A wreath product group approach to signal and image processing .II. Convolution, correlation, and applications

This paper continues the investigation of the use of spectral analysis on certain noncommutative nite groups|wreath product groups|in digital signal processing. We describe here the generalization of discrete cyclic convolution to convolution over these groups and show how it reduces to multiplication in the spectral domain. Finite group-based convolution is de ned in both the spatial and spectral domains and its properties established. We pay particular attention to wreath product cyclic groups and further describe convolution properties from a geometric view point, in terms of operations with speci c signals and lters. Group-based correlation is de ned in a natural way and its properties follow from those of convolution. We nally consider an application of convolution: the detection of similarity of perceptually similar signals, and an application of correlation: the detection of similarity of group transformed signals. Several examples using images are included to demonstrate the ideas pictorially. EDICS numbers: SP 2-HIER and SP 4-MTHE Department of Electrical and Computer Engineering, University of Vermont, Burlington VT 05405. Supported by NASA/JOVE yDepartment of Mathematics and Statistics, University of Vermont, Burlington VT 05405. Supported by NASA VT Space Grant zDepartment of Mathematics, Dartmouth College, Hanover, NH 03755. Supported by NSF, ONR, ARPA, National Center of Atmospheric Research xDepartment of Mathematics, Dartmouth College, Hanover NH 03755. Supported by ARPA {Department of Mathematics, Univ. of Florida, Gainesville, FL. Supported by ONR