This Article 
 Bibliographic References 
 Add to: 
N-Dimensional Tensor Voting and Application to Epipolar Geometry Estimation
August 2001 (vol. 23 no. 8)
pp. 829-844

Abstract—We address the problem of epipolar geometry estimation efficiently and effectively, by formulating it as one of hyperplane inference from a sparse and noisy point set in an 8D space. Given a set of noisy point correspondences in two images of a static scene without correspondences, even in the presence of moving objects, our method extracts good matches and rejects outliers. The methodology is novel and unconventional, since, unlike most other methods optimizing certain scalar, objective functions, our approach does not involve initialization or any iterative search in the parameter space. Therefore, it is free of the problem of local optima or poor convergence. Further, since no search is involved, it is unnecessary to impose simplifying assumption (such as affine camera or local planar homography) to the scene being analyzed for reducing the search complexity. Subject to the general epipolar constraint only, we detect wrong matches by a novel computation scheme, 8D Tensor Voting, which is an instance of the more general N-dimensional Tensor Voting framework. In essence, the input set of matches is first transformed into a sparse 8D point set. Dense, 8D tensor kernels are then used to vote for the most salient hyperplane that captures all inliers inherent in the input. With this filtered set of matches, the normalized Eight-Point Algorithm can be used to estimate the fundamental matrix accurately. By making use of efficient data structure and locality, our method is both time and space efficient despite the higher dimensionality. We demonstrate the general usefulness of our method using example image pairs for aerial image analysis, with widely different views, and from nonstatic 3D scenes (e.g., basketball game in an indoor stadium). Each example contains a considerable number of wrong matches.

[1] E.L. Allgower and S. Gnutzmann, “Simplicial Pivoting for Mesh Generation of Implicitly Defined Surfaces,” Computer Aided Geometric Design, vol. 8, no. 4, pp. 305-325, 1991.
[2] J. Chai and S.D. Ma, “Robust Fundamental Matrix Estimation from Uncalibrated Images,” Pattern Recognition Letters, vol. 19, no. 9, pp. 829-838, 1998.
[3] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[4] Q. Ke, G. Xu, and S.D. Ma, “Recovering Epipolar Geometry by Reactive Tabu Search,” Proc. IEEE Int'l Conf. Computer Vision, pp. 767-771, 1998.
[5] O.D. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint.Cambridge, Mass.: MIT Press, 1993.
[6] G. Guy and G. Medioni, “Inference of Surfaces, 3D Curves, and Junctions from Sparse, Noisy, 3D Data,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 11, pp. 1265-1277, Nov. 1997.
[7] P.V.C. Hough, “Methods and Means for Recognizing Complex Patterns,” US Patent 3 069 654, Dec. 1962.
[8] R.I. Hartley, In Defense of the 8-Point Algorithm IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 6, pp. 580-593, June 1997.
[9] M.-S. Lee, “Tensor Voting for Salient Feature Inference in Computer Vision,” Ph.D. thesis, Univ. of Southern California, 1998.
[10] H.C. Longuet-Higgins, “A Computer Algorithm for Reconstructing a Scene from Two Projections,” Nature, vol. 293, pp. 133-135, 1981.
[11] W.E. Lorensen and H.E. Cline, “Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Computer Graphics (SIGGRAPH '87 Proc.), vol. 21, pp. 163-169, 1987.
[12] G. Medioni, M.-S. Lee, and C.-K. Tang, A Computational Framework of Feature Extraction and Segmentation. Elsevier Science, 2000.
[13] G.M. Nielson and B. Hamann, The Asymptotic Decider: Removing the Ambiguity in Marching Cubes Proc. Visualization '91, pp. 83-91, 1991.
[14] P. Pritchett and A. Zisserman, “Wide Baseline Stereo Matching,” Proc. IEEE Int'l Conf. Computer Vision, pp. 754-760, 1998.
[15] C.-K. Tang and G. Medioni, “Inference of Integrated Surface, Curve, and Junction Descriptions from Sparse, Noisy 3D Data,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 11, pp. 1206-1223, Nov. 1998.
[16] C.-K. Tang, G. Medioni, and M.-S. Lee, “Epipolar Geometry Estimation by Tensor Voting in 8D,” Proc. IEEE Int'l Conf. Computer Vision, pp. 502-509, 1999.
[17] C.-K. Tang, “Tensor Voting for Feature Extraction, Integration, and Higher Dimensional Inference,” PhD thesis, Univ. of Southern California, 2000.
[18] P.H.S. Torr and D.W. Murray, “Statistical Detection of Independent Movement from a Moving Camera,” Image and Vision Computing, vol. 1, no. 4, 1993.
[19] P. Torr and D. Murray, “The Development and Comparison of Robust Methods for Estimating the Fundamental Matrix,” Int'l J. Computer Vision, vol. 3, no. 24, pp. 271-300, 1997.
[20] C. Weigle and D.C. Banks, “Complex-Valued Contour Meshing,” Proc. IEEE Visualization Conf., pp. 173-179, 1996.
[21] G. Welch and G. Bishop, “An Introduction to the Kalman Filter,” Technical Report TR 95-041, Univ. of North Carolina at Chapel Hill. 2001. .
[22] Z. Zhang, “Determining the Epipolar Geometry and Its Uncertainty—A Review,” Int'l J. Computer Vision, vol. 27, no. 2, pp. 161-195, 1998.

Index Terms:
Tensor, hyperplane inference, epipolar geometry, matching, robust estimation.
Chi-Keung Tang, Gérard Medioni, Mi-Suen Lee, "N-Dimensional Tensor Voting and Application to Epipolar Geometry Estimation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 8, pp. 829-844, Aug. 2001, doi:10.1109/34.946987
Usage of this product signifies your acceptance of the Terms of Use.