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Wavelets: a versatile tool Signal Processing: “Adaptive” affine time-frequency representation Statistics: existence test of moments Paulo Gonçalves INRIA.

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Presentation on theme: "Wavelets: a versatile tool Signal Processing: “Adaptive” affine time-frequency representation Statistics: existence test of moments Paulo Gonçalves INRIA."— Presentation transcript:

1 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

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

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

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

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

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

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

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

9 Smoothing to enhance readability Quadratic classes NON ADAPTIVE SMOOTHING

10 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)

11 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

12 Adaptive Smoothing Anisotropic Diffusion

13

14 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

15 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

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

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

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

19 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,…

20 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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