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Optimal Motion Estimation from Multiple View Normalized Epipolar Constraint

This paper presents a statistically optimal method for estimating camera motion and 3D point location from multiple images based on epipolar constraints. The algorithm utilizes a multiview version of the normalized epipolar constraint and employs Lagrange and Riemannian optimization techniques. Experimental results demonstrate the effectiveness and robustness of the algorithm.

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Optimal Motion Estimation from Multiple View Normalized Epipolar Constraint

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  1. Optimal Motion Estimation from Multiple View Normalized Epipolar Constraint René Vidal (UCB) Yi Ma (UIUC), Shawn Hsu (UCB), Shankar Sastry (UCB) ECE Department University of Illinois at Urbana/Champaign EECS Department University of California at Berkeley

  2. Introduction: Problem Statement Input: Corresponding image points in multiple images Output: Camera motion, 3D point location.

  3. Introduction: Related Literature Discrete Case Continuous Case Zhuang et. al. ’84 ’88 Heeger, Jepson ’92 ’93 Kanatani ’93 (Shi, Tomasi ’94) Ma, Kosecka, Sastry ‘98 Longuet-Higgins ’81 Tsai, Huang ’84, Toscani, Faugeras ’86 Two-view Linear Horn ’90, Weng, Ahuja, Huang ’93 Luong, Faugeras ’96 Hartley, Sturm ’97 Zhang ’98 Ma, Kosecka, Sastry ‘99 Bruss, Horn ’83 Maybank ’93 Soatto, Brockett ’97 Zhang, Tomasi ’99 Two-view Nonlinear Tomasi, Kanade ’92 (orth.) Szeliski, Kang ’93 Taylor, Kriegman ’95 (rot.) Triggs ‘95 Heyden, Astrom ’97 (mult.) Oliensis ’99 Multiview Linear/ Nonlinear

  4. Introduction: What is this paper about? Generalization from two views to multiple views of a statistically optimal method for motion and structure estimation based on epipolar constraints From Ma et. al. [ICCV’99] • Epipolar constraints are enough if motion is not rectilinear • A minimal set of constraints is the set of epipolar constraints among three consecutive images Derive a multiview version of the normalized epipolar constraint using Lagrange optimization Use Riemannian optimization on manifolds to get camera motion

  5. Problem Formulation: Multiview Geometry Problem:How to obtain optimal estimates of camera motion and scene structure from multiple images of a cloud of 3D feature points?

  6. Problem Formulation: Algebraic Dependency Theorem: (Algebraic Dependency)Trilinear and quadrilinear constraints are algebraically dependent on bilinear ones, except when the camera centers lie on a straight line(rectilinear motion). [Ma ICCV’99]

  7. Objective Function: Constrained Optimization m images of n points: Noise model, i.i.d. Gaussian: Minimize the objective: subject to constraints: where Remark: Motion and structure estimation is a constrained nonlinear optimization problem which minimizes the reprojection error.

  8. Objective Function: Normalized Epipolar Constraint “Observability Grammian” is invertible, except for points on the line. Using Lagrange optimization and some algebra we obtain: This is an statistically optimal objective function for motion and structure estimation with a nice geometric structure. Hessian of this analytic formula can be used for sensitivity analysis. If “m=2” we obtain the well known normalized epipolar constraint.

  9. Objective Function: Normalized Epipolar Constraint Motion estimation objective function: Normalized epipolar constraint(multiview version): Remark: For the uncalibrated case we just need to replace by

  10. Geometric Optimization: Optimal Triangulation • Initialize • Given structure solve for motion • Given motion solve for structure • Goto 2 until converge

  11. Geometric Optimization: Riemannian Newton • M is not a regular Euclidean space • Can do Euclidean optimization, but need to project to M • There are optimization techniques for Stiefel Manifolds • Quadratic convergence is guaranteed if Hessian is non-degenerate [Smith and Brockett ’93]

  12. Experimental Results: Motion XX-YY, Rot=20o

  13. Experimental Results: Noise 3 pix., Rot=20o

  14. Experimental Results: Real Images, Sequence 1

  15. Experimental Results: Real Images, Sequence 2 Our algorithm Oliensis [IJCV ’99]

  16. Conclusions • Generalization from two views to multiple views of a statistically optimal method for motion and structure estimation based on epipolar constraints • Epipolar constraints are enough if motion is not rectilinear • A minimal set of constraints is the set of epipolar constraints among three consecutive images • Derivation of multiview normalized epipolar constraint • Analytic expression with a nice geometric structure • Use of Lagrange and Riemannian optimization for SFM • Experiments show that the algorithm gives correct estimation of the motion

  17. Current Research • Can improve existing recursive methods if multiview normalized epipolar constraint is used as the objective • Sensitivity analysis of the multiview case using the Hessian of the normalized epipolar constraint • Occlusion: points need to be tracked for 3 frames • Experiments show the algorithm is robust: it works in rectilinear case as well

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