School of Information Science and Engineering, Northeastern University, China

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Abstract—The fundamental matrix captures the intrinsic geometric properties of two images of a same 3D scene. It should be of rank two for all the epipolar lines to intersect in a unique epipole. Traditional methods of enforcing the rank two property of the matrix are to parameterize the fundamental matrix during the estimation. This usually results in a system of nonlinear multivariable polynomial equations of higher degree. The solution of which is then hand over to some numerical techniques. Numerical precision analysis and convergence proof of these solutions are needed but neglected. This paper studies the structure of the typical nonlinear multivariable polynomial equations encountered in the fundamental matrix estimation with rank constraint. An algebraic method is presented to solve this type of equations. The method is based on the classical Lagrange multipliers method. After careful transformations of the problem, we reduce the problem to the solution of a single variable polynomial equation.

Index Terms—computer vision, epipolar geometry, fundamental matrix, nonlinear multivariable equations

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Cite: Yuanbin Wang, Bin Zhang, and Tianshun Yao, " An Algebraic Method for Estimating the Fundamental Matrix with Rank Constraint," Journal of Computers vol. 5, no. 7, pp. 1027-1037, 2010.