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

This paper continues the investigation in digital signal processing of spectral analysis on certain non-commutative nite groups|wreath product groups. 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 deened 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 speciic signals and lters. Group-based correlation is deened 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 that of correlation: the detection of similarity of group transformed signals. Several examples using images are included to demonstrate the ideas pictorially.