Analyse harmonique (mathématiques)

Résumé

thumb|upright=1.2|Analyseur harmonique mécanique de Lord Kelvin datant de 1878. L'analyse harmonique est la branche des mathématiques qui étudie la représentation des fonctions ou des signaux comme superposition d'ondes de base. Elle approfondit et généralise les notions de série de Fourier et de transformée de Fourier. Les ondes de base s'appellent les harmoniques, d'où le nom de la discipline. Durant ces deux derniers siècles, elle a eu de nombreuses applications en physique sous le nom d'analyse spectrale, et connaît des applications récentes notamment en traitement des signaux, mécanique quantique, neurosciences, stratigraphie… Des analyseurs harmoniques mécaniques ont vu le jour vers 1920 et permettaient d'obtenir graphiquement jusqu'au d'un développement de Fourier. L'analyse harmonique, historiquement liée au développement de la théorie des séries de Fourier, a reçu un ensemble de généralisations modernes, notamment grâce aux travaux de l'école russe de Gelfand, qui la situe d

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https://fr.wikipedia.org/wiki/Analyse_harmonique_(math%C3%A9matiques)

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We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.

Springer2019

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3D Spectral Nonrigid Registration of Facial Expression Scans

Gabriel Louis Cuendet, Christophe René Joseph Ecabert, Hazim Kemal Ekenel, Jean-Philippe Thiran, Marina Zimmermann

In this paper, we introduce a new template-based spectral nonrigid registration method in which the target is represented using multilevel partition of unity (MPU) implicit surfaces and the template is embedded in a discrete Laplace-Beltrami based spectral representation using the manifold harmonics transform (MHT). The implicit surface parametrization of the target allows us to avoid computing correspondences during the registration as in classical nonrigid iterative closest point (ICP) techniques. It also allows us to denoise the 3D scans and fill the holes by interpolating the noisy 3D data and to incorporate different types of 3D surfaces into our model, independently of their original parametrization. We take advantage of spectral geometry processing methods to compute a spectral embedding of the template and use it as a parametric surface deformation model. We optimize the nonrigid deformation directly in the spectral domain, thus effectively reducing the size of the parameter space as compared with the classical per vertex affine transformation deformation model. In addition, we introduce a new 3D facial expressions database, EPFL3DFace, on which we apply the proposed method to nonrigidly register 3D face scans that contain different expressions. This database consists of 3D scans of 120 subjects posing 35 different facial expressions. These include various standard prototypical facial expressions as well as individual action units, visemes, and the facial movement of biting one’s own upper lip, which are suitable for a large variety of applications.

Institute of Electrical and Electronics Engineers

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On the Averaged Green’s Function of an Elliptic Equation with Random Coefficients

Marius Christopher Lemm

We consider a divergence-form elliptic difference operator on the lattice Zd, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green’s function of this model. Our main contribution is a refinement of Bourgain’s approach which improves the key decay rate from −2d+ϵ to −3d+ϵ. (The optimal decay rate is conjectured to be −3d.) As an application, we derive estimates on higher derivatives of the averaged Green’s function which go beyond the second derivatives considered by Delmotte–Deuschel and related works.

2019

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