A Nonlocal Laplacian-Based Model for Bituminous Surfacing Crack Recovery and its MPI Implementation
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A Nonlocal Laplacian-Based Model for Bituminous Surfacing Crack Recovery and its MPI Implementation Noémie Debroux1 · Carole Le Guyader2 · Luminita A. Vese3 Received: 2 July 2019 / Accepted: 30 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is devoted to the challenging problem of fine structure detection with applications to bituminous surfacing crack recovery. Drogoul (SIAM J Imag Sci 7(4):2700–2731, 2014) shows that such structures can be suitably modeled by a sequence of smooth functions whose Hessian matrices blow up in the perpendicular direction to the crack, while their gradient is null. This observation serves as the basis of the introduced model that also handles the natural dense and highly oscillatory texture 2 2 2 2 exhibited by the images: We propose weighting ∂∂ xu2 + ∂∂ xu2 , u denoting the reconstructed image, by a variable that 1
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annihilates great expansion of this quantity, making then a connection with the elliptic approximation of the Blake–Zisserman functional. Extending then the ideas developed in the case of first-order nonlocal regularization to higher-order derivatives, we derive and analyze a nonlocal version of the model, and provide several theoretical results among which there are a Γ -convergence result as well as a detailed algorithmic approach and an MPI implementation based on a natural domain decomposition approach. Keywords Γ -convergence · Viscosity solutions · Nonlocal second-order operators · Message passing interface (MPI) parallelization Mathematics Subject Classification 35D40 · 45E · 49J45 · 65D18 · 68W10 · 68U10 · 90C06
1 Introduction
This article is dedicated to our wonderful friend and researcher, Mila Nikolova.
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Luminita A. Vese [email protected] Noémie Debroux [email protected] Carole Le Guyader [email protected]
1
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
2
Laboratoire de Mathématiques de l’INSA Rouen Normandie, Institut National des Sciences Appliquées de Rouen, Normandie Université, 76000 Rouen, France
3
Department of Mathematics, University of California Los Angeles, Hilgard Avenue, Los Angeles, CA 90095-1555, USA
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