FILTER CONSTRUCTION IN THE FOURIER DOMAIN

This chapter explains the problem of finding filters (filter pairs) in the Fourier domain. It characterizes the Daubechies system in terms of Fourier series. The chapter utilizes the Fourier series to identify other desirable conditions satisfied by the lowpass filter and in so doing, creates a new family of orthogonal filters. The new conditions, suggested by Ronald Coifman, led Daubechies to develop the so‐called Coiflet filters. The chapter provides Daubechies' general formulation and explicitly constructs two members of the Coiflet filter family. It also developes the entire family of biorthogonal spline filter pairs. The Cohen‐Daubechies‐Feauveau biorthogonal filter pair is utilized by the Federal Bureau of Investigation for compressing digitized images of fingerprints and by the JPEG2000 committee for the JPEG2000 Image Compression Standard. Unser and Blu report that the Cohen‐Daubechies‐Feauveau 9/7 biorthogonal filter pair produces a wavelet transformation matrix that is closer to orthogonal than that generated by the (9, 7) biorthogonal spline filter pair.