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Wavelets and Adaptive Smoothing in Time-Frequency Signal Processing

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This work explores the versatile application of wavelet transforms in time-frequency representations of signals, emphasizing adaptive smoothing techniques that enhance readability and analysis. It addresses the existence of moments and the implications of diffusion equations for signal processing. Notably, topics include non-stationarity, short-time Fourier transform, and quadratic classes. The research highlights the computational efficiency and various applications of wavelets, such as in data compression, denoising, and multifractal analysis.

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Wavelets and Adaptive Smoothing in Time-Frequency Signal Processing

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  1. 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

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

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

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

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

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

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

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

  9. Smoothing to enhance readability Quadratic classes NON ADAPTIVE SMOOTHING

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

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

  12. Adaptive SmoothingAnisotropic Diffusion

  13. Adaptive SmoothingAnisotropic Diffusion

  14. Combining time and frequencyWavelet 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)

  15. Combining time and frequencyWavelet 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)

  16. Covariance: time-scale shifts Affine classTime-scale shifts covariance

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

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

  19. Affine diffusionOpen 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) • Wavelets: Theory and ApplicationsA. 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

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