Harmonic Analysis : Motions and Pattern Analysis on Motion Groups and Their Homogeneous Spaces

Fourier-like transforms that are defined on several groups of motions on the plane and on the plane itself (viewed as the homogeneous space of these motion groups) are presented. These transforms should have many applications in the area of pattern recognition, detection, and the representation of motions in pattern analysis. The method is based upon group-representation theory and abstract harmonic analysis on semidirect product groups. However, most of the ideas are generalizations and interpretations of classical concepts and methods such as the Fourier descriptor method used for boundary recognition. Computational aspects are investigated, and examples are given. >

[1]  Wesley E. Snyder,et al.  Application of the one-dimensional Fourier transform for tracking moving objects in noisy environments , 1982, Comput. Vis. Graph. Image Process..

[2]  King-Sun Fu,et al.  Shape Discrimination Using Fourier Descriptors , 1977, IEEE Trans. Syst. Man Cybern..

[3]  S. Winograd,et al.  Hecke's theorem in quadratic reciprocity, finite nilpotent groups and the cooley-tukey algorithm , 1982 .

[4]  Louis Auslander.  A Factorization Theorem for the Fourier Transform of a Separable Locally Compact Abelian Group , 1984 .

[5]  I. Segal.  AN EXTENSION OF PLANCHEREL'S FORMULA TO SEPARABLE UNIMODULAR GROUPS , 1950 .

[6]  Garth Warner,et al.  Harmonic Analysis on Semi-Simple Lie Groups II , 1972 .

[7]  King-Sun Fu,et al.  Shape Discrimination Using Fourier Descriptors , 1977, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[8]  F. I. Mautner.  Note on the Fourier inversion formula on groups , 1955 .

[9]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[10]  D Marr,et al.  Theory of edge detection , 1979, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[11]  Azriel Rosenfeld,et al.  A Fourier approach to cloud motion estimation , 1978 .

[12]  S. L. Daniel,et al.  Algorithm 511: CDC 6600 Subroutines IBESS and JBESS for Bessel Functions Iυ(x) and Jυ(x), x≥0, υ ≥0 [S18] , 1977, TOMS.

[13]  Wesley E. Snyder,et al.  Track Acquisition of Sub-Pixel Targets , 1983 .

[14]  A. Kirillov.  Elements of the theory of representations , 1976 .

[15]  Wesley E. Snyder,et al.  Calculation of displacement fields by means of the motion detection transform , 1984, ICASSP.