Phase field method for mean curvature flow with boundary constraints

International audienceThis paper is concerned with the numerical approximation of mean curvature flow $t \to \Omega(t)$ satisfying an additional inclusion-exclusion constraint $\Omega_1 \subset \Omega(t) \subset \Omega_2$. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a $\Gamma$-convergence result and then show some numerical comparisons of these two different models

Divergence-free Wavelets for Navier-Stokes

In this paper, we investigate the use of compactly supported divergence-free wavelets for the representation of the Navier-Stokes solution. After reminding the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for 2D and 3D incompressible flows. In order to compute the nonlinear term, we propose a new method which provides in practice with the Hodge decomposition of any flow: this decomposition enables us to separate the incompressible part of the flow from its orthogonal complement, which corresponds to the gradient component of the flow. Finally we show numerical tests to validate our approach.Comment: novembre 200

The monogenic synchrosqueezed wavelet transform: a tool for the decomposition/demodulation of AM–FM images

The synchrosqueezing method aims at decomposing 1D functions into superpositions of a small number of “Intrinsic Modes”, supposed to be well separated both in time and frequency. Based on the unidimensional wavelet transform and its reconstruction properties, the synchrosqueezing transform provides a powerful representation of multicomponent signals in the time–frequency plane, together with a reconstruction of each mode. In this paper, a bidimensional version of the synchrosqueezing transform is defined, by considering a well-adapted extension of the concept of analytic signal to images: the monogenic signal. We introduce the concept of “Intrinsic Monogenic Mode”, that is the bidimensional counterpart of the notion of Intrinsic Mode. We also investigate the properties of its associated Monogenic Wavelet Decomposition. This leads to a natural bivariate extension of the Synchrosqueezed Wavelet Transform, for decomposing and processing multicomponent images. Numerical tests validate the effectiveness of the method on synthetic and real images

The Fourier-based Synchrosqueezing Transform

The short-time Fourier transform (STFT) and continuous wavelet transform (CWT) are intensively used to analyze and process multicomponent signals, ie superpositions of mod- ulated waves. The synchrosqueezing is a post-processing method which circumvents the uncertainty relations, inherent to these linear transforms, by reassigning the coefficients in scale or frequency. Originally introduced in the setting of the continuous wavelet transform, it provides a sharp, con- centrated representation, while remaining invertible. This technique received a renewed interest with the recent publi- cation of an approximation result, which provides guarantees for the decomposition of a multicomponent signal. This paper adapts the formulation of the synchrosqueezing to the STFT, and states a similar theoretical result. The emphasis is put on the differences with the CWT-based synchrosqueezing, and all the content is illustrated through numerical experiments

On the Mode Synthesis in the Synchrosqueezing Method

Publication in the conference proceedings of EUSIPCO, Bucharest, Romania, 201

Comment un genre de sites construit des niches professionnelles

De la même façon que leurs ancêtres imprimés, audio et visuels, les médias numériques sont accompagnés de discours qui suivent leur naissance et leur développement en termes d’usage et de modèles d’activité aussi bien qu’en termes de modèles de communication. Récemment, les blogs, à la suite des sites web, ont été présentés dans ces discours comme étant très innovants et révolutionnaires pour les journalistes et pour le journalisme, allant jusqu’à changer les règles de communication dans l’ensemble. Ces discours produisent des « imaginaires » sur eux-mêmes. Une fois de plus, en 2009, on a entendu les mêmes thèmes en ce qui concerne les soi-disant « réseaux sociaux » comme la marque nommée Twitter. Cet article analyse les effets de ce cycle discursif sur le journalisme et sur ses pratiques.In the same manner as their printed, audio and visual ancestors, digital media are escorted by discourses, that follow their birth and development, in terms of usage and business models, as well as communication models. Recently, weblogs, following websites, have been presented, in these discourses, as being very innovative and revolutionary for journalists and journalism, even changing the rules of communication as a whole. These discourses produce “imaginaries” about them. Again, in 2009, the same themes were heard, concerning the so-called “social networks,” such as the brand named Twitter. The article analyses the effects of this discursive cycle on journalism and its practices

Mod\'elisations de textures par champ gaussien \`a orientation locale prescrite

This paper presents two new models of oriented texture, based on a new class of Gaussian fields, called locally anisotropic fractional Brownian fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation with the Cholesky method or the turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation.Comment: in Frenc

Convex Super-Resolution Detection of Lines in Images

International audienceIn this paper, we present a new convex formulation for the problem of recovering lines in degraded images. Following the recent paradigm of super-resolution, we formulate a dedicated atomic norm penalty and we solve this optimization problem by means of a primal–dual algorithm. This parsimonious model enables the reconstruction of lines from lowpass measurements, even in presence of a large amount of noise or blur. Furthermore, a Prony method performed on rows and columns of the restored image, provides a spectral estimation of the line parameters, with subpixel accuracy

Texture Modeling by Gaussian fields with prescribed local orientation

International audienceThis paper presents a new framework for oriented texture modeling. We introduce a new class of Gaussian fields, called Locally Anisotropic Fractional Brownian Fields, with prescribed local orientation at any point. These fields are a local version of a specific class of anisotropic self-similar Gaussian fields with stationary increments. The simulation of such textures is obtained using a new algorithm mixing the tangent field formulation and a turning band method, this latter method having proved its efficiency for generating stationary anisotropic textures. Numerical experiments show the ability of the method for synthesis of textures with prescribed local orientation

From CNNs to Shift-Invariant Twin Models Based on Complex Wavelets

We propose a novel antialiasing method to increase shift invariance and prediction accuracy in convolutional neural networks. Specifically, we replace the first-layer combination "real-valued convolutions + max pooling" ($\mathbb{R}$Max) by "complex-valued convolutions + modulus" ($\mathbb{C}$Mod), which is stable to translations. To justify our approach, we claim that $\mathbb{C}$Mod and $\mathbb{R}$Max produce comparable outputs when the convolution kernel is band-pass and oriented (Gabor-like filter). In this context, $\mathbb{C}$Mod can be considered as a stable alternative to $\mathbb{R}$Max. Thus, prior to antialiasing, we force the convolution kernels to adopt such a Gabor-like structure. The corresponding architecture is called mathematical twin, because it employs a well-defined mathematical operator to mimic the behavior of the original, freely-trained model. Our antialiasing approach achieves superior accuracy on ImageNet and CIFAR-10 classification tasks, compared to prior methods based on low-pass filtering. Arguably, our approach's emphasis on retaining high-frequency details contributes to a better balance between shift invariance and information preservation, resulting in improved performance. Furthermore, it has a lower computational cost and memory footprint than concurrent work, making it a promising solution for practical implementation