A minimal parameterization of the trifocal tensor

Abstract

The paper describes a minimal set of 18 parameters that can represent any trifocal tensor consistent with the internal constraints. 9 parameters describe three orthogonal matrices and 9 parameters describe 10 elements of a sparse tensor T̃ with 17 elements in well-defined positions equal to zero. Any valid trifocal tensor is then given as some specific T̃ transformed by the orthogonal matrices in the respective image domain. The paper also describes a simple approach for estimating the three orthogonal matrices in the case of a general 3 × 3 × 3 tensor, i.e., when the internal constraints are not satisfied. This can be used to accomplish a least squares approximation of a general tensor to a tensor that satisfies the internal constraints. This type of constraint enforcement, in turn, can be used to obtain an improved estimate of the trifocal tensor based on the normalized linear algorithm, with the constraint enforcement as a final step. This makes the algorithm more similar to the corresponding algorithm for estimation of the fundamental matrix. An experiment on synthetic data shows that the constraint enforcement of the trifocal tensor produces a significantly better result than without enforcement, expressed by the positions of the epipoles, given that the constraint enforcement is made in normalized image coordinates.