Yosemite National Park, California The Wigner Function Representation of Multi-Resolution Analysis as a Key to Optimal Wavelet Basis Selection Bedros B. Afeyan Polymath Research Inc., Pleasanton, CA, M. R. Douglas and R. B. Spielman, Sandia National Laboratories, Albuquerque, NM Presented at: YES 2000 Yosemite National Park, California 10/29-11/1, 2000

Outline What are Wavelets, MRD, Wavelet Transforms and Wigner Functions in Phase Space? DWT-SP (PRI) Mathematica NoteBooks: An Introduction Analyzing a Double Gaussian with Two Carriers (generic oscillatory band limited 1D Signal). Comparison to each Gaussian with its own carrier cases. What are the advantages of a Wigner Representation of the MRD for Best Wavelet Basis Selection? Comparisons with other techniques that are less illuminating and less reliable.

What Are Wavelets. Start @ (www. wavelets. org) surf (Mathsoft, amara, Mallat, Meyer, Daubechies, Beylkin, Coifman, Strang, Sweldens, Jawerth... Wavelets are localized kernels or atoms in PHASE SPACE. You may think of them as basis functions with prescribed dilation and translation properties. They may or may not be orthonormal or have compact support or be differentiable everywhere, or be fractal, or have many zero momemts. Wavelets are like breathing wave packets which can home in on structures in phase space better than FT or WFT ever could. When the scale is decreased translation steps between wavelets should likewise be decreased

What is MRD or Multi-resolution Decomposition? Multiresolution: Zoom in and out on a number of successively finer scales in a sequence of nested approximation subspaces {Vj}j in Z. In general, get an overcomplete basis set in L2(R). Approximate (or truncate) by bounding the scales of interest. Scaling functions and the scaling equation: The Wavelets: Low pass filter High pass filter These filters decompose a sampled signal into 2 sub-sampled channels: the coarse approximation of the signal and the missing details at finer scales. The original signal can be reconstructed from these channels by interpolation.

The Scaling Function and Wavelet for Haar or Daubechies 1 in X-Space

The Scaling Function and Wavelet for Haar or Daubechies 1 in K- Space

The Scaling Functions and Wavelets for Daubechies 2-6 in X-Space

The Scaling Functions and Wavelets for Daubechies 2-6 in k-Space

A Little Wavelet Analysis: Pick a Signal, Any Signal!

Here’s MRD in Action for 6 Cases (Multi-Resolution Decomposition)

Data Compression Does Depend on Wavelet Choice

Cumulative Energy In # of Wavelets Kept Helps Discrimination Too

How Does Daubechies 5 Do It?

DWT-SP with Daubechies 5 The Actual Wavelet Coefficients

How Much Better Do You Do With Daubechies 5 than Daubechies 3?

The Least Square Error vs Number of Wavelets Kept for Daubechies 1-6

Comparisons of the Least Square Errors Between Daubechies 3 & 5

Least Square Error Reduction as More Levels Are Added to the MRD for Daubechies 1-6

The Relative Least Squares Error Does Not Go to Zero Very Fast Between Daubechies 3 & 5

How Well Do Box Windowed Fourier Transforms Do?

How Well Do Tapered Windowed Fourier Transforms Do?

What Do Coiflet Scaling Functions Look Like in X-Space?

What Do Coiflet Scaling Functions Look Like in K-Space?

What Do Coiflet Wavelets Look Like in X-Space?

What Do Coiflet Wavelets Look Like in K-Space?

What Do Least Asymmetric Daubechies Filter Scaling Functions Look Like in X-Space?

What Do Least Asymmetric Daubechies Filter Scaling Functions Look Like in K-Space?

What Do Least Asymmetric Daubechies Filter Wavelets Look Like in X-Space?

What Do Least Asymmetric Daubechies Filter Wavelets Look Like in k-Space?

We Can Now Do the Same Analysis with Coiflets and Least Asymmetric Daub Filters As We Did With Daubchies 1-6

The MRD Using Coifflets and Least Asymmetric Daub Filters

Data Compression via Coiflets and Asymmetric Daub Filters

Cumulative Energy Build Up via Coiflets and Least Asymmetric Daub Filters

This Is the MRD Using Least Asymmetric Daub Filter 4

The Actual Wavelet Coefficients Used in the MRD Using Least Asymmetric Daub Filter 4

How Does LADF4 Do Better Than LADF6? It Doesn’t!

RMS Error Decrease as a Function of Number of Wavelets Kept

Reconstruction Using LADF4 Adding a Few Wavelets at a Time

The Cumulative Error During LADF4 MRD Reconstruction

K-Space Reconstruction Using LADF4 A Few Wavelets at a Time

The Cumulative Error in K-Space During LADF4 MRD Reconstruction

What Do Biorthogonal Spline Scaling Functions of Odd and Even Order Look Like in X-Space?

What Do Biorthogonal Spline Scaling Functions of Odd and Even Order Look Like in K-Space?

What Do Biorthogonal Spline Wavelets of Odd and Even Order Look Like in X-Space?

What Do Biorthogonal Spline Wavelets of Odd and Even Order Look Like in K-Space?

What Do Spline Scaling Functions Look Like in X-Space?

What Do Spline Scaling Functions Look Like in K-Space?

What Do Spline Wavelets Look Like in X-Space?

What Do Spline Wavelets Look Like in K-Space?

What Do Shannon Scaling Functions Look Like in X-Space?

What Do Shannon Scaling Functions Look Like in K-Space?

What Do Shannon Wavelets Look Like in X-Space?

What Do Shannon Wavelets Look Like in K-Space?

We Can Now Do the Same Analysis with Biorthogonal Spline, Spline and Shanon Filters, as We Did With Daub1-6, LADF and Coiflets

The MRD Using Biorthogonal Spline, Spline and Shanon Filters

Data Compression via Biorthogonal Spline, Spline and Shanon Filters

Cumulative Energy Build Up via Biorthogonal Spline, Spline and Shanon Filters

This Is the MRD Using a Shannon Filter

Wavelet Coefficients Used in the MRD Using a Shannon Filter

How Does a Shannon Filter Do Better Than Spline Filter[2,8]

RMS Error Decrease as a Function of Number of Wavelets Kept

Scale Dependent Accumulation of RMS Error for Biorthognal Spline, Spline and Shannon MRD

Reconstruction Using a Shannon Filter Adding a Few Wavelets at a Time

Cumulative Error During Shannon Filter MRD Reconstruction

K-Space Reconstruction Using Shannon A Few Wavelets at a Time

The Cumulative Error in K-Space During Shannon MRD Reconstruction

What is A Suitable Phase Space Distribution Function Corresponding to a Signal f(x)? (the Wigner Function) The Wigner function is a bilinear functional of the signal f(x) which represents the signal in phase space- which is to say- simultaneously in position and wavenumber space.

The Wigner Function Representation of our Gaussian Enveloped Two Separate Carriers Signal

Contour Plots of the Wigner Function Representation of our Two Gaussians with Two Carriers Signal

Mean Square Error in Daubechies 1 or Haar MRD’s Performance in Wigner Land (8/57%, 15/36%, 24/12%, 30/7%, 38/11%)

Mean Square Error in Daubechies 3 MRD’s Performance in Wigner Land (8/60%, 15/33%, 23/12%, 30/7%, 38/4%)

Mean Square Error in Daubechies 5 MRD Using the Wigner Function Representation (8/27%, 15/13%, 23/6%, 30/4%, 38/2%)

Mean Square Error in Phase Space for Daubechies 6 MRD Using the Wigner Function (6/58%, 13/23%, 19/9.2%, 25/5%, 32/3%, 38/1.7%)

Wigner Function Representation of the MRD Using LADF2 (6/77%,13/50%,19/30%,26/18%,32/11%,38/7%)

Wigner Function Representation of the MRD Using LADF4 (6/67%,13/30%,19/17%,26/11%,32/6.7%,38/4%)

Wigner Function Representation of the MRD Using LADF6 (6/66%,13/33%,19/14%,26/6%,32/3%,38/2%)

Spline Filters Do a Great Job Also Even Competes with Daubechies 5

Wigner Function Representation of the MRD Using Shannon (6/49%,13/37%,19/30%,25/24%,32/21%,38/21%)

A Single Gaussian with a LF Carrier Is (Less than) Half the Story

How Does Daubechies 5 Break Down the LHS Gaussian?

The WLT Coefficients Corresponding to the Daub 5 MRA of LHS Wavepacket Are

The Wigner Function of One of the Gaussians with a LF Carrier

Haar Has Lots of Trouble Even with Just a Single Scale Gaussian (8/40%,18/16%,23/8%,40/3.3%, 38/0.7%)

Daubechies 3 Is Not Ideal But Does Consistently Beat Haar (8/30%, 18/10%, 23/3%, 30/1%, 38/.5%)

Daubechies 5 Starts with Least Amount of Pollution at Lowest Levels of MRA (8/12%, 18/5%, 23/1%,30/0.3%,38/0.1%)

How About the High Frequency Carrier RHS Wavepacket Acting Alone?

How Does Daubechies 5 Break Down the RHS Gaussian?

The WLT Coefficients Corresponding to the Daub 5 MRA of RHS Wavepacket Are

The Wigner Function of the RHS Wavepacket with a High frequency Carrier

Haar/Daub 1 Has Trouble with High Frequency Carrier Wave Packets (8/39%, 18/31%, 23/15%,30/9.5%,38/4.4%)

Daub 3 Reconstruction is Initially Even Worse But Quickly Recovers (8/52%, 18/19%, 23/10%,30/4%,38/2.5%)

Daub 5 Does Not Even Blink an Eye to Catch the HF Signal (8/21%, 18/10%, 23/6%,30/4%,38/2%)

Haar (or Daub 1)Wavelet’s Wigner Representation In Phase Space

Daubechies 3 Wavelet’s Wigner Representation In Phase Space

Daubechies 5 Wavelet’s Wigner Representation In Phase Space

Daubechies 6 Wavelet’s Wigner Representation In Phase Space

Least Asymmetric Daub Filter 2 Wavelet’s Wigner Representation

Least Asymmetric Daub Filter 4 Wavelet’s Wigner Representation

Least Asymmetric Daub Filter 6 Wavelet’s Wigner Representation

Quadratic Spline Filter Wavelet’s Wigner Representation

Shannon Wavelet’s Wigner Representation

A PRI Prescription The Combined application of Wigner function phase space techniques and those of Multi Resolution Decomposition with Wavelets allows us to detect patterns across scales which would be lost otherwise. By analysing the time evolution of a signal on different scales in space, we can detect changes in patterns that are random from those that are systematic, coherent or resonant. Specially constructed phase space functionals beyond Wigner (bilinear) ones can capture even more pertinent information such as signaling the onset of 3 wave interactions, inverse cascades, etc. These, combined with MRA with the optimum wavelet family for that signal type lead to optimally useful representations.