0.2 A multiresolution formulation of wavelet systems  (Page 7/8)

Experimentation with these displays can be very informative in terms of the properties and capabilities of the wavelet transform, the effects ofparticular wavelet systems, and the way a wavelet expansion displays the various attributes or characteristics of a signal.

Examples of wavelet expansions

In this section, we will try to show the way a wavelet expansion decomposes a signal and what the components look like at different scales.These expansions use what is called a length-8 Daubechies basic wavelet (developed in Chapter: Regularity, Moments, and Wavelet System Design ), but that is not the main point here. The local nature of the wavelet decomposition is the topic of this section.

These examples are rather standard ones, some taken from David Donoho's papers and web page. The first is a decomposition of a piecewise linearfunction to show how edges and constants are handled. A characteristic of Daubechies systems is that low order polynomials are completely containedin the scaling function spaces $V_j$ and need no wavelets. This means that when a section of a signal is a section of a polynomial (such as a straight line), there are no wavelet expansion coefficients $d_j\left(k\right)$ , but when the calculation of the expansion coefficients overlaps an edge,there is a wavelet component. This is illustrated well in [link] where the high resolution scales gives a very accurate location of the edges and this spreads out over $k$ at the lower scales. This gives a hint of how the DWT could be used for edge detection and howthe large number of small or zero expansion coefficients could be used for compression.

[link] shows the approximations of the skyline signal in the various scaling function spaces $V_j$ . This illustrates just how the approximations progress, giving more and more resolution at higherscales. The fact that the higher scales give more detail is similar to Fourier methods, but the localization is new. [link] illustrates the individual wavelet decomposition by showing the components of the signal that exist in the wavelet spaces $W_j$ at different scales $j$ . This shows the same expansion as [link] , but with the wavelet components given separately rather than being cumulatively added to the scalingfunction. Notice how the large objects show up at thelower resolution. Groups of buildings and individual buildings are resolved according to their width.The edges, however, are located at the higher resolutions and are located very accurately.

The second example uses a chirp or doppler signal to illustrate how a time-varying frequency is described by the scale decomposition. [link] gives the coefficients of the DWT directly as a function of $j$ and $k$ . Notice how the location in $k$ tracks the frequencies in the signal in a way the Fourier transform cannot.  [link] and [link] show the scaling function approximations and the wavelet decomposition of this chirp signal. Again, notice inthis type of display how the “location" of the frequencies are shown.