Solution Space Analysis of Essential Matrix Based on Algebraic Error Minimization

Gaku Nakano ;

"This paper reports on a solution space analysis of the essential matrix based on algebraic error minimization. Although it has been known since 1988 that an essential matrix has at most 10 real solutions for five-point pairs, the number of solutions in the least-squares case has not been explored. We first derive that the Karush-Kuhn-Tucker conditions of algebraic errors satisfying the Demazure constraints can be represented by a system of polynomial equations without Lagrange multipliers. Then, using computer algebra software, we reveal that the simultaneous equation has at most 220 real solutions, which can be obtained by the Gauss-Newton method, Groebner basis, and homotopy continuation. Through experiments on synthetic and real data, we quantitatively evaluate the convergence of the proposed and the existing methods to globally optimal solutions. Finally, we visualize a spatial distribution of the global and local minima in 3D space."