Wavelet Transform and Multiscale Analysis

One of the most fundamental problems in signal processing is to find a suitable representation of the data that will facilitate an analysis procedure. One way to achieve this goal is to use transformation, or decomposition of the signal on a set of basis functions prior to processing in the transform domain. Transform theory has played a key role in image processing for a number of years, and it continues to be a topic of interest in theoretical as well as applied work in this field. Image transforms are used widely in many image processing fields, including image enhancement, restoration, encoding, and description .

Historically, the Fourier transform has dominated linear time-invariant signal processing. The associated basis functions are complex sinusoidal waves eUDt that correspond to the eigenvectors of a linear time-invariant operator. A signal f (t) defined in the temporal domain and its Fourier transform /(«), defined in the frequency domain, have the following relationships [12, 13]:

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Fourier transform characterizes a signal f (t) via its frequency components. Since the support of the bases function eimt covers the whole temporal domain (i.e. infinite support), f (rn) depends on the values of f (t) for all times. This makes the Fourier transform a global transform that cannot analyze local or transient properties of the original signal f (t).

In order to capture frequency evolution of a nonstatic signal, the basis functions should have compact support in both time and frequency domains. To achieve this goal, a windowed Fourier transform (WFT) was first introduced with the use of a window function w(t) into the Fourier transform :

The energy of the basis function gT%%(t) = w(t — t)e—i%t is concentrated in the neighborhood of time t over an interval of size at, measured by the standard deviation of |g|2. Its Fourier transform is gT%%(m) = W(m — %)e—iT%), with energy in frequency domain localized around %, over an interval of size am. In a time-frequency plane (t, m), the energy spread of what is called the atom gT%% (t) is represented by the Heisenberg rectangle with time width at and frequency width am. The uncertainty principle states that the energy spread of a function and its Fourier transform cannot be simultaneously arbitrarily small, verifying:

The shape and size of Heisenberg rectangles of a WFT determine the spatial and frequency resolution offered by such transform.

Examples of spatial-frequency tiling with Heisenberg rectangles are shown in Fig. 6.1. Notice that for a windowed Fourier transform, the shapes of the time-frequency boxes are identical across the whole time-frequency plane, which means that the analysis resolution of a windowed Fourier transform remains the same across all frequency and spatial locations.

To analyze transient signal structures of various supports and amplitudes in time, it is necessary to use time-frequency atoms with different support sizes for different temporal locations. For example, in the case of high-frequency structures, which vary rapidly in time, we need higher temporal resolution to accurately trace the trajectory of the changes; on the other hand, for lower frequency, we will need a relatively higher absolute frequency resolution to give a better measurement of the value of frequency. We will show in the next section that wavelet transform provides a natural representation which satisfies these requirements, as illustrated in Fig. 6.1(d).