1.1 Discrete-time signals  (Page 10/10)

The second almost universal requirement is that the wavelet system generates a multiresolution analysis (MRA). This means that alow resolution function (low scale $j$ ) can be expanded in terms of the same function at a higher resolution (higher $j$ ). This is stated by requiring that the generator of a MRA waveletsystem, called a scaling function $\phi \left(t\right)$ , satisfies

This equation, called the refinement equation or the MRA equation or basic recursion equation , is similar to a differential equation in that its solution is what defines thebasic scaling function and wavelet [link] , [link] .

The current state of the art is that most of the necessary and sufficient conditions on the coefficients $h\left(n\right)$ are known for the existence, uniqueness, orthogonality, and other propertiesof $\phi \left(t\right)$ . Some of the theory parallels Fourier theory and some does not.

A third important feature of a MRA wavelet system is a discrete wavelet transform (DWT) can be calculated by a digitalfilter bank using what is now called Mallat's algorithm. Indeed, this connection with digital signal processing (DSP) hasbeen a rich source of ideas and methods. With this filter bank, one can calculate the DWT of a length-N digital signal withorder N operations. This means the number of multiplications and additions grows only linearly with the length of the signal.This compares with $Nlog\left(N\right)$ for an FFT and $N^2$ for most methods and worse than that for some others.

These basic ideas came from the work of Meyer, Daubechies, Mallat, and others but for a time looked like a solution lookingfor a problem. Then a second phase of research showed there are many problems to which the wavelet is an excellent solution. Inparticular, the results of Donoho, Johnstone, Coifman, Beylkin, and others opened another set of doors.

Generalization of the basic wavelet system

After (in some cases during) much of the development of the above basic ideas,a number of generalizations [link] were made. They are listed below:

1. A larger integer scale factor than $M=2$ can be used to give a more general M-band refinement equation [link]
   $$\phi \left(t\right)=\sum _{n}h\left(n\right)\phi (Mt-n)$$
   than the “dyadic" or octave based Equation 4 from Rational Function Approximation . This also gives more than two channels in the accompanying filter bank. Itallows a uniform frequency resolution rather than the resulting logarithmic one for $M=2$ .
2. The wavelet system called a wavelet packet is generated by “iterating" the wavelet branches of the filter bank to give afiner resolution to the wavelet decomposition. This was suggested by Coifman and it too allows a mixture of uniform andlogarithmic frequency resolution. It also allows a relatively simple adaptive system to be developed which has anautomatically adjustable frequency resolution based on the properties of the signal.
3. The usual requirement of translation orthogonality of the scaling function and wavelets can be relaxed to give what iscalled a biorthogonal system [link] . If the expansion basis is not orthogonal, a dual basis can be created that willallow the usual expansion and coefficient calculations to be made. The main disadvantage is the loss of a Parseval's theoremwhich maintains energy partitioning. Nevertheless, the greater flexibility of the biorthogonal system allows superiorperformance in many compression and denoising applications.
4. The basic refinement Equation 4 from Rational Function Approximation gives the scaling function in terms of a compressed version of itself(self-similar). If we allow two (or more) scaling functions, each being a weighted sum of a compress version of both, a moregeneral set of basis functions results. This can be viewed as a vector of scaling functions with the coefficients being a matrixnow. Once again, this generalization allows more flexibility in the characteristics of the individual scaling functions andtheir related multi-wavelets. These are called multi-wavelet systems and are still being developed.
5. One of the very few disadvantages of the discrete wavelet transform is the fact it is not shift invariant. In otherwords, if you shift a signal in time, its wavelet transform not only shifts, it changes character! For many applications indenoising and compression, this is not desirable although it may be tolerable. The DWT can be made shift-invariant by calculating the DWT of a signal for all possible shifts andadding (or averaging) the results. That turns out to be equivalent to removing all of the down-samplers in theassociated filter bank (an undecimated filter bank ), which is also equivalent to building an overdetermined or redundant DWT from a traditional wavelet basis. Thisovercomplete system is similar to a “tight frame" and maintains most of the features of an orthogonal basis yet is shiftinvariant. It does, however, require $Nlog\left(N\right)$ operations.
6. Wavelet systems are easily modified to being an adaptive system where the basis adjusts itself to the properties of the signalor the signal class. This is often done by starting with a large collection or library of expansion systems and bases. Asubset is adaptively selected based on the efficiency of the representation using a process sometimes called pursuit . In other words, a set is chosen that will result in the smallestnumber of significant expansion coefficients. Clearly, this is signal dependent, which is both its strength and its limitation.It is nonlinear.
7. One of the most powerful structures yet suggested for using wavelets for signal processing is to first take the DWT, then doa point-wise linear or nonlinear processing of the DWT, finally followed by an inverse DWT. Simply setting some of thewavelet domain expansion terms to zero results in linear wavelet domain filtering, similar to what would happen if the same weredone with Fourier transforms. Donoho [link] , [link] and others have shown by using some form of nonlinear thresholding of the DWT, one can achieve nearoptimal denoising or compression of a signal. The concentrating or localizing character of the DWT allows this nonlinearthresholding to be very effective.

The present state of activity in wavelet research and application shows great promise based on the abovegeneralizations and extensions of the basic theory and structure [link] . We now have conferences, workshops, articles, newsletters, books, and email groups that are moving the stateof the art forward. More details, examples, and software are given in [link] , [link] , [link] .