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 TransformFf 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