## Wavelet Filter Selection

where \$(x), ^(y), and ^(z) contain the low-pass filter h in the x, y, and z directions, respectively (see Eq. (5)). The seven wavelet functions have the form contain the high-pass filter g in the x, y and z directions, respectively. During each level of the transform, the scaling function and the seven wavelet functions are applied, respectively, to the smooth (or the original) image at that level, forming a total of eight images, one smooth and seven detailed images. The wavelet coefficients are the voxel values of the eight images after the transform.

Figure 6 shows two levels of 3D wavelet transform on an image volume data set. The first level decomposes the data into

The selection of wavelet filters is a crucial step that dictates the performance of image compression. A good filter bank should be reasonably fast, should provide a transform where most of the energy is packed in a small number of coefficients, and should not introduce distortions. The following characteristics address these points.

(1) Compact support: Compact support wavelets and scaling functions have values inside a finite range and are zero outside the range, therefore they can be implemented with finite impulse response (FIR) filters. Since filtering with FIR filters is generally much faster than filtering with infinite impulse response (IIR) filters, compact support wavelets provide fast processing.

(2) Smoothness: The smoothness of a wavelet is commensurate with the highest order at which its moment vanishes such that J xm^(x)dx = 0 for m = 0, 1,..., N — 1. Wavelets with high degree of smoothness minimize distortions and will likely lead to better representation with most of the energy packed in a smaller number of coefficients.

FIGURE 6 A two-level 3D wavelet transform on an image volume data set.

(3) Symmetry: Symmetry is a desirable characteristic of a filter because it provides a linear phase in the frequency response of the filter. Filters with linear phase will produce less distortion.

(4) Filter length: A short filter allows fast computation of the wavelet transform, but smoothness is obtained with long filters. Image compression is based on a trade-off between the filter length and compression performance. In general, the following types of wavelet filters perform well for 3D MR and CT images

(a) Daubechies orthogonal wavelet from D4-D20 [1]

(c) Biorthogonal wavelet filters [6-8].

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