Singly-Bordered Block-Diagonal Form for Minimal Problem Solvers

Abstract

The Gröbner basis method for solving systems of polynomial equations became very popular in the computer vision community as it helps to find fast and numerically stable solutions to difficult problems. In this paper, we present a method that potentially significantly speeds up Gröbner basis solvers. We show that the elimination template matrices used in these solvers are usually quite sparse and that by permuting the rows and columns they can be transformed into matrices with nice block-diagonal structure known as the singly-bordered block-diagonal (SBBD) form. The diagonal blocks of the SBBD matrices constitute independent subproblems and can therefore be solved, i.e. eliminated or factored, independently. The computational time can be further reduced on a parallel computer by distributing these blocks to different processors for parallel computation. The speedup is visible also for serial processing since we perform $O(n^3)$ Gauss-Jordan eliminations on smaller (usually two, approximately ${n \over 2} \times {n \over 2}$ and one ${n \over 3} \times {n \over 3}$) matrices. We propose to compute the SBBD form of the elimination template in the preprocessing offline phase using hypergraph partitioning. The final online Gröbner basis solver works directly with the permuted block-diagonal matrices and can be efficiently parallelized. We demonstrate the usefulness of the presented method by speeding up solvers of several important minimal computer vision problems.