# A New Characterization of the Trifocal Tensor

@inproceedings{Papadopoulo1998ANC, title={A New Characterization of the Trifocal Tensor}, author={Th{\'e}odore Papadopoulo and Olivier D. Faugeras}, booktitle={ECCV}, year={1998} }

This paper deals with the problem of characterizing and parametrizing the manifold of trifocal tensors that describe the geometry of three views like the fundamental matrix characterizes the geometry of two. The paper contains two new results. First a new, simpler, set of algebraic constraints that characterizes the set of trifocal tensors is presented. Second, we give a new parametrization of the trifocal tensor based upon those constraints which is also simpler than previously known…

## 37 Citations

### A metric parametrization for trifocal tensors with non-colinear pinholes

- Computer Science2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
- 2015

This work investigates a new parametrization of the trifocal tensor for calibrated cameras with non-colinear pinholes obtained from a quotient Riemannian manifold, and incorporates this formulation into state-of-the art methods for optimization on manifolds.

### The tensors of three affine views

- Mathematics, Computer ScienceProceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149)
- 1999

This paper shows how the estimation of the tensors from point correspondences is achieved through factorization, and discusses the estimation from line correspondences.

### A Geometric Analysis of the Trifocal Tensor

- MathematicsIVCNZ
- 1998

This paper gives a novel expression for the trifocal tensor, derive constraints on its geometrical structure and investigate its reconstruction ability computationally, showing that the reconstruction quality is not directly related to the self-consistency of thetrifocal Tensor.

### A minimal parameterization of the trifocal tensor

- Computer Science2009 IEEE Conference on Computer Vision and Pattern Recognition
- 2009

The paper describes a minimal set of 18 parameters that can represent any trifocal tensor consistent with the internal constraints, and describes a simple approach for estimating the three orthogonal matrices in the case of a general 3 ×3 × 3 tensor, i.e., when theinternal constraints are not satisfied.

### A Geometric Approach to the Trifocal Tensor

- MathematicsJournal of Mathematical Imaging and Vision
- 2010

Two different sets of constraints, in the entries of T, that must be satisfied by trifocal tensors are found, which are not complete, but have the property that it is possible to extract from it a set of only eight equations that are generically complete.

### On Some Properties of Calibrated Trifocal Tensors

- MathematicsJournal of Mathematical Imaging and Vision
- 2017

This paper defines a new notion—the trifocal essential matrix, a generalization of the ordinary (bifocal) essential matrix that is closely related to the calibratedtrifocal tensor, and proves the two necessary and sufficient conditions that characterize the set of trIfocal essential matrices.

### Internal Constraints of the Trifocal Tensor

- EngineeringArXiv
- 2011

A second set of minimal and sucient constraints that is simpler is derived and it is shown how this can be used in a new parameterization of the trifocal tensor, which has become an integral part of many projective reconstruction algorithms.

### A MINIMAL SET OF CONSTRAINTS AND A MINIMAL PARAMETERIZATION FOR THE TRIFOCAL TENSOR

- Engineering
- 2002

The topic of this paper is the so-called trifocal tensor (TFT), which describes the relative orientation of three uncalibrated images. The TFT is made up of 27 homogenous elements but only has 18…

### A Minimal Set of Constraints for the Trivocal Tensor

- MathematicsECCV
- 2000

It is shown that, in general, eight nonlinear algebraic constraints are enough to constitute a trifocal tensor by turning attention from correlation to homographic slices, simple geometric considerations yield the desired result.

### The Key to Three-View Geometry

- MathematicsInternational Journal of Computer Vision
- 2011

A set of canonical transformations of the image spaces that make the description of three-view geometry very simple are described and the standard linear method for estimation of the trifocal tensor is extended to include a constraint enforcement as a final step, similar to the constraint enforcement of the fundamental matrix.

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