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Dominant-Subspace Invariants
July 2000 (vol. 22 no. 7)
pp. 649-662

Abstract—Object recognition requires robust and stable features that are unique in feature space. Lie group analysis provides a constructive procedure to determine such features, called invariants, when they exist. Absolute invariants are rare in general, so quasi-invariants relax the restrictions required for absolute invariants and, potentially, can be just as useful in real-world applications. This paper develops the concept of a dominant-subspace invariant, a particular type of quasi-invariant, using the theory of Lie groups. A constructive algorithm is provided that fundamentally seeks to determine an integral submanifold which, in practice, is a good approximation to the orbit of the Lie group action. This idea is applied to the long-wave infrared problem and experimental results are obtained supporting the approach. Other application areas are cited.

[1] G. Arnold, “Analysis of Invariants for Thermophysical Models in Infrared Image Understanding,” PhD thesis, Univ. of Virginia, 1999.
[2] P.J. Olver, Applications of Lie Groups to Differential Equations. New York: Springer-Verlag, 1993.
[3] D.H. Sattinger and O.L. Weaver, Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. New York: Springer-Verlag, 1986.
[4] P.F. Stiller, “General Approaches to Recognizing Geometric Configurations from a Single View,” Proc. SPIE Int'l Conf., Vision Geometry VI, vol. 3,168, pp. 262-273, July 1997.
[5] I. Weiss, “Model-Based Recognition of 3D Objects from One View,” Proc. DARPA Image Understanding Workshop, pp. 641-652, 1998.
[6] L. Quan, Invariants of Six Points and Projective Reconstruction from Three Uncalibrated Images IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 1, pp. 34-46, Jan. 1995.
[7] A. Shashua, “Algebraic Functions for Recognition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 8, pp. 779-789, 1995.
[8] D. Slater and G. Healey, “Exploiting an Atmospheric Model for Automated Invariant Material Classification in Hyperspectral Imagery,” Proc. SPIE Int'l Conf., Apr. 1998.
[9] N. Nandhakumar and J.K. Aggarwal, “Integrated Analysis of Thermal and Visual Images for Scene Interpretation,” Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 4, pp. 469-481, July 1988.
[10] J. Michel, “Thermophysical Algebraic Invariance for Infrared Image Interpretation,” PhD thesis, Univ. of Virginia, 1996.
[11] D. Kapor, Y.N. Lakshman, and T. Saxena, “Computing Invariants Using Elimination Methods,” Proc. IEEE Int'l Symp. Computer Vision, pp. 97-102, 1995.
[12] G. Arnold, K. Sturtz, and V. Velten, “Lie Group Analysis in Object Recognition,” Proc. DARPA Image Understanding Workshop, May 1997.
[13] T.O. Binford and T.S. Levitt, “Quasi-Invariants: Theory and Exploitation,” Proc. DARPA Image Understanding Workshop, O. Firschein, ed., pp. 819-830, 1993.
[14] C.W. Therrien, Decision Estimation and Classification: An Introduction to Pattern Recognition and Related Topics. John Wiley&Sons, 1989.
[15] R.C. Gonzalez and R.E. Woods, Digital Image Processing, 2nd ed., Addison-Wesley, 2002, Chapter 8.
[16] S. MacLane and G. Birkhoff, Algebra. New York: Chelsea Publishing Company, 1993.
[17] J.L. Mundy and A. Zisserman, Geometric Invariance in Computer Vision.Cambridge, Mass.: MIT Press, 1992.

Index Terms:
Dominant-subspace invariants, Lie group analysis, principal basis, quasi-invariants, thermophysical invariance, thermophysical model.
D. Gregory Arnold, Kirk Sturtz, Vince Velten, N. Nandhakumar, "Dominant-Subspace Invariants," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 7, pp. 649-662, July 2000, doi:10.1109/34.865182
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