Abstract
Complex geometric features such as oriented points, lines or 3D frames are increasingly used in image processing and computer vision. However, processing these geometric features is far more difficult than processing points, and a number of paradoxes can arise. We establish in this article the basic mathematical framework required to avoid them and analyze more specifically three basic problems: (1) what is a random distribution of features, (2) how to define a distance between features, (3) and what is the “mean feature” of a number of feature measurements?
We insist on the importance of an invariance hypothesis for these definitions relative to a group of transformations that models the different possible data acquisitions. We develop general methods to solve these three problems and illustrate them with 3D frame features under rigid transformations.
The first problem has a direct application in the computation of the prior probability of a false match in classical model-based object recognition algorithms. We also present experimental results of the two other problems for the statistical analysis of anatomical features automatically extracted from 24 three-dimensional images of a single patient's head. These experiments successfully confirm the importance of the rigorous requirements presented in this article.
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References
S. L. Altmann, Rotations, Quaternions, and Double Groups, Clarendon Press: Oxford, 1986.
N. Ayache, Artificial Vision for Mobile Robots—Stereo-Vision and Multisensor Perception, MIT Press, 1991.
N. Ayache and O. D. Faugeras, “Hyper: A new approach for the recognition and positioning of two-dimensional objects,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 8, No.1, pp. 44–54, 1986.
P. J. Besl and N. McKay, “A method for registration of 3D shapes,” PAMI, Vol. 14, No.2, pp. 239–256, 1992.
H. F. Durrant-Whyte, Integration, Coordination and Control of Multi-Sensor Robot Systems, Kluwer Academic Publishers, 1988.
J. Feldmar and N. Ayache, “Rigid, affine and locally affine registration of free-form surfaces,” IJCV, Vol. 18, No.2, pp. 99–119, 1996.
M. Fidrich and J. P. Thirion, “Multiscale representation and analysis of features from medical images,” in Computer Vision, Virtual Reality and Robotics in Medicine (Proc. first CVRMed'95), N. Ayache (Ed.), number 905 in LNCS, INRIA, Springer, 1995, pp. 358–364.
M. Fréchet, “Les éléments aléatoires de nature quelconque dans un espace distancié,” Ann. Inst. H. Poincaré, Vol. 10, pp. 215–310, 1948.
W. E. L. Grimson, Object Recognition by Computer—The Role of Geometric Constraints, MIT Press, 1990.
W. E. L. Grimson and D. P. Huttenlocher, “On the sensitivity of geometric hashing,” in Proc. Third ICCV, 1990, pp. 334–338.
W. E. L. Grimson, D. P. Huttenlocher, and D. W Jacobs, “A study of affine matching with bounded sensor error,” Int. Journ. of Comput. Vision, Vol. 13, No.1, pp. 7–32, 1994.
G. Hochschild, The Structure of Lie Groups, Holden-Day, 1965.
D. P. Huttenlocher and S. Ullman, “Object recognition using alignment,” in Proc. of ICCV, 1987, pp. 72–78.
H. Karcher, “Riemannian center of mass and mollifier smoothing,” Comm. Pure Appl. Math, Vol. 30, pp. 509–541, 1977.
W. S. Kendall, “Probability, convexity, and harmonic maps with small image i: uniqueness and fine existence,” in Proc. London Math. Soc., 1990, Vol. 61, No.2, pp. 371–406.
M. G. Kendall and P. A. P. Moran, Geometrical Probability, Number 10 in Griffin's Statistical Monographs and Courses, Charles Griffin & Co. Ltd., 1963.
F. Klein, Erlangen program, Inaugural address at the University of Erlangen, 1872.
Y. Lamdan and H. J. Wolfson, “Geometric hashing: A general and efficient model-based recognition scheme,” in Proc. of Second ICCV, 1988, pp. 238–289.
Y. Lamdan and H. J. Wolfson, “On the error analysis of geometric hashing,” in IEEE Int. Conf. on Comput. Vis. and Patt. Recog., 1991, pp. 22–27.
X. Pennec, “Correctness and robustness of 3D rigid matching with bounded sensor error,” Research Report 2111, INRIA, November 1993.
X. Pennec and N. Ayache, “An O(n 2) algorithm for 3D substructure matching of proteins,” in Shape and Pattern Matching in Computational Biology, A. Califano, I. Rigoutsos, and H. J. Wolson (Eds.), Proc. First Int. Workshop, Seattle, Wash, June 20, 1994, Plenum Publishing, 1996.
X. Pennec, “L'incertitude dans les problèmes de reconnaissance et de recalage—Applications en imagerie médicale et biologie moléculaire,” Ph. D. Thesis, Ecole Polytechnique, Palaiseau (France), December 1996. http://www. inria. fr/epidaure/ personnel/ pennec/These. html
X. Pennec and J. P. Thirion, “Validation of 3D registration methods based on points and frames,” in Proceedings of the 5th Int. Conf on Comp. Vision (ICCV'95), Cambridge, MA, June 1995, pp. 557–562, Part of the INRIA Research Report no 2470.
X. Pennec and J. P. Thirion, “A framework for uncertainty and validation of 3D registration methods based on points and frames,” Int. Journal of Computer Vision, 1997, Vol. 25, No.3, pp. 203–229.
H. Poincaré, Calcul des Probabilités, 2nd edition, Paris, 1912.
I. Rigoutsos and R. Hummel, “Distributed Bayesian object recognition,” in Proceedings of Int. Conf on Comput. Vis. and Pat. Recog, IEEE Computer Society Press, June 1993, pp. 180–186.
L. A. Santalo, Integral Geometry and Geometric Probability, volume 1 of Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Company, 1976.
G. Subsol, J. Ph. Thirion, and N. Ayache, “A general scheme for automatically building 3D morphometric anatomical atlases: application to a skull atlas,” in MRCAS'95, November 1995.
J. P. Thirion, “New feature points based on geometric invariants for 3D image registration,” Int. Journ. Comp. Vis., Vol. 18, No.2, pp. 121–137, 1996.
H. J. Wolfson, “Model-based recognition by geometric hashing,” in Proc. of 1st Europ. Conf. on Comput. Vision (ECCV 90), O. Faugeras (Ed.), Springer-Verlag, Lecture Notes in Computer Science 427, April 1990, pp. 526–536.
Z. Zhang, “Iterative point matching for registration of free-form curves and surfaces,” Int. Journ. Comp. Vis., Vol. 13, No.2, pp. 119–152, 1994, Also Research Report No. 1658, INRIA Sophia-Antipolis, 1992.
Z. Zhang and O. Faugeras, 3D Dynamic Scene Analysis: A Stereo Based Approach, volume 27 of Springer Series in Information Science, chapter 2: Uncertainty Manipulation and Parameter Estimation, Springer Verlag, 1992, pp. 9–27.
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Pennec, X., Ayache, N. Uniform Distribution, Distance and Expectation Problems for Geometric Features Processing. Journal of Mathematical Imaging and Vision 9, 49–67 (1998). https://doi.org/10.1023/A:1008270110193
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DOI: https://doi.org/10.1023/A:1008270110193