Quaternionic Wavelets

Abstract

Quaternion-valued functions have been used as a model for colour images and have recently been studied using various Fourier-type transforms. We develop some fundamental wavelet theory for quaternionic signals using the Fourier kernel introduced by Brackx, De Schepper, and Sommen in [1]. We present several analogs to classical wavelet theory, such as the quadrature mirror filter condition. We also include necessary design conditions for a wavelet basis to have desired regularity and sufficient design conditions which will guarantee compact support. Due to the difficulty in constructing orthogonal wavelet bases, we present some theory for biorthogonal wavelet bases and construct a few examples.

Keywords:

• Biorthogonal Wavelets
• Color Images
• Fourier Transform
• Orthogonal Wavelets
• Quaternions

Mathematics Subject Classification:

• Primary 42C40
• Secondary 94A12

Acknowledgments

J. A. Hogan thanks Roy and H.G.