Wavelet Filtering

The term image enhancement is also a very general term, which encompasses many techniques. We have recognized that it is often used to describe the outcome of filtering. If the definition of enhancement is applying a process to the image that results in a "better" overall image appearance, then the term is a misnomer. Linear filtering blocks a portion of the true signal in most applications, which is probably not best defined as enhancement. In this section we provide a qualitative description of filtering. The only assumption we make here is that the reader understands Fourier analysis in one dimension. If so, the extension to two dimensions will be easily accomplished.

Things that change rapidly in the signal domain give rise to larger amplitudes for the sine waves that wiggle more quickly (high frequencies) in the Fourier expansion of the signal. Likewise, things that have long-range structure in the signal domain, give rise to larger amplitudes for the sine waves that wiggle slowly (low frequencies) in the Fourier expansion. Of course, there are many structures that lie in the middle of these extremes. The reader should keep in mind that the above descriptors are relative terms. Signals that are delta-function like will give rise to Fourier components across the entire spectrum. There is more to the story, because we are working in two dimensions. Let's consider a two-dimensional function or contrived image that is nothing more than an infinite vertical line (^-direction) of a few pixels in width in the other direction (r-direction) embedded in an empty 2-D field. Note that in the vertical direction there is no variation along the line indicating it will look as a low frequency signal in this direction (this may be considered as a sine wave with an infinite period, which may be considered as a DC component). If we approach this line from the horizontal direction, it appears as an abrupt change, for an instance, then it is flat for a few pixels, then another abrupt change takes place, and it is gone; that is, any horizontal slice will look like the rectangle function that is used in many Fourier analysis textbooks for examples. It takes two frequency coordinates to describe this signal or any 2-D signal (image). Specifically, this signal is purely a DC signal in the y-direction when considering its Fourier composition and a sinc-type function in the other direction. Consequently, the transform is a sinc function along the fx coordinate axis and about zero elsewhere; this may be deduced by considering the (fx,fy) coordinates and noting that the fx component is zero everywhere except at fy = 0. The following may be observed: (1) Linear structures in the vertical direction are likely to give rise to Fourier signatures in the fx direction. The narrower the width the more spread out the contribution is in the Fourier fx direction and the wider in width, the more contracted along the fx direction. (2) Linear structures in the horizontal direction are likely to have significant Fourier signatures in the fy direction and less in the fx direction. (3) Taking this a step further, spots give rise to components in both coordinate directions. These examples are idealizations that may inspire the newcomer to Fourier analysis to observe what exactly the Fourier Transform is telling the user.

Filtering can be applied to set the stage for detection or segmentation. The basic idea is that there is some structure that we define as the signal of interest, which in our case is the localized calcified areas in mammograms termed "calcifications." These signals are surrounded (or embedded in) by other signals (in this case normal breast tissue) that may interfere with the ability to automatically detect them. In the best scenario, the signal of interest will have a frequency signature that is somewhat different than the background. If this is the case, filtering the image will pass the signal of interest (perhaps not intact) and block a portion of the background tissue. If this is successful, the filtered image will show a relatively more pronounced calcified area and a somewhat subdued background when compared with the raw image.

A simple somewhat contrived example of the usefulness behind filtering is proved here. Suppose we have a white 2-D (n x n) noise field with variance a2 and filter it with a perfect band pass filter. Can we say anything about the resulting noise power? The answer is yes. White noise by definition is a flat power spectrum (more correctly a constant power spectral density). For illustration, we will apply a perfect half-band filter to this field and calculate the resulting noise power. In the Fourier domain, the half band filter looks like a square box centered about zero of unit height with its sidewalls intersecting the frequency coordinates and the midway point. Fourier components within the box are passed when filtering and everything outside is blocked. Thus the total area in the Fourier domain is n2, the pass-band area is (n/2)2, and the blocked portion of the Fourier domain is n2 — (n/2)2 = 4n2. Considering the transform properties, the resulting noise power is 4 a2. The important point here is that if the signal of interest has strong signatures in the lower part of the frequency spectrum, they will be passed almost intact while the noise would be heavily damped. This is an idealized situation that helps to understand the reasons for filtering.

In the following, a very brief description of wavelet analysis is presented. There is no way that we can give proper justice to this elegant theory of signal decomposition. Beware it took great minds many years to put wavelet analysis on such a beautiful foundation, which is now discussed as commonplace.

When considering the actual wavelet application, the wavelets are not expressed explicitly in an analytical form, but are expressed as two filter kernels corresponding to a weighted differencing operator and a weighted smoothing operator, which are complementary operators. In the literature these are referred to as the mother wavelet and associated scaling function. The forward transform is applied by alternative applications of the two kernels with down sampling interleaved between the applications. This procedure generates the wavelet expansion coefficients. The inverse transform is also achieved by repeated convolutions with two related filter kernels with up-sampling interleaved between the convolutions. The filtering aspect of the analysis is implemented by applying the forward transform and setting the desired coefficients to zero before applying the inverse transform.

The procedure described above may also be presented using the equivalent terms of dilation (or contraction) and translation of the mother wavelet function. For a given wavelet basis, there is really only one wavelet function or mother wavelet. The entire basis is constructed from translations and dilations of this wavelet. Spreading it out reduces the resolution and translating provides spatial information. The translations and dilations are not arbitrary, but are picked in a certain way from an orthogonal basis at multiple resolutions.

A way to view this is that the wavelet coefficients are really correlation figures of merit indicating how well the signal and a given region correlates with the particular version of the mother wavelet. The important thing to note here is that when the wavelet is spread out, the inner product with the signal at a given region (or spatial location) encompasses the length of the dilated wavelet. This implies that the wavelets coefficient holds information about the entire spatial region. As the wavelet spreads out, the frequency-band narrows. Thus, the analysis is better localized in frequency but worse localized in space.

The reverse argument applies when the wavelet is most contracted implying better spatial location but spread out in frequency. These ideas are fundamental to understanding both Fourier and wavelet analysis. For the purpose of this discussion, a very simple wavelet interpretation was developed and presented in the following paragraphs.

The wavelet expansion may be considered as a band pass filter network. The intact signal (raw image) is put into the mill and out come many filtered versions of the image. The orthogonal wavelet gives an expansion of the form:

where F0 is the raw image, the dj s represent band pass versions of the raw image, and Fj is a blurred version of the raw image, which contains the DC and slow varying image attributes. These images are linearly independent, which amounts to perfect reconstruction and is one of the great strengths of wavelet analysis compared with just any band pass filter network. Each image of these expansion images may be divided further into three complimentary components expressed as vertical, horizontal, and diagonal components, which are also not correlated. Roughly speaking, the ds represent an octave sectioning (or fine to coarse image representation) of the frequency domain information. This can be observed by taking the Fourier transform of each expansion component individually and noting where each has appreciable Fourier amplitudes. Figure 13.4 shows the idealized division of the Fourier domain relative to the image expansion images.

Figure 13.4: Idealized graphical representation of the first four band pass splittings of the raw image.

There are many orthogonal wavelet coefficients to choose from, and the band pass nature is not generally sharp indicating that the expansion components will have some shared frequency attributes; these cancel when performing the addition. Here is a simple rule of thumb: the shorter the wavelet filter kernel the less sharp the cutoffs are in the Fourier domain and the longer the support length the sharper the cutoffs. The strength of the two-channel or quadrature wavelet filter is the orthogonality of the expansion images. The price paid for orthogonality is the fixed-way the associated information is divided (octave sectioning).

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