Optimal Motion and Structure Estimation

Juyang Weng, Narendra Ahuja, Thomas S Huang

Research output: Contribution to journalArticlepeer-review


The existing linear algorithms exhibit various high sensitivities to noise. The analysis presented in this paper provides insight into the causes for such high sensitivities. It is shown in this paper 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. Then, we introduce methods for optimal motion and structure estimation under two situations of noise distribution: 1) known and 2) unknown. Computationally, the optimal estimation amounts to minimizing a nonlinear function. For the correct convergence of this nonlinear minimization, we use a two-step approach. 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. In order to assess the accuracy of the optimal solution, the error in the solution of the optimal estimation algorithm is compared with a theoretical lower error bound—Cramer-Rao bound. The simulations have shown that with Gaussian noise added to the coordinates of the image points, the actual error in the optimal solution is very close to the bound. In addition, we also use the Cramer-Rao bound to indicate the inherent instability of motion estimation from small image disparities, such as motion from optical flow. Finally, it is known that given the same nonlinear objective function and the same initial guess, different minimization methods may lead to different solutions. We investigate the performance difference between a batch least-squares technique (Levenberg-Marquardt) and a sequential least-squares technique (iterated extended Kalman filter) for this motion estimation problem, and the simulations showed that the former gives better results.

Original languageEnglish (US)
Pages (from-to)864-884
Number of pages21
JournalIEEE transactions on pattern analysis and machine intelligence
Issue number9
StatePublished - Sep 1993


  • Cramer-Rao bound
  • extended Kalman filter
  • maximum likelihood estimation
  • minimum variance estimation
  • motion estimation
  • nonlinear least-squares
  • structure from motion

ASJC Scopus subject areas

  • Software
  • Computer Vision and Pattern Recognition
  • Computational Theory and Mathematics
  • Artificial Intelligence
  • Applied Mathematics


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