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Optimal Motion and Structure Estimation
September 1993 (vol. 15 no. 9)
pp. 864-884

The causes of existing linear algorithms exhibiting various high sensitivities to noise are analyzed. It is shown that even a small pixel-level perturbation may override the epipolar information that is essential for the linear algorithms to distinguish different motions. This analysis indicates the need for optimal estimation in the presence of noise. Methods are introduced for optimal motion and structure estimation under two situations of noise distribution: known and unknown. Computationally, the optimal estimation amounts to minimizing a nonlinear function. For the correct convergence of this nonlinear minimization, a two-step approach is used. The first step is using a linear algorithm to give a preliminary estimate for the parameters. The second step is minimizing the optimal objective function starting from that preliminary estimate as an initial guess. A remarkable accuracy improvement has been achieved by this two-step approach over using the linear algorithm alone.

[1] G. Adiv, "Determining three-dimensional motion and structure from optical flow generated by several moving objects,"IEEE Trans. Patt. Anal. Machine Intell.vol. PAMI-7, pp. 348-401, 1985.
[2] J. Aisbett, "An iterated estimation of the motion parameters of a rigid body from noisy displacement vectors,"IEEE Trans. Patt. Anal. Machine Intell., vol. 12, no. 11, pp. 1092-1098, 1990.
[3] B. D. Anderson and J. B. Moore,Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979.
[4] H. H. Baker, "Multiple-image computer vision," inProc. 41st Photogrammetric Week(Stuttgart, West Germany), Sept. 1987, pp. 7-19.
[5] T. Broida and R. Chellappa, "Estimation of object motion parameters from noisy images,"IEEE Trans. Pattern Anal. Machine Intell, vol. PAMI-8, no. 1, Jan. 1986.
[6] K. M. Brown and J. E. Dennis, "Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation,"Numeriche Mathematik, vol. 18, pp. 289-297, 1972.
[7] A. R. Bruss and B. K. Horn, "Passive navigation,"Comput. Vision Graphics Image Processing, vol. 21, pp. 3-20, 1983.
[8] H. Cramér,Mathematical Methods of Statistics. Princeton, NJ: Princeton Univ. Press, 1946.
[9] N. Cui, J. Weng, and P. Cohen, "Extended structure and motion analysis from monocular image sequences," inProc. Third Int. Conf. Comput. Vision(Osaka, Japan), Dec. 1990, pp. 222-229.
[10] O. D. Faugeras, N. Ayache, B. Faverjon, and F. Lutsman, "Building visual maps by combining noisy stereo measurements," inProc. IEEE Conf. Robotics Automat.pp. 1433-1438, 1987.
[11] O. D. Faugeras, F. Lustman, and G. Toscani, "Motion and structure from point and line matches," inProc. Int. Conf. Comput. Vision(London, England), June, 1987.
[12] R. J. Fitzgerald, "Divergence of the Kalman filter,"IEEE Trans. Automat. Contr., vol. AC-16, pp. 736-747, Dec. 1971.
[13] A. Gelb (Ed.),Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974.
[14] A. A. Giordano and F. M. Hsu,Least Squares Estimation with Applications to Digital Signal Processing, New York: Wiley, 1985.
[15] R. Haralick and J. Lee, "The facet approach to optical flow," inProc. Image Understanding Workshop(Arlington, VA), June 1983.
[16] D. Heeger, "Optical flow from spatiotemporal filters," inProc. First Int. Conf. Comput. Vision(London, England), June 1987, pp. 181-190.
[17] B. K. Horn and B. G. Schunck, "Determining optical flow,"Artificial Intell., vol. 17, pp. 185-204, 1981.
[18] T. S. Huang and O. D. Faugeras, "Some properties of theEmatrix in two-view motion estimation,"IEEE Trans. Patt. Anal. Machine Intell., vol. 11, no. 12, pp. 1310-1312, 1989.
[19] P. J. Huber,Robust Statistics. New York: Wiley, 1981.
[20] R. E. Kalman,A New Approach to Linear Filtering and Prediction Problems, J. Basic Eng., 1960, pp. 35-45, Series 82D.
[21] J.K. Kearny, W.B. Thompson, and D.L. Boley, "Optical flow estimation: An error analysis of gradient-based methods with local optimization,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, pp. 229-244, Mar. 1987.
[22] K. Levenberg, "A method for the solution of certain nonlinear problems in least squares,"Quart. Appl. Math., vol. 2, pp. 164-168, 1944.
[23] H. C. Longuet-Higgins, "A computer program for reconstructing a scene from two projections,"Nature, vol. 293, pp. 133-135, Sept. 1981.
[24] H. C. Longuet-Higgins, "The reconstruction of a scene from two projections--Configurations that defeat the 8-point algorithm," inProc. 1st Conf. Artificial Intell. Applications(Denver, CO), Dec. 5-7, 1984, pp. 395-397.
[25] D. G. Luenberger, Optimization byVector Space Methods. New York: Wiley, 1969.
[26] D. G. Luenberger,Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 1982, 2nd ed.
[27] D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters,"SIAM J. Appl. Math., vol. 11, pp. 431-441, 1963.
[28] P. S. Maybeck,Stochastic Models, Estimation, and Control. New York: Academic, 1979, vol. 1.
[29] P. S. Maybeck,Stochastic Models, Estimation, and Control. New York: Academic, 1982, vol. 2.
[30] A. Mitiche and J. K. Aggarwal, "A computational analysis of time-varying images,"Handbook of Pattern Recognition and Image Processing(T. Y. Young and K. S. Fu, Eds.). New York: Academic, 1986.
[31] L. Matthies, R. Szeliski, and T. Kanade, "Kalman filter-based algorithm for estimating depth from image sequences," inProc. IEEE Conf. Comput. Vision Patt. Recogn.(Ann Arbor, MI), June 5-9, 1988, pp. 199-213.
[32] E. H. Moore,General Analysis, 1935, Memoirs, Amer. Philosoph. Soc. 1.
[33] H.-H. Nagel and W. Enkelmann, "An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences,"IEEE Trans. Patt. Anal. Machine Intell., vol. PAMI-8, no. 5, pp. 565-593, Sept. 1986.
[34] J. Ortega and W. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic, 1970.
[35] R. Penrose, "A generalized inverse for matrices,"Cambridge Philosoph. Soc., vol. 51, pp. 406-413, 1955.
[36] R. Penrose, "On best approximate solutions of linear matrix equations,"Cambridge Philosoph. Soc., vol. 52, pp. 17-19, 1956.
[37] C. R. Rao,Linear Statistical Inference and Its Applications. New York: Wiley, 1973, 2nd ed.
[38] P. Rives, E. Breuil, and B. Espiau, "Recursive estimation of 3-D features using optical flow and camera motion," inProc. Conf. Intell. Autonomous Syst.(Amsterdam), Dec. 1986, pp. 522-532.
[39] J. W. Roach and J. K. Aggarwal, "Determining the movement of objects from a sequence of images,"IEEE Trans. Patt. Anal. Machine Intell., vol. 2, no. 6, pp. 554-562, 1980.
[40] F. H. Schlee, C. J. Standish, and N. F. Tota, "Divergence in the Kalman Filter,"AIAA J., vol. 5, pp. 1114-1120, June 1967.
[41] H. W. Sorenson,Parameter Estimation: Principles and Problems. New York: Marcel Dekker, 1980.
[42] H. W. Sorenson (Ed),Kalman Filtering: Theory and Application. New York: IEEE Press, 1985.
[43] M. E. Spetsakis and J. Aloimonos, "Optimal motion estimation," inProc. Workshop Visual Motion(Irvine, CA), Mar. 1989, pp. 229-237.
[44] G. Toscani and O. D. Faugeras, "Structure from motion using the reconstruction and reprojection technique,"Proc. IEEE Workshop Comput. Vision(Miami, FL), Nov. 1987, pp. 345-348.
[45] H. L. Van Trees,Detection, Estimation, and Modulation Theory, Vol. I. New York: Wiley, 1968.
[46] R. Y. Tsai and T. S. Huang, "Uniqueness and estimation of 3-D motion parameters of rigid bodies with curved surfaces,"IEEE Trans. Patt. Anal. Machine Intell., vol. 6, no. 1, pp. 13-27, 1984.
[47] A. Verri and T. Poggio, "Against quantitative optical flow," inProc. First Int. Conf. Comput. Vision(London, England), June 1987, pp. 171-180.
[48] A. M. Waxman, B. Kamgar-Parsi, and M. Subbarao, "Closed-form solutions to image flow equations," inProc. First Int. Conf. Comput. Vision(London, England), June 1987, pp. 12-24.
[49] J. Weng, N. Ahuja, and T. S. Huang, "Error analysis of motion parameters estimation from image sequences," inProc. Int. Conf. Comput. Vision(London, England), June, 1987.
[50] J. Weng, T.S. Huang, and N. Ahuja, "3-D motion estimation, understanding, and prediction from noisy image sequences,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-9, no. 3, 1987.
[51] J. Weng, T. S. Huang, and N. Ahuja, "A two-step approach to optimal motion and structure estimation," inProc. IEEE Workshop Computer Vision(Miami, FL), Nov. 1987, pp. 355-357.
[52] J. Weng, N. Ahuja, and T. S. Huang, "Closed form solution + maximum likelihood: A robust approach to motion and structure estimation," inProc. IEEE Conf. Computer vision and Pattern Recognition, June 1988.
[53] J. Weng, N. Ahuja, and T. S. Huang, "Matching two perspective views,"IEEE Trans. Patt. Anal. Machine Intell., vol. 14, no. 8, pp. 806-825, Aug. 1992.
[54] J. Weng, N. Ahuja, and T. S. Huang, "Optimal motion and structure estimation," inProc. IEEE Conf. Comput. Vision Patt. Recogn.(San Diego, CA), 1989, pp. 144-152.
[55] J. Weng, T. S. Huang, and N. Ahuja, "Motion and structure from two perspective views: algorithms, error analysis and error estimation," inIEEE Trans. Patt. Anal Machine Intell., vol. 11, no. 5, pp. 451-476, 1989.
[56] S. S. Wilks,Mathematical Statistics. New York: Wiley, 1962.
[57] B. L. Yen and T. S. Huang, "Determining 3-D motion and structure of a rigid body using the spherical projection,"Comput. Vision Graphics Image Processing, vol. 21, pp. 21-32, 1983.
[58] S. Zacks,The Theory of Statistical Inference. New York: Wiley, 1971.
[59] X. Zhuang, T. S. Huang, and R. M. Haralick, "Two-view motion analysis: A unified algorithm,"J. Opt. Soc. Amer., vol. 3, no. 9, pp. 1492-1500, Sept. 1986, ser. A.
[60] X. Zhuang, T. S. Huang, and N. Ahuja, "A simplified linear optical flowmotion algorithm,"Compt. Vision Graphics, Image Processing, vol. 42, pp. 334-344, 1988.

Index Terms:
optimal motion estimation; nonlinear function minimization; structure estimation; epipolar information; noise distribution; minimisation; motion estimation
Citation:
J. Weng, N. Ahuja, T.S. Huang, "Optimal Motion and Structure Estimation," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 864-884, Sept. 1993, doi:10.1109/34.232074
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