Complex Wavelet Filters

Introduction

Complex wavelet analysis requires two Hilbert-pairs of wavelets. A Hilbert-pair of wavelets consists of two wavelets which have a phase-shift of 90° to each other. The two Hilbert-pairs are designed to form a filter bank to decompose the signal into high and low frequencies. The to Hilbert-pairs can be designed to also fulfil the perfect-reconstruction condition. Selesnick has developed a method for designing Hilbert transform pairs of biorthogonal wavelet bases.

Implementation

The implementation is part of HornetsEye. Since the Ruby code is stand-alone you can download it here if you do not want to download the whole HornetsEye package: selesnick.rb

Example

Biorthogonal Hilbert-pairs of wavelet bases (k=ks=5,l=3)

Overview of the filter bank	H0	H1	G0	G1
	H0	H1	G0	G1

See Also

• HornetsEye
• Hypercomplex Wavelets

External Links

• Ivan Selesnick's homepage
  • I. W. Selesnick: The design of approximate Hilbert transform pairs of wavelet bases, IEEE Trans. on Signal Processing, 50(5):1144-1152, May 2002
• Nick Kingsbury's homepage
• Perfect Reconstruction (Nick Kingsbury)