The non-Riemannian dislocated crystal: A tribute to Ekkehart Kröner (1919-2000)
Nicolas Van Goethem
Journal of Geometric Mechanics 2010, 2(3): 303-320 doi: 10.3934/jgm.2010.2.303
This expository paper is a tribute to Ekkehart Kröner's results on the intrinsic non-Riemannian geometrical nature of a single crystal filled with point and/or line defects. A new perspective on this old theory is proposed, intended to contribute to the debate around the still open Kröner's question: "what are the dynamical variables of our theory?"
keywords: disclinations strain incompatibility Single crystals dislocations linear elasticity non-Riemannian geometry.
Minimal partitions and image classification using a gradient-free perimeter approximation
Samuel Amstutz Antonio André Novotny Nicolas Van Goethem
Inverse Problems & Imaging 2014, 8(2): 361-387 doi: 10.3934/ipi.2014.8.361
In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.
keywords: deblurring Image classification optimal partitions perimeter approximation.

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