Download presentation

Presentation is loading. Please wait.

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) ESGCO Italy - April 19-22, 2002

3
**Outline Atomic linear decompositions**

Classes of energetic distributions Smoothing to enhance readability Diffusion equations: adaptive smoothing Open issues

4
**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

5
**Combining time and frequency Non Stationarity: Intuitive**

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

6
**Combining time and frequency Short-time Fourier Transform**

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

7
**Combining time and frequency Wavelet Transform**

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

8
**Combining time and frequency Quadratic classes**

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

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**

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

12
**Adaptive Smoothing Anisotropic Diffusion**

13
**Adaptive Smoothing Anisotropic Diffusion**

14
**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)

15
**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)

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**

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

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) 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

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google