A Wreath Product Group Approach to Signal and Image Processing:
Part I -- Multiresolution Analysis
R. Foote, G. Mirchandani, D. Rockmore, D. Healy and T. Olson
IEEE.Trans. on Signal Processing, Vol. 48, No. 1, pp.102-132, Jan.2000
Paper (pdf) / Paper (ps)
We propose the use of spectral analysis on certain noncommutative finite
groups in digital signal processing, and in particular, image processing.
We pay significant attention to groups constructed as wreath products of
cyclic groups. Within this large class of groups our approach recovers the
DFT, Haar wavelet transform, various multichannel pyramid filter banks and
other aspects of multiresolution analysis as special cases of a more
general phenomenon. In addition, the group structure provides a rich
algebraic structure which can be exploited for the analysis and
manipulation of signals. Our approach relies on a synthesis of ideas
found in the early work of Holmes, Karpovsky, Trachtenberg and others on
noncommutative filtering, as well as Diaconis's spectral analysis approach
to understanding data.