Automatic Solver Generator for Systems of Laurent Polynomial Equations
This addresses the need for efficient polynomial system solvers in geometric computer vision, though it appears to be an incremental improvement over existing elimination template methods.
The paper tackles the problem of automatically generating fast solvers for families of Laurent polynomial systems with shared monomial structure, which frequently arise in computer vision applications. The proposed automatic solver generator produces solvers that exceed state-of-the-art speeds in most tested cases, including for problems like optimal 3-view triangulation and semi-generalized hybrid pose estimation.
In computer vision applications, the following problem often arises: Given a family of (Laurent) polynomial systems with the same monomial structure but varying coefficients, find a solver that computes solutions for any family member as fast as possible. Under appropriate genericity assumptions, the dimension and degree of the respective polynomial ideal remain unchanged for each particular system in the same family. The state-of-the-art approach to solving such problems is based on elimination templates, which are the coefficient (Macaulay) matrices that encode the transformation from the initial polynomials to the polynomials needed to construct the action matrix. Knowing an action matrix, the solutions of the system are computed from its eigenvectors. The important property of an elimination template is that it applies to all polynomial systems in the family. In this paper, we propose a new practical algorithm that checks whether a given set of Laurent polynomials is sufficient to construct an elimination template. Based on this algorithm, we propose an automatic solver generator for systems of Laurent polynomial equations. The new generator is simple and fast; it applies to ideals with positive-dimensional components; it allows one to uncover partial $p$-fold symmetries automatically. We test our generator on various minimal problems, mostly in geometric computer vision. The speed of the generated solvers exceeds the state-of-the-art in most cases. In particular, we propose the solvers for the following problems: optimal 3-view triangulation, semi-generalized hybrid pose estimation and minimal time-of-arrival self-calibration. The experiments on synthetic scenes show that our solvers are numerically accurate and either comparable to or significantly faster than the state-of-the-art solvers.