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On the Optimization Criteria Used in Two-View Motion Analysis
July 1998 (vol. 20 no. 7)
pp. 717-729
Abstract—The three best-known criteria in two-view motion analysis are based, respectively, on the distances between points and their corresponding epipolar lines, on the gradient-weighted epipolar errors, and on the distances between points and the reprojections of their reconstructed points. The last one has a better statistical interpretation, but is, however, significantly slower than the first two. In this paper, I show that, given a reasonable initial guess of the epipolar geometry, the last two criteria are equivalent when the epipoles are at infinity, and differ from each other only a little even when the epipoles are in the image, as shown experimentally. The first two criteria are equivalent only when the epipoles are at infinity and when the observed object/scene has the same scale in the two images. This suggests that the second criterion is sufficient in practice because of its computational efficiency. Experiments with several thousand computer simulations and four sets of real data confirm the analysis. The result is valid for both calibrated and uncalibrated images.
[1] J.K. Aggarwal and N. Nandhakumar, “On the Computation of Motion from Sequences of Images: A Review,” Proc. IEEE, vol. 76, no. 8, pp. 917-935, 1988.
[2] T. Huang and A. Netravali,“Motion and structure from feature correspondences: A review,” Proc. IEEE, vol. 82, pp. 252-268, Feb. 1994.
[3] O. Faugeras, "What Can Be Seen in Three Dimensions With an Uncalibrated Stereo Rig," Proc. Second European Conf. Computer Vision, pp. 563-578,Santa Margherita Ligure, Italy, May 1992.
[4] R. Hartley, R. Gupta, and T. Chang, “Stereo from Uncalibrated Cameras,” Proc. Conf. Computer Vision and Pattern Recognition, pp. 761-764, June 1992.
[5] Z. Zhang, "Determining the Epipolar Geometry and Its Uncertainty: A Review," Tech. Rep. 2927, INRIA Sophia-Antipolis, France, July 1996. Int'l J. Computer Vision, vol. 27, no. 2, pp. 161-195, 1998.
[6] Z. Zhang, "On the Epipolar Geometry Between Two Images With Lens Distortion," Int'l Conf. Pattern Recognition, vol. 1, pp. 407-411,Vienna, Aug. 1996.
[7] G.P. Stein, “Lens Distortion Calibration Using Point Correspondences,” Proc. 1997 Conf. Computer Vision and Pattern Recognition, pp. 143-148, June 1997.
[8] H. Longuet-Higgins, "A Computer Algorithm for Reconstructing a Scene From Two Projections," Nature, vol. 293, pp. 133-135, 1981.
[9] R. Tsai and T. Huang, "Uniqueness and Estimation of Three-Dimensional Motion Parameters of Rigid Objects With Curved Surface," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 6, no. 1, pp. 13-26, Jan. 1984.
[10] Q.-T. Luong, Matrice Fondamentale et Calibration Visuelle sur l'Environnement-Vers une plus grande autonomie des systèmes robotiques. PhD thesis, Universitéde Paris-Sud, Centre d'Orsay, Dec. 1992.
[11] O. Faugeras, "Stratification of 3-D Vision: Projective, Affine, and Metric Representations," J. Optical Soc. Am. A, vol. 12, pp. 465-484, Mar. 1995.
[12] Q.-T. Luong and O.D. Faugeras, "The Fundamental Matrix: Theory, Algorithms and Stability Analysis," Int'l J. Computer Vision, vol. 17, pp. 43-76, Jan. 1996.
[13] 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.
[14] O. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint.Cambridge, Mass.: MIT Press, 1993.
[15] L. Shapiro, A. Zisserman, and M. Brady, "3D Motion Recovery via Affine Epipolar Geometry," Int'l J. Computer Vision, vol. 16, pp. 147-182, 1995.
[16] K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice.Amsterdam: Elsevier, 1996.
[17] J. Weng, N. Ahuja, and T. Huang, "Optimal Motion and Structure Estimation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 864-884, 1993.
[18] C. Tomasi and T. Kanade, "Shape and Motion From Image Streams Under Orthography: A Factorization Method," Int'l J. Computer Vision, vol. 9, no. 2, pp. 137-154, 1992.
[19] R. Hartley, "Euclidean Reconstruction From Uncalibrated Views," Applications of Invariance in Computer Vision, J. Mundy and A. Zisserman, eds., vol. 825, Lecture Notes in Computer Science. Berlin: Springer-Verlag, 1993, pp. 237-256.
[20] J. Oliensis, "A Multi-Frame Structure From Motion Algorithm Under Perspective Projection," tech. rep., NEC Research Institute, Apr. 1997. (Revised version, Mar. 1998.)
[21] G. Xu and Z. Zhang, Epipolar Geometry in Stereo, Motion and Object Recognition. Kluwer Academic Publishers, 1996.
[22] Z. Zhang and G. Xu, "A General Expression of the Fundamental Matrix for Both Perspective and Affine Cameras," Proc. 15th Int'l Joint Conf. Artificial Intelligence, pp. 1,502-1,507,Nagoya, Japan, Aug. 1997.
[23] Z. Zhang, R. Deriche, O. Faugeras, and Q.-T. Luong, "A Robust Technique for Matching Two Uncalibrated Images Through the Recovery of the Unknown Epipolar Geometry," Artificial Intelligence J., vol. 78, pp. 87-119, Oct. 1995.
[24] I. Reid and D. Murray, "Active Tracking of Foveated Feature Clusters Using Affine Structure," Int'l J. Computer Vision, vol. 18, no. 1, pp. 41-60, 1996.
Index Terms:
Motion analysis, multiple-view geometry, 3D reconstruction, optimization criteria, algorithmic comparison, structure from motion, uncalibrated images.
Citation:
Zhengyou Zhang, "On the Optimization Criteria Used in Two-View Motion Analysis," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, no. 7, pp. 717-729, July 1998, doi:10.1109/34.689302
