This Article 
 Bibliographic References 
 Add to: 
How to Put Probabilities on Homographies
October 2005 (vol. 27 no. 10)
pp. 1666-1670
We present a family of "normal” distributions over a matrix group together with a simple method for estimating its parameters. In particular, the mean of a set of elements can be calculated. The approach is applied to planar projective homographies, showing that using priors defined in this way improves object recognition.

[1] M. Berger, A Panoramic View of Riemannian Geometry. Berlin: Springer, 2003.
[2] M.A. Fischler and R.C. Bolles, “Random Sampling Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography,” Comm. ACM, vol. 24, no. 6, pp. 381-395, 1981.
[3] P. Fletcher, C. Lu, and S. Joshi, “Statistics of Shape via Principal Geodesic Analysis on Lie Groups,” Proc. IEEE Computer Vision and Pattern Recognition, vol. 1, pp. 95-101, 2003.
[4] P. Fletcher, C. Lu, S. Joshi, and S. Pizer, “Gaussian Distributions on Lie Groups and Their Application to Statistical Shape Analysis,” Proc. Information Processing in Medical Imaging, pp. 450-462, 2003.
[5] V.M. Govindu, “Lie-Algebraic Averaging for Globally Consistent Motion Estimation,” Proc. IEEE Computer Vision and Pattern Recognition, vol. 1, pp. 684-691, 2004.
[6] U. Grenander, Probabilities on Algebraic Structures. New York: Wiley, 1963.
[7] W. Hays and R. Winkler, Statistics: Probability, Inference, and Decision. New York: Holt, Rinehart and Winston, 1970.
[8] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978.
[9] P. Buser and H. Karcher, Gromov's Almost Flat Manifolds. Asterisque, 1981.
[10] D.G. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints,” Int'l J. Computer Vision, vol. 60, no. 2, pp. 91-110, 2004.
[11] E. Miller and C. Chef'dhotel, “Practical Non-Parametric Density Estimation on a Transformation Group for Vision,” Proc. IEEE Computer Vision and Pattern Recognition, vol. 2, pp. 114-121, 2003.
[12] X. Pennec, “Probabilities and Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements,” Proc. Nonlinear Signal and Image Processing, pp. 194-198, 1999.
[13] R. Rao and D. Ruderman, “Learning Lie Groups for Invariant Visual Perception,” Proc. Advances in Neural Information Processing Systems 11, pp. 810-816, 1999.
[14] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations. Englewood Cliffs, N.J.: Prentice-Hall, 1974.
[15] H.C. Wang, “Discrete Nilpotent Subgroups of Lie Groups,” J. Differential Geometry, vol. 3, pp. 481-492, 1969.

Index Terms:
Index Terms- Homography, lie groups, normal distribution, Bayesian statistics, geodesics.
Evgeni Begelfor, Michael Werman, "How to Put Probabilities on Homographies," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 10, pp. 1666-1670, Oct. 2005, doi:10.1109/TPAMI.2005.200
Usage of this product signifies your acceptance of the Terms of Use.