A wreath product group approach to signal and image processing .II. Convolution, correlation, and applications

For pt.I see ibid., vol.48, no.1, p.102-32 (2000). 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 the generalization of discrete cyclic convolution in 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 (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.

[1]  Azriel Rosenfeld,et al.  Computer Vision , 1988, Adv. Comput..

[2]  J. S. Bomba,et al.  Alpha-numeric character recognition using local operations , 1959, IRE-AIEE-ACM '59 (Eastern).

[3]  M. Newman,et al.  Interpolation and approximation , 1965 .

[4]  Andreas Steffen Digital pulse compression using multirate filter banks , 1991 .

[5]  Rosalind W. Picard,et al.  Finding similar patterns in large image databases , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[6]  L. Goddard Approximation of Functions , 1965, Nature.

[7]  Dennis M. Healy,et al.  A wreath product group approach to signal and image processing .I. Multiresolution analysis , 2000, IEEE Trans. Signal Process..

[8]  Robert Bregovic,et al.  Multirate Systems and Filter Banks , 2002 .

[9]  M. Unser Local linear transforms for texture measurements , 1986 .

[10]  P. Vaidyanathan Multirate Systems And Filter Banks , 1992 .

[11]  C. P. Horne,et al.  Digital pulse compression , 1988 .

[12]  Yasuhito Suenaga,et al.  Robust face identification scheme: KL expansion of an invariant feature space , 1992, Other Conferences.

[13]  Hadar I. Avi-Itzhak,et al.  Multiple Subclass Pattern Recognition: A Maximin Correlation Approach , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[14]  Luc Van Gool,et al.  An Extended Class of Scale-Invariant and Recursive Scale Space Filters , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  P. E. Anuta,et al.  Spatial Registration of Multispectral and Multitemporal Digital Imagery Using Fast Fourier Transform Techniques , 1970 .

[16]  Harold S. Stone,et al.  Progressive wavelet correlation using Fourier methods , 1999, IEEE Trans. Signal Process..

[17]  R. Tolimieri,et al.  The tensor product: a mathematical programming language for FFTs and other fast DSP operations , 1992, IEEE Signal Processing Magazine.

[18]  Martin Vetterli Running FIR and IIR filtering using multirate filter banks , 1988, IEEE Trans. Acoust. Speech Signal Process..

[19]  A. W. M. van den Enden,et al.  Discrete Time Signal Processing , 1989 .

[20]  G.,et al.  A Wreath Product Group Approach to Signal and Image Processing : Part I | Multiresolution AnalysisR , 1999 .

[21]  Robert D. Nowak,et al.  Multiscale Modeling and Estimation of Poisson Processes with Application to Photon-Limited Imaging , 1999, IEEE Trans. Inf. Theory.

[22]  G. Turin,et al.  An introduction to matched filters , 1960, IRE Trans. Inf. Theory.