The Wavelet Transform and Wavelet-Based Filtering (Biomedical Image Analysis)

The wavelet transform belongs to the class of integral transforms, such as the Fourier transform. Whereas the Fourier transform uses sine and cosine functions as basis functions, the wavelet transform uses special functions with finite support, termed wavelets. The most fundamental difference between wavelets and the Fourier transform is the scalability of wavelets. Whereas the Fourier transform allows us to analyze the frequency components only globally, shifting and scaling operations used in the wavelet transform allow us to analyze local properties of a signal or image. The wavelet transform is therefore useful in analyzing signals and images with large discontinuities and nonperiodic signals (the Fourier transform assumes all signals to be periodic). The wavelet transform gives rise to powerful and flexible filters, to the analysis of images at different scales (multiscale analysis), and to lossy image compression methods.

To understand the wavelet transform, let us recall the definition of convolution (Section 2.3) of a time function f(t) with a kernel g(t):


The integral in Equation (4.1) needs to be evaluated for all possible values of t .If the kernel has finite support [e.g., g(t) is completely zero for all t < — T and for all t > T], the integration boundaries may be finite, too. In this example, integration would take place from t – T to T — t . For each value of t for which Equation (4.1) is evaluated, the integral returns one scalar value. Let us call this value W(t ). Also, let us choose a wavelet function ^ (t) for the kernel g(t). Wavelet functions are a specific class of functions that are explained below. With these definitions, Equation (4.1) can be rewritten as


We can read Equation (4.2) as computing one convolution value W(t ) of the function to be transformed, f, by evaluating the integral in Equation (4.2) at the value t , which causes the wavelet to be centered onf(T). Now let us introduce another parameter, s, which allows us to stretch (s > 1) or compress (0 < s < 1) the wavelet function. We now have two selectable parameters, t and s, and Equation (4.2) extends into


With Equation (4.3) we transform f (t) into a two-dimensional space (s,t ). The parameter t selects the focus where the wavelet function is centered on f (t), and the parameter s determines the sharpness of the focus, since for small s, only small sections of f are included in the integration, and for large s, longer sections of f are included in the integration. With an additional normalization factor, we arrive at the definition of the one-dimensional, continuous wavelet transform W(s,t ){f} of a function f (t):


withtmp2F-160_thumbbeing the Fourier transform oftmp2F-161_thumb(t). For a functiontmp2F-162_thumb(t) to be a wavelet, it is necessary thattmp2F-163_thumbbe finitetmp2F-164_thumband that the mean value oftmp2F-165_thumb(t) vanishestmp2F-166_thumb. Furthermore,tmp2F-167_thumb(t) must have finite support, that is, beyond a certain value oftmp2F-168_thumb(t) = 0 for t < —11 and fortmp2F-169_thumb. The simplest example of a wavelet function is the Haar wavelet (Figure 4.1), defined by wheretmp2F-170_thumb(t) is the wavelet function, s and t are the scaling and shift parameters, respectively, andtmp2F-171_thumbis defined as


The wavelet properties, namely, the zero mean value and the finite support (0 to 1), can be seen immediately. The finite value for c^ can be proven by computing the Fourier transform of the Haar wavelet and applying Equation (4.5). For the Haar wavelet, c^ = 2 ln 2. Intuitively, application of the Haar wavelet in the wavelet transform [Equation (4.4)] can be interpreted as the computation of the finite difference of two range averages: from 0 to 0.5 and from 0.5 to 1. This intuitive interpretation of the wavelet transform as the computation of a weighted finite difference will become important when we introduce the bandpass and lowpass components of the wavelet analysis filter.

In wavelet analysis, the unscaled wavelet ^ (t) is referred to as the mother wavelet, and the scaled and translated wavelets ^ s,T (t) that are obtained from the mother wavelet,


are called the wavelet basis. From the wavelet properties described above, it can be concluded that the mother wavelet has a frequency spectrum that exhibits band-passlike characteristics, rapidly dropping off toward w = 0 and gradually dropping off toward high frequencies (Figure 4.1B). Translation in time (parameter t ) only changes the phase of the Fourier transform, but changing the scale s changes the bandwidth of the Fourier transform in the opposite direction:


With suitable scaling steps (e.g., doubling s for each wavelet), the wavelet basis acts like a bandpass filter bank, each expanded wavelet providing a lower (and compressed) frequency band.

The Haar wavelet (A) and the magnitude of its Fourier transform (B).

FIGURE 4.1 The Haar wavelet (A) and the magnitude of its Fourier transform (B).

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