0.10 Wavelet-based signal processing and applications  (Page 13/13)

The usefulness of wavelets in representing functions in these and several other classes stems from the fact that for most of these spaces thewavelet basis is an unconditional basis , which is a near-optimal property.

To complete this discussion, we have to motivate the property of an unconditional basis being asymptotically optimal for a particular problem,say data compression [link] . [link] suggests why a basis in which the coefficients are solid and orthosymmetric may bedesired. The signal class is defined to be the interior of the rectangle bounded by the lines $x=\pm a$ and $y=\pm b$ . The signal corresponding to point $A$ is the worst-case signal for the two bases shown in the figure; the residual error (with $n=1$ ) is given by $asin\left(\theta \right)+bcos\left(\theta \right)$ for $\theta \in \left\{0,,,\alpha \right\}$ and is minimized by $\theta =0$ , showing that the orthosymmetric basis is preferred. This result is really aconsequence of the fact that $a\ne b$ (which is typically the case why one uses transform coding—if $a=b$ , it turns out that the “diagonal” basis with $\theta =\frac{\pi }{4}$ is optimal for $n=1$ ). The closer the coefficient body is to a solid, orthosymmetric body with varying sidelengths, the less the individual coefficients are correlated with each other and the greater the compression in this basis.

In summary, the wavelet bases have a number of useful properties:

1. They can represent smooth functions.
2. They can represent singularities
3. The basis functions are local. This makes most coefficient-based algorithms naturally adaptive to inhomogeneities in the function.
4. They have the unconditional basis (or near optimal in a minimax sense) property for a variety of function classes implying that if one knowsvery little about a signal, the wavelet basis is usually a reasonable choice.

   

5. They are computationally inexpensive—perhaps one of the few really useful lineartransform with a complexity that is $O\left(N\right)$ —as compared to a Fourier transform, which is $Nlog\left(N\right)$ or an arbitrary linear transform which is $O\left(N^2\right)$ .
6. Nonlinear soft-thresholding is near optimal for statistical estimation.
7. Nonlinear soft-thresholding is near optimal for signal recovery.
8. Coefficient vector truncation is near optimal for data compression.

Applications

Listed below are several application areas in which wavelet methods have had some success.

Numerical solutions to partial differential equations

The use of wavelets as basis functions for the discretization of PDEs has had excellent success. They seem to give a generalization of finiteelement methods with some characteristics of multigrid methods. It seems to be the localizing ability of wavelet expansions that give rise tosparse operators and good numerical stability of the methods [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] .

Seismic and geophysical signal processing

One of the exciting applications areas of wavelet-based signal processing is in seismic and geophysical signal processing. Applications ofdenoising, compression, and detection are all important here, especially with higher-dimensional signals and images. Some of the references can befound in [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] [link] , [link] , [link] , [link] .

Medical and biomedical signal and image processing

Another exciting application of wavelet-based signal processing is in medical and biomedical signal and image processing. Again, applications ofdenoising, compression, and detection are all important here, especially with higher dimensional signals and images. Some of the references can befound in [link] , [link] , [link] .

Application in communications

Some applications of wavelet methods to communications problems are in [link] , [link] , [link] , [link] , [link] .

Fractals

Wavelet-based signal processing has been combined with fractals and to systems that are chaotic [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . The multiresolution formulation of the wavelet and the self-similarcharacteristic of certain fractals make the wavelet a natural tool for this analysis. An application to noise removal from music is in [link] .

Other applications are to the automatic target recognition (ATR) problem, and many other questions.

Wavelet software

There are several software packages available to study, experiment with, and apply wavelet signal analysis. There are several Matlab programs at the end of this book. MathWorks, Inc. has a Wavelet Toolbox [link] ; Donoho's group at Stanford has WaveTool; the Yale group has XWPL and WPLab [link] ; Taswell at Stanford has WavBox [link] , a group in Spain has Uvi-Wave; MathSoft, Inc. has S+WAVELETS; Aware, Inc. has WaveTool; and the DSP group at Rice has a Matlab wavelet toolbox available over the internet at http://www-dsp.rice.edu. There is a good description and list of severalwavelet software packages in [link] . There are several Matlab programs in Appendix C of this book. They were used to create the various examples and figures in this book and should be studied whenstudying the theory of a particular topic.