Solving polynomial equations for minimal problems in computer vision
K Zuzana · 2007
Many vision tasks require efficient solvers of sys- tems of polynomial equations. Epipolar geometry and rela- tive camera pose computation are tasks which can be formu- lated as minimal problems which lead to solving systems of algebraic equations. Often, these systems are not trivial and therefore special algorithms have to be designed to achieve numerical robustness and computational efficiency. In this work we suggest improvements of current techniques for solving systems of polynomial equations suitable for some vision problems. We introduce two tricks. The first trick helps to reduce the number of variables and degrees of the equations. The second trick can be used to replace compu- tationally complex construction of Gr¨ obner basis by a sim- pler procedure. We demonstrate benefits of our technique by providing a solution to the problem of estimating radial dis- tortion and epipolar geometry from eight correspondences in two images. Unlike previous algorithms, which were able to solve the problem from nine correspondences only, we enforce the determinant of the fundamental matrix be zero. This leads to a system of eight quadratic and one cubic equation. We provide an efficient and robust solver of this problem. The quality of the solver is demonstrated on syn- thetic and real data.