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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) ESGCO Italy - April 19-22, 2002

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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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**Combining time and frequency Fourier transform**

s(t) s(t) = < s(.) , δ(.-t) > s(t) = < S(.) , ei2πt. > |S(f)| S(f) = < s(.) , ei2πf. > S(f) = < S(.) , δ(.-f) > “Blind” to non stationnarities! u θ

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**Combining time and frequency Non Stationarity: Intuitive**

x(t) Fourier X(f) time frequency Musical Score

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**Combining time and frequency Short-time Fourier Transform**

Ff Tt < s(.) , δ(. - t) > < s(.) , gt,f(.) > = Q(t,f) = <s(.) , TtFf g0(.) >

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**Combining time and frequency Wavelet Transform**

Ψ0( (u–t)/a ) Tt Da frequency Ψ0(u) time < s(.) , TtDa Ψ0 > = O(t,f = f0/a)

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**Combining time and frequency Quadratic classes**

(Affine Class) Wigner dist.: Quadratic class: (Cohen Class) Wigner dist.:

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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**

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

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**Adaptive Smoothing Anisotropic Diffusion**

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**Adaptive Smoothing Anisotropic Diffusion**

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**Combining time and frequency Wavelet Transform**

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) STFT: Constant bandwidth analysis STFT: redundant decompositions (Balian Law Th.) Good for: compression, coding, denoising, statistical analysis Good for: Regularity spaces characterization, (multi-) fractal analysis Computational Cost in O(N) (vs. O(N log N) for FFT)

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**Combining time and frequency Wavelet Transform**

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) STFT: Constant bandwidth analysis STFT: redundant decompositions (Balian Law Th.) Good for: compression, coding, denoising, statistical analysis Good for: Regularity spaces characterization, (multi-) fractal analysis Computational Cost in O(N) (vs. O(N log N) for FFT)

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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**

Wavelet Transform < s(.) , TtDa Ψ0 > Affine Diffusion scheme

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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) 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 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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Transforms: Basis to Basis Normal Basis Hadamard Basis Basis functions Method to find coefficients (“Transform”) Inverse Transform.

Transforms: Basis to Basis Normal Basis Hadamard Basis Basis functions Method to find coefficients (“Transform”) Inverse Transform.

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