Abstract
A group theoretic approach to image representation and analysis is presented. The concept of a wavelet transform is extended to incorporate different types of groups. The wavelet approach is generalized to Lie groups that satisfy conditions of compactness and commutability and to groups that are determined in a particular way by subgroups that satisfy these conditions. These conditions are fundamental to finding the invariance measure for the admissibility condition of a mother wavelet-type transform. The following special cases of interest in image representation and in biological and computer vision are discussed: 2-and 3-D rigid motion, similarity and Lorentzian groups, and 2-D projective groups obtained from 3-D camera rotation.
This research was supported by U.S.-Israel Binational Science Foundation grant 8800320, by the Franz Ollendorff Center of the Department of Electrical Engineering, and by the Fund for Promotion of Research at the Technion. J. Segman is a VATAT (Israel National Committee for Planning and Budgeting Universities) Fellow at the Technion.
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References
A. Grossmann and J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Appl. Math., vol. 15, 1984, pp. 723–736.
J. Kovacevic, Filter Banks and Wavelets: Extensions and Applications, Ph. D dissertation, Center for Telecommunications Research, Department of Electrical Engineering, Colombia University, New York, 1992.
D. Marr and S. Ullman, “Directional selectivity and its use in early visual Processing,” Proc. Roy. Soc. London Sen B., vol. 211, 1981, pp. 151–180.
E.C. Hildreth, The Measurement of Visual Motion, Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, MA 1983.
H. Greenspan, M. Porat, and Y.Y. Zeevi, “Image analysis in a position-orientation space,” IEEE, Trans. Patt. Anal. Mach. Intell., to appear.
T. Caelli, W. C. Hoffman, and H. Lindman, “Subjective Lorentz transformation and the perception of motion,” J. Opt. Soc. Am., vol. 68, 1978, pp. 402–411.
K. Kanatani, Group-Theoretical Methods in Image Understanding, Springer-Verlag, Berlin, 1990.
M.K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Informat. Theory, vol. II-8, 1962, pp. 179–187.
EA. Sadjadi and E.L. Hall, “Three-dimensional moment invariants,” IEEE Trans. Patt. Anal. Mach. Intell., vol. 2, 1980, pp. 127–136.
G. Salmon, Lessons Introductory to Modern Higher Algebra, Dodges, Foster, Dublin, 1876.
J. Grace and A. Young, The Algebra of Invariants, Cambridge U. Press, Cambridge, England, 1903; reprint, G.E. Stechert, New York, 1941.
J. Segman, J. Rubinstein, and Y.Y. Zeevi, The Canonical Coordinates Method for Pattern Deformation: Theoretical and Computation Aspects, EE Pub. No. 735, Technion, Israel Institute of Technology, Haifa, Israel, 1989; IEEE Trans. Patt. Anal. Mach. Intell., to appear.
J. Rubinstein, J. Segman, and Y.Y. Zeevi, “Recognition of distorted patterns by invariance kernels,” J. Patt. Recog., vol. 24, 1991, pp. 959–967.
C.E. Heil and D.E Walnut, “Continuous and discrete wavelet transforms,” SIAM J. Appl. Math., vol. 31, 1989, pp. 628–666.
TV. Papathomas and B. Julesz, “Lie differential operators in animal and machine vision,” in From Pixels to Features, J.C. Simon, ed., Elsevier, New York, 1989.
E.L. Schwartz, “Computational anatomy and functional architecture of striate cortex: A special map** approach to perceptual coding,” Vis. Res., vol. 20, 1980, pp. 645–669.
S.G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE, Trans. Patt. Anal. Mach. Intell., vol. 37, 1989, pp. 2091–2110.
S.G. Mallat, “Multiresolution approximation and wavelet orthonormal bases of L2,” Trans. Amer. Math. Soc., vol. 3-15, Sept. 1989, pp. 69–87.
J. Segman and Y.Y. Zeevi, Inverse, Isometry and Convolution Properties of the Integral Transform Associated with the Invariance Kernels, EE. Pub. 803, Technion-Israe/ Institute of Technology. Haifa, Israel, 1991.
P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
N.Y. Vilenkin, Special Functions and the Theory of Group Representations, American Mathematical Society, Providence, RI, 1968.
W. Miller, Jr., Symmetry Groups and Their Application, Academic Press, New York, 1972.
A. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965.
W. Miller, Jr., Symmetry and Separation of Variables, Addison-Wesley, Reading, MA, 1977.
E. Wigner, Group Theory and Its Application to the Quantum of Atomic Spectra, Academic Press, New York, 1959.
J.D. Talman, Special Functions: A Group Theoretic Approach, W.A. Benjamin, New York, 1968.
J. Segman and Y.Y. Zeevi, “Approximating images deformed by the rotation projective group,” to appear.
G.L. Turin, “An introduction to matched filters,” IRE Trans. Informat. Theory, vol. IT-6, 1960, pp. 311–329.
D.A. Pintsov, “Invariant pattern recognition, symmetry, and radon transform,” J. Opt. Soc. Am. A., vol. 6, 1989, pp. 1544–1554.
J. Segman, “Fourier cross-correlation and invariance transformations for an optimal recognition of functions deformed by affine groups,” J. Opt. Soc. Am. A, vol. 9, 1992, pp. 895–902.
I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math., vol. 41, 1988, pp. 909–996.
Y.Y. Zeevi and I. Gertner, “The finite Zak transform: An efficient tool for image representation and analysis,” J. Vis. Commun. Image Rep., vol. 3, 1992, pp. 13–23.
M. Zibulski and Y.Y. Zeevi, Over Sampling in the Gabor Scheme, EE Pub. 801, Technion-Israel Institute of Technology, Haifa, Israel, 1991.
W. Schempp, “Radar ambiguity functions the Heisenberg group, and holomorphic theta series,” Proc. Amer. Math. Soc., vol. 92, 1984, pp. 103–110.
W. Schempp, Harmonic Analysis on the Heisenberg Nilpotent Lie Group with Applications to Signal Theory, Pitman Research Notes in Mathematical Sciences 147, Longman Scientific and Technical, Harlow, Essex, U.K. 1986.
E.G. Kaluins and W. Miller, Jr., A Note on Group Contractions and Radar Ambiguity Functions, IMA Preprints Series 682, University of Minnesota, Minneapolis, MN, 1990.
I. Gertner and R. Talimieri, “The group theoretic approach to image representation, J. Vis. Commun. Image Rep., vol. 1, 1990, pp. 67–82.
J. Segman and W. Schempp, “Two ways to incorporate scale in the Heisenberg group with an intertwining operator,” J. Math. Imag. Vis., this issue.
J.E. Campbell, Introductory Treatise on Lie’s Theory of Finite Continuous Transformation Groups, Chelsea, New York, 1966.
A.A. Sagle and R.E. Walde, Introduction to Lie Groups and Lie Algebra, Academic Press, New York, 1973.
F. John, Partial Differential Equations, Springer-Verlag, New York, 1982.
M. Hamermesh, Group Theory and Its Application to Physical Problems, Addison-Wesley, Reading, MA, 1962.
Y. Meyer, Principle d’incertitude, bases Hilbertiennes et algèbres d’opérateurs, Séminaire Bourbaki, 1985–1986.
I.H. Sneddon, The Use of Integral Transform, McGraw-Hill, New York, 1972.
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Segman, J., Zeevi, Y.Y. (1993). Image Analysis by Wavelet-Type Transforms: Group Theoretic Approach. In: Laine, A. (eds) Wavelet Theory and Application. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3260-6_4
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DOI: https://doi.org/10.1007/978-1-4615-3260-6_4
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