Wavelets: a versatile tool

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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
leave _at_ IST ISR (2003-2004)
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)

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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 frequencyFourier transform

• s(t)
• s(t) lt s(.) , d(.-t) gt
• s(t) lt S(.) , ei2pt. gt

• S(f)
• S(f) lt s(.) , ei2pf. gt
• S(f) lt S(.) , d(.-f) gt

Blind to non stationnarities!
u
?
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Combining time and frequencyNon Stationarity
Intuitive
Fourier
x(t)
X(f)
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Combining time and frequencyShort-time Fourier
Transform
lt s(.) , d(. f) gt

• lt s(.) , d(. - t) gt

lt s(.) , gt,f(.) gt Q(t,f)
lts(.) , TtFf g0(.) gt
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Combining time and frequencyWavelet Transform
frequency
time
lt s(.) , TtDa ?0 gt O(t,f f0/a)
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Combining time and frequencyQuadratic classes
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Smoothing to enhance readability Quadratic
classes
NON ADAPTIVE SMOOTHING
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SmoothingHeat Equation and Diffusion
Uniform gaussian smoothing as solution of the
Heat Equation (Isotropic diffusion)
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Adaptive SmoothingAnisotropic Diffusion
Locally control the diffusion rate with a signal
dependant time-frequency conductance
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Adaptive SmoothingAnisotropic Diffusion
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Adaptive SmoothingAnisotropic Diffusion
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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)
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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)
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Affine classTime-scale shifts covariance
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Affine diffusionTime-scale covariant heat
equations
Axiomatic approach of multiscale analysis (L.
Alvarez, F. Guichard, P.-L. Lions, J.-M. Morel)
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Affine diffusionTime-scale covariant heat
equations
Affine Diffusion scheme
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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,

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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 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