Recent Developments on Direct Relative Orientation

This paper presents a novel version of the five-point relative orientation algorithm given in Nistér (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the minimal five correspondences the algorithm returns up to ten real solutions. The algorithm can also operate on many points. Like the previous version of the five-point algorithm, our method can operate correctly even in the face of critical surfaces, including planar and ruled quadric scenes. The paper presents comparisons with other direct methods, including the previously developed five-point method, two different six-point methods, the seven-point method, and the eight-point method. It is shown that the five-point method is superior in most cases among the direct methods. The new version of the algorithm was developed from the perspective of algebraic geometry and is presented in the context of computing a Gröbner basis. The constraints are formulated in terms of polynomial equations in the entries of the fundamental matrix. The polynomial equations generate an algebraic ideal for which a Gröbner basis is computed. The Gröbner basis is used to compute the action matrix for multiplication by a single variable monomial. The eigenvectors of the action matrix give the solutions for all the variables and thereby also relative orientation. Using a Gröbner basis makes the solution clear and easy to explain.