A Wreath Product Group Approach to Signal and Image Processing:
Part II -- Convolution, Correlation, and Applications
G. Mirchandani, R. Foote, D. Rockmore, D. Healy and T. Olson,
IEEE.Trans. on Signal Processing, vol.48, No.3, pp.749-767, March 2000
Paper (pdf) / Paper (ps)
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.