A wreath product group approach to signal and image processing .I. Multiresolution analysis

We propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and, in particular, image processing. We pay significant attention to groups constructed as wreath products of cyclic groups. Within this large class of groups, our approach recovers the discrete Fourier transform (DFT), the Haar wavelet transform, various multichannel pyramid filter banks, and other aspects of multiresolution analysis as special cases of a more general phenomenon. In addition, the group structure provides a rich algebraic structure that can be exploited for the analysis and manipulation of signals. Our approach relies on a synthesis of ideas found in the early work of Holmes (1987, 1990), Karpovsky and Trachtenberg (1985), and others on noncommutative filtering, as well as Diaconisʹs (1989) spectral analysis approach to understanding data.