliptic approximations, $G(\mathbb{R}^2)$-space of oscillating functions, infinity Laplacian, viscosity solutions, $\Gamma$-convergence, convergence analysis, partial minimizer, nonlocal second order operators, fractional Sobolev space, tempered distributions, Fourier transform, fine structure segmentation

2010 Mathematics Subject Classification

Primary 45Exx, 49Jxx, 68U10. Secondary 35D40, 65D18.

Full Text (PDF format)

The project is co-financed by the European Union with the European regional development fund (ERDF, HN0002137) and by the Normandie Regional Council via the M2NUM project. The authors would like to thank Denis Jouin and Cyrille Fauchard (CEREMA, France) for providing the bituminous surfacing images.

Received 5 July 2017

Published 27 March 2018