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H 1 Wavelets and stochastic processes: how the Gaussian world became sparse

Auteurs : Unser, Michael (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : We start with a brief historical account of wavelets and of the way they shattered some of the preconceptions of the 20th century theory of statistical signal processing that is founded on the Gaussian hypothesis. The advent of wavelets led to the emergence of the concept of sparsity and resulted in important advances in image processing, compression, and the resolution of ill-posed inverse problems, including compressed sensing. In support of this change in paradigm, we introduce an extended class of stochastic processes specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. Starting from first principles, we prove that the solutions of such equations are either Gaussian or sparse, at the exclusion of any other behavior. Moreover, we show that these processes admit a representation in a matched wavelet basis that is "sparse" and (approximately) decoupled. The proposed model lends itself well to an analytic treatment. It also has a strong predictive power in that it justifies the type of sparsity-promoting reconstruction methods that are currently being deployed in the field.

    Keywords: wavelets - fractals - stochastic processes - sparsity - independent component analysis - differential operators - iterative thresholding - infinitely divisible laws - Lévy processes

    Codes MSC :
    42C40 - Wavelets and other special systems
    60G18 - Self-similar processes
    60G20 - Generalized stochastic processes
    60H40 - White noise theory
    60G22 - Fractional processes, including fractional Brownian motion

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 01/04/15
      Date de captation : 24/01/15
      Collection : Special events ; 30 Years of Wavelets
      Format : quicktime ; audio/x-aac
      arXiv category : Classical Analysis and ODEs ; Numerical Analysis
      Domaine : Analysis and its Applications
      Durée : 00:38:34
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2015-01-24_Unser.mp4

    Informations sur la rencontre

    Nom de la rencontre : 30 years of wavelets / 30 ans des ondelettes
    Organisateurs de la rencontre : Feichtinger, Hans G. ; Torrésani, Bruno
    Dates : 23/01/15 - 24/01/15
    Année de la rencontre : 2015
    URL Congrès : https://www.chairejeanmorlet.com/1523.html

    Citation Data

    DOI : 10.24350/CIRM.V.18723003
    Cite this video as: Unser, Michael (2015). Wavelets and stochastic processes: how the Gaussian world became sparse. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18723003
    URI : http://dx.doi.org/10.24350/CIRM.V.18723003


    1. Unser, M., & Tafti, Pouya D. (2014). An introduction to sparse stochastic processes. Cambridge: Cambridge University Press - www.cambridge.org/9781107058545

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