A Wreath Product Group Approach to Signal Processing

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 DFT, 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 which 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, Karpovsky, Trachtenberg and others on noncommutative filtering, as well as Diaconis’s spectral analysis approach to understanding data. EDICS numbers: SP 2.4.4 and SP 4.1.3 Department of Mathematics and Statistics, University of Vermont, Burlington VT 05405 Department of Electrical and Computer Engineering, University of Vermont, Burlington VT 05405. Supported by NASA\ JOVE Department of Mathematics, Dartmouth College, Hanover, NH 03755. Supported by NSF, ONR, ARPA Department of Mathematics, Dartmouth College, Hanover NH 03755. Supported by ARPA ¶Department of Mathematics, Univ. of Florida, Gainesville, FL. Supported by ONR