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Wavelets: a versatile tool Signal Processing: “Adaptive” affine time-frequency representation Statistics: existence test of moments Paulo Gonçalves INRIA Rhône-Alpes, France On IST – ISR ( ) IST-ISR January 2004

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PDEs applied to Time Frequency Representations Julien Gosme (UTT, France) Pierre Borgnat (IST-ISR) Etienne Payot (Thalès, France)

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Outline Atomic linear decompositions Classes of energetic distributions Smoothing to enhance readability Diffusion equations: adaptive smoothing Open issues

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s(t) s(t) = Combining time and frequency Fourier transform |S(f)| S(f) = “Blind” to non stationnarities! u θ

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time frequency Combining time and frequency Non Stationarity: Intuitive x(t)X(f) Fourier Musical Score time frequency

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= Q(t,f) Combining time and frequency Short-time Fourier Transform = FfFf TtTt

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Combining time and frequency Wavelet Transform time frequency = O(t,f = f 0 /a) Ψ 0 (u) Ψ 0 ( (u–t)/a ) DaDa TtTt

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Quadratic class: (Cohen Class) Wigner dist.: Quadratic class: (Affine Class) Wigner dist.: Combining time and frequency Quadratic classes

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Smoothing to enhance readability Quadratic classes NON ADAPTIVE SMOOTHING

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Smoothing… Heat Equation and Diffusion Uniform gaussian smoothing as solution of the Heat Equation (Isotropic diffusion) Anisotropic (controlled) diffusion scheme proposed by Perona & Malik (Image Processing)

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Adaptive Smoothing Anisotropic Diffusion Preserves time frequency shifts covariance properties of the Cohen class Locally control the diffusion rate with a signal dependant time-frequency conductance

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Adaptive Smoothing Anisotropic Diffusion

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Frequency dependent resolutions (in time & freq.) (Constant Q analysis) Orthonormal Basis framework (tight frames) Unconditional basis and sparse decompositions Pseudo Differential operators Fast Algorithms (Quadrature filters) Combining time and frequency Wavelet Transform STFT: Constant bandwidth analysis STFT: redundant decompositions (Balian Law Th.) Good for: compression, coding, denoising, statistical analysis Computational Cost in O(N) (vs. O(N log N) for FFT) Good for: Regularity spaces characterization, (multi-) fractal analysis

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Frequency dependent resolutions (in time & freq.) (Constant Q analysis) Orthonormal Basis framework (tight frames) Unconditional basis and sparse decompositions Pseudo Differential operators Fast Algorithms (Quadrature filters) Combining time and frequency Wavelet Transform STFT: Constant bandwidth analysis STFT: redundant decompositions (Balian Law Th.) Good for: compression, coding, denoising, statistical analysis Computational Cost in O(N) (vs. O(N log N) for FFT) Good for: Regularity spaces characterization, (multi-) fractal analysis

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Affine class Time-scale shifts covariance Covariance: time-scale shifts

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Affine diffusion Time-scale covariant heat equations Axiomatic approach of multiscale analysis (L. Alvarez, F. Guichard, P.-L. Lions, J.-M. Morel)

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Affine diffusion Time-scale covariant heat equations Affine Diffusion scheme Wavelet Transform

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Affine diffusion Open Issues Corresponding Green function (Klauder)? Corresponding operator linear? integral? affine convolution? Stopping criteria? (Approached) reconstruction formula? Matching pursuit, best basis selection Curvelets, edgelets, ridgelets, bandelets, wedgelets,…

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Wavelet And Multifractal Analysis (WAMA) Summer School in Cargese (Corsica), July 19-31, 2004 (P. Abry, R. Baraniuk, P. Flandrin, P. Gonçalves, S. Jaffard) 1. Wavelets: Theory and Applications A. Aldroubi, A. Antoniadis, E. Candes, A. Cohen, I. Daubechies, R. Devore, A. Grossmann, F. Hlawatsch, Y. Meyer, R. Ryan, B. Torresani, M. Unser, M. Vetterli 2.Multifractals: Theory and Applications A. Arnéodo, E. Bacry, L. Biferale, S. Cohen, F. Mendivil, Y. Meyer, R. Riedi, M. Teich, C. Tricot, D. Veitch

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