Wavelets: a versatile Wavelets: a versatile tooltool

• Signal Processing: “Adaptive” affine

time-frequency representation

• Statistics: existence test of momentsPaulo Gonçalves

INRIA Rhône-Alpes, FranceOn leave @ IST – ISR (2003-

2004)

IST-ISR January 2004

PDEs applied to Time PDEs applied to Time Frequency RepresentationsFrequency Representations

Julien Gosme (UTT, France)Pierre Borgnat (IST-ISR)

Etienne Payot (Thalès, France)

Outline

Atomic linear decompositionsClasses of energetic distributions

Smoothing to enhance readability Diffusion equations: adaptive smoothing

Open issues

s(t)

s(t) = < s(.) , δ(.-t)

>

s(t) = < S(.) , ei2πt. >

Combining time and frequencyFourier transform

|S(f)|

S(f) = < s(.) , ei2πf.

>

S(f) = < S(.) , δ(.-f)

>

“Blin

d” to non st

ationnariti

es!

t)-δ(u

f)-δ(θ

u

θ

time

frequency

Combining time and frequencyNon Stationarity: Intuitive

x(t) X(f)Fourier

Musical Score

-25

-20

-15

-10

-5

time

frequency

< s(.) , gt,f(.) > = Q(t,f)

< s(.) , δ(. - t) >

Combining time and frequencyShort-time Fourier Transform

< s(.) , δ(. – f) >

= <s(.) , TtFf g0(.) >

Ff

Tt

222

4π1 f Δ Δt

Combining time and frequencyWavelet Transform

time

frequency

< s(.) , TtDa Ψ0 > = O(t,f = f0/a)

Ψ0(u)

Ψ0( (u–t)/a )

D

a

Tt

dθ du f)-θt,-Π(u θ(u,W Π) ; f(t, C ss Quadratic class: (Cohen Class)

dσ } f σ exp{-i2π σ/2)-s(t σ/2)s(t : f)(t,WsWigner dist.:

Quadratic class: (Affine Class)

dθ du ) aθ , at-u Π( θ)(u,W Π) ; a(t,Ω ss

dσ } f σ exp{-i2π σ/2)-s(t σ/2)s(t : f)(t,WsWigner dist.:

Combining time and frequencyQuadratic classes

xE

xE dt |s(t)| 2

df |S(f)| 2

| |

df dt | g , s | 2 ft,

t f

212ft,1ft,212ft, dt dt } )(t g )(t g { )s(t )s(t | g , s |

f)t, ; t,(t Π 21

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)

);,(),,(

),()0,,(

,

ftWftW

ftWftW

ft

),(),();,( ftGftWftW

Anisotropic (controlled) diffusion scheme proposed by Perona & Malik (Image Processing)

Adaptive SmoothingAnisotropic Diffusion

));,();,((),,(

),()0,,(

ftWftcdivftW

ftWftW

x

),(),("");,( ),( ftGftWftW ft

Preserves time frequency shifts covariance properties of the Cohen class

Locally control the diffusion rate with a signal dependant time-frequency conductance

Adaptive SmoothingAnisotropic Diffusion

Adaptive SmoothingAnisotropic Diffusion

0 10 20 30 40 50 60 70 80 90 1008.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

10.2

10.4Wigner dist.

Spectrogram (short win.)

Spectrogram (large win.)

Smoothed Pseudo Wigner

Optimal diffused dist.

Entropy based stopping criterium

iteration step

• 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)

Combining time and frequencyWavelet Transform

STFT: Constant bandwidth analysis

STFT: redundant decompositions (Balian Law Th.)

Good for: compression, coding, denoising, statistical analysis

Computational Cost in O(N) (vs. O(N log N) for FFT)

Good for: Regularity spaces characterization, (multi-) fractal analysis

• 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)

Combining time and frequencyWavelet Transform

STFT: Constant bandwidth analysis

STFT: redundant decompositions (Balian Law Th.)

Good for: compression, coding, denoising, statistical analysis

Computational Cost in O(N) (vs. O(N log N) for FFT)

Good for: Regularity spaces characterization, (multi-) fractal analysis

Affine classTime-scale shifts covariance

dθ du aθ , at-uΠ θ)(u,W Π)a;(t,Ω )(ss

Covariance: time-scale shifts

s(t) )(0

0

0 at-ts

a1

a)(t,Ωs )( 00

0s a.a ,a

t-tΩ

);,(),,( 2

atadiv

at

);,(0

0),,(2

2

ftW

f

fdiv

ftW

);,(),,( 2

atdiva

at

);,(),,(

atadiva

at

Affine diffusionTime-scale covariant heat equations

),()0,,( 1 fatWat

Axiomatic approach of multiscale analysis (L. Alvarez, F. Guichard, P.-L. Lions, J.-M. Morel)

Affine diffusionTime-scale covariant heat equations

Affine Diffusion scheme

Wavelet Transform

< s(.) , TtDa Ψ0 >

Affine diffusionOpen Issues

• Corresponding Green function (Klauder)?• Corresponding operator

• linear?• integral?• affine convolution?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)

1. 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

2. Multifractals: Theory and Applications

A. Arnéodo, E. Bacry, L. Biferale, S. Cohen, F. Mendivil, Y. Meyer, R. Riedi, M. Teich, C. Tricot, D. Veitchhttp://wama2004.org