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 finite 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 defined 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 specific signals and filters. Group-based correlation is defined in a natural way, and its properties follow from those of convolution.We finally 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.