We have constructed orthogonal filters in Chapters 4 and 5 and biorthogonal filter pairs in Chapter 7. The constructions were ad hoc and somewhat limiting in that it was difficult to produce a method for finding filters of arbitrary length. Equipped with Fourier series from Chapter 8, we can reproduce and indeed generalize the filter (pair) construction of previous chapters in a much more systematic way.

In Section 9.1 we reformulate the problem of finding filters (filter pairs) in the Fourier domain. The orthogonality relationships satisfied by the Daubechies filters of Chapter 5 can be reduced to three identities involving their Fourier series, and similar results hold true for biorthogonal spline filter pairs and their associated highpass filters. Conditions necessary to ensure g images are highpass filters that annihilate polynomial data are reformulated in the Fourier domain. We characterize the Daubechies system (5.65) in terms of Fourier series in Section 9.1. In the section that follows, we learn how to modify this system to produce a new family of orthogonal filters called Coiflet filters. The entire family of biorthogonal spline filter pairs introduced in Chapter 7 is developed in Section 9.4. The Cohen–Daubechies–Feauveau biorthogonal filter pair is utilized by the Federal Bureau of Investigation for compressing digitized images ...

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