On Some Properties of Calibrated Trifocal Tensors

TL;DR

This paper investigates the algebraic properties of calibrated trifocal tensors in three-view geometry, introducing the concept of trifocal essential matrices and deriving algebraic constraints that characterize their structure.

Contribution

It introduces the notion of trifocal essential matrices and provides necessary and sufficient algebraic conditions for their characterization, advancing understanding of calibrated trifocal tensors.

Findings

Derived two necessary and sufficient conditions for trifocal essential matrices.

Proposed 15 quartic and 99 quintic polynomial constraints on calibrated trifocal tensors.

Showed that in real cases, the 15 quartic constraints are sufficient.

Abstract

In two-view geometry, the essential matrix describes the relative position and orientation of two calibrated images. In three views, a similar role is assigned to the calibrated trifocal tensor. It is a particular case of the (uncalibrated) trifocal tensor and thus it inherits all its properties but, due to the smaller degrees of freedom, satisfies a number of additional algebraic constraints. Some of them are described in this paper. More specifically, we define a new notion --- the trifocal essential matrix. On the one hand, it is a generalization of the ordinary (bifocal) essential matrix, and, on the other hand, it is closely related to the calibrated trifocal tensor. We prove the two necessary and sufficient conditions that characterize the set of trifocal essential matrices. Based on these characterizations, we propose three necessary conditions on a calibrated trifocal tensor. They have a form of 15 quartic and 99 quintic polynomial equations. We show that in the practically significant real case the 15 quartic constraints are also sufficient.