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In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is twofold. One one hand, we provide Lipschitz regularity results for weakly elliptic non-local HJB, extending the results previously obtained in [8]. On the other hand, we establish a convergence result, based on half relaxed limits and a comparison principle for the effective problem. The latter strongly relies on the regularity and the ellipticity properties of the effective Hamiltonian, for which a fine Lipschitz estimate of the corrector plays a crucial role.

In the fields of nanoscience and nanotechnology, it is important to be able to functionalize surfaces chemically for a wide variety of applications. Scanning tunneling microscopes (STMs) are important instruments in this area used to measure the surface structure and chemistry with better than molecular resolution. Self-assembly is frequently used to create monolayers that redefine the surface chemistry in just a single-molecule-thick layer (Love et al. in Chem Rev 105(4):1103–1170, 2005; Nuzzo and Allara in J Am Chem Soc 105(13):4481–4483, 1983; Smith et al. in Prog Surf Sci 75(1):1–68, 2004). Indeed, STM images reveal rich information about the structure of self-assembled monolayers since they convey chemical and physical properties of the studied material. In order to assist in and to enhance the analysis of STM and other images (Thomas et al. in ACS Nano 10(5):5446–5451, 2016; Thomas et al. in ACS Nano 9(5):4734–4742, 2015), we propose and demonstrate an image processing framework that produces two image segmentations: One is based on intensities (apparent heights in STM images) and the other is based on textural patterns. The proposed framework begins with a cartoon + texture decomposition, which separates an image into its cartoon and texture components. Afterward, the cartoon image is segmented by a modified multiphase version of the local Chan–Vese model (Wang et al. in Pattern Recognit 43(3):603–618, 2010), while the texture image is segmented by a combination of 2D empirical wavelet transform and a clustering algorithm. Overall, our proposed framework contains several new features, specifically in presenting a new application of cartoon + texture decomposition and of the empirical wavelet transforms and in developing a specialized framework to segment STM images and other data. To demonstrate the potential of our approach, we apply it to raw STM images of various monolayers and present their corresponding segmentation results.

International audience; In the fields of nanoscience and nanotechnology, it is important to be able to functionalize surfaces chemically for a wide variety of applications. Scanning tunneling microscopes (STMs) are important instruments in this area used to measure the surface structure and chemistry with better than molecular resolution. Self-assembly is frequently used to create monolayers that redefine the surface chemistry in just a single-molecule-thick layer. Indeed, STM images reveal rich information about the structure of self-assembled monolayers since they convey chemical and physical properties of the studied material. In order to assist in and to enhance the analysis of STM and other images, we propose and demonstrate an image-processing framework that produces two image segmentations: one is based on intensities (apparent heights in STM images) and the other is based on textural patterns. The proposed framework begins with a cartoon+texture decomposition, which separates an image into its cartoon and texture components. Afterward, the cartoon image is segmented by a modified multiphase version of the local Chan-Vese model, while the texture image is segmented by a combination of 2D empirical wavelet transform and a clustering algorithm. Overall, our proposed framework contains several new features, specifically in presenting a new application of cartoon+texture decomposition and of the empirical wavelet transforms and in developing a specialized framework to segment STM images and other data. To demonstrate the potential of our approach, we apply it to actual STM images of cyanide monolayers on Au\{111\} and present their corresponding segmentation results.

Cyanide monolayers on Au{111} restructure from a hexagonal close-packed lattice to a mixed-orientation "ribbon" structure through thermal annealing. The new surface structure loses most of the observed surface features characterizing the initial as-adsorbed system with "ribbon" domain boundaries isolating rotationally offset surface regions where the orientation is guided by the underlying gold lattice. A blue shift to higher frequencies of the CN vibration to 2235cm-1 with respect to the as-adsorbed CN/Au{111} vibration at 2146 cm-1 is observed. In addition, a new low-frequency mode is observed at 145 cm-1, suggesting a chemical environment change similar to gold-cyanide crystallization. We discuss this new structure with respect to a mixed cyanide/isocyanide monolayer and propose a bonding scheme consisting of Au-CN and Au-NC bound molecules that are oriented normal to the Au{111} surface.

International audience; We detail in this paper the numerical implementation of the so-called image curvature microscope , an algorithm that computes accurate image curvatures at subpixel resolution, and yields a curvature map conforming with our visual perception. In contrast to standard methods, which would compute the curvature by a finite difference scheme, the curvatures are evaluated directly on the level lines of the bilinearly interpolated image, after their independent smoothing, a step necessary to remove pixelization artifacts. The smoothing step consists in the affine erosion of the level lines through a geometric scheme, and can be applied in parallel to all level lines. The online algorithm allows the user to visualize the image of curvatures at different resolutions, as well as the set of level lines before and after smoothing. Source Code The ANSI C++ implementation reviewed by IPOL is given in the file curv.cpp. The complete source code, code documentation, and online demo are accessible at the IPOL web page of this article 1. The level lines extraction (files levelLine.cpp, lltree.cpp) and level lines smoothing (file gass.cpp) are independent algorithms, out of the scope of this article, and details for their implementation are planned for separate IPOL publications.

This article refers to algorithms based on finite difference schemes for computing mean and affine curvature evolutions of digital images, introduced by Alvarez and Morel [L. Alvarez, J.M. Morel, “Formalization and computational aspects of image analysis”, Acta Numerica, pp. 159, 1994]. We discuss consistency, stability and convergence. Our analysis focuses on some possible choices of the parameters, choices that generate multiple variants in the implementations. Meaningful visual examples on how the algorithms actually work are provided.

International audience; We are interested in the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. This problem can be reformulated, using level sets, as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous non-coercive Hamiltonian. The periodic setting was earlier studied by Cardaliaguet, Lions and Souganidis (2009). Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton-Jacobi equation converge in $L^\infty$-weak $*$ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media.

International audience; In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of ''mixed operators'' for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.

In this paper we define the continuous Level Lines Shortening evolution of a two-dimensional image as the Curve Shortening operator acting simultaneously and independently on all the level lines of the initial data, and show that it computes a viscosity solution for the mean curvature motion. This provides an exact analytical framework for its numerical implementation, which runs online on any image at http://www.ipol.im/. Analogous results hold for its affine variant version, the Level Lines Affine Shortening., 23 pages

We establish new Hölder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii–Lionsʼs method. We thus extend the Hölder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local–nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

International audience; This paper presents a review, analysis, and comparison of numerical methods implementing the curvature motion and the affine curvature motion for two-dimensional (2D) images, shapes, and curves. These curvature scale spaces allow, in principle, one to compute an accurate multiscale curvature in digital images. The fastest and most invariant of them can be used in a complete image processing chain. This numerical chain simulates the accurate subpixel evolution of an image by mean curvature motion or by affine invariant curvature motion. To do so, it lets all the level lines of the image evolve by curvature shortening (of affine shortening), computes the image curvature directly on the smoothed level lines, and reconstructs the evolved image and its curvatures in an intrinsic, grid independent representation. The paper describes a careful implementation of this chain and analyzes its effects on many examples. The microscopic visualization of an image curvature map reveals after processing many image details. This image process improves graphic images and gets rid of compression and aliasing effects. It also gives an accurate tool to explore the validity of Attneave's and Julesz's theories on shape perception and texture discrimination. The "curvature microscope'' runs online for any image at http://www.ipol.im/pub/algo/cmmm_image_curvature_microscope/.

International audience; This paper presents an image processing algorithm simulating a sub-pixel evolution of an image by mean curvature motion or by affine curvature motion. The sub-pixel algorithm computes the image curvature directly on the smoothed level lines, and yields a microscopic visualization of the curvature map revealing many image details, and getting rid of aliasing effects. This "curvature microscope" showing curvatures in false colors runs on line on any image proposed by users at http://www.ipol.im/pub/algo/cmmm_image_curvature_microscope/.