Wavelets: a versatile tool Signal Processing: “Adaptive” affine time-frequency representation Statistics: existence test of moments Paulo Gonçalves INRIA Rhône-Alpes, France On leave @ IST – ISR (2003-2004) IST-ISR January 2004

PDEs applied to Time Frequency Representations Julien Gosme (UTT, France) Pierre Borgnat (IST-ISR) Etienne Payot (Thalès, France) ESGCO Italy - April 19-22, 2002

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

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

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

Combining time and frequency Short-time Fourier Transform Ff Tt < s(.) , δ(. - t) > < s(.) , gt,f(.) > = Q(t,f) = <s(.) , TtFf g0(.) >

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

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

Smoothing to enhance readability Quadratic classes NON ADAPTIVE SMOOTHING

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)

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

Adaptive Smoothing Anisotropic Diffusion

Adaptive Smoothing Anisotropic Diffusion

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)

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)

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

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

Affine diffusion Time-scale covariant heat equations Wavelet Transform < s(.) , TtDa Ψ0 > Affine Diffusion scheme

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,…

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 http://wama2004.org