A wreath product group approach to signal and image processing-part II: Convolution, correlation, and applications

This paper continues the investigation of the use of spectral analysis on c ertain noncommutative finite groups-wreath product groups-in digital signal processing. We describe here the generalization of discrete cyclic convolu tion to convolution over these groups and show hom it reduces to multiplica tion in the spectral domain, Finite group-based convolution is defined in b oth 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 w ith specific signals and filters. Group-based correlation is defined in a n atural way, and its properties follow: from those of convolution, We finall y consider an application of convolution (the detection of similarity of pe rceptually similar signals) and an application of correlation (the detectio n of similarity of group-transformed signals). Several examples using image s are included to demonstrate the ideas pictorially.