# On Some Properties of Calibrated Trifocal Tensors

arXiv: https://arxiv.org/abs/1601.01467 · HTML: https://mlsanity.com/paper/1601.01467/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2016.

## Built on by (most cited first, 4 of 4)

- [Snapshot of Algebraic Vision](https://mlsanity.com/paper/2210.11443/) (2022)
- [Practical solutions to the relative pose of three calibrated cameras](https://mlsanity.com/paper/2303.16078/) (2023)
- [Tensor-Based Self-Calibration of Cameras via the TrifocalCalib Method](https://mlsanity.com/paper/2509.17620/) (2025)
- [Multiprojective Geometry of Compatible Triples of Fundamental and Essential Matrices](https://mlsanity.com/paper/2602.23450/) (2026)

## Abstract (arXiv)

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.