Internal Constraints of the Trifocal Tensor

TL;DR

This paper provides a new, simpler set of minimal constraints for the trifocal tensor, enhancing theoretical understanding and potentially improving estimation algorithms in multi-view geometry.

Contribution

It introduces a second, simpler set of minimal and sufficient internal constraints for the trifocal tensor and proposes a new parameterization based on these constraints.

Findings

Derived a new set of 8 minimal constraints for the trifocal tensor.

Presented a new parameterization of the trifocal tensor using these constraints.

Enhanced theoretical understanding of the trifocal tensor's internal constraints.

Abstract

The fundamental matrix and trifocal tensor are convenient algebraic representations of the epipolar geometry of two and three view configurations, respectively. The estimation of these entities is central to most reconstruction algorithms, and a solid understanding of their properties and constraints is therefore very important. The fundamental matrix has 1 internal constraint which is well understood, whereas the trifocal tensor has 8 independent algebraic constraints. The internal tensor constraints can be represented in many ways, although there is only one minimal and sufficient set of 8 constraints known. In this paper, we derive a second set of minimal and sufficient constraints that is simpler. We also show how this can be used in a new parameterization of the trifocal tensor. We hope that this increased understanding of the internal constraints may lead to improved algorithms for estimating the trifocal tensor, although the primary contribution is an improved theoretical understanding.