A New Characterization of the Trifocal Tensor

  title={A New Characterization of the Trifocal Tensor},
  author={Th{\'e}odore Papadopoulo and Olivier D. Faugeras},
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… 

A metric parametrization for trifocal tensors with non-colinear pinholes

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

  • T. ThórhallssonD. W. Murray
  • Mathematics, Computer Science
    Proceedings. 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

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

  • K. Nordberg
  • Computer Science
    2009 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

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

  • E. Martyushev
  • Mathematics
    Journal 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

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.


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

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

  • K. Nordberg
  • Mathematics
    International 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.



A nonlinear method for estimating the projective geometry of 3 views

A new way of deriving the trifocal tensor based on Grassmann-Cayley algebra is given that sheds some new light on its structure and leads to a complete characterization of its geometric and algebraic properties which is fairly institute, i.e. geometric.

Canonic Representations for the Geometries of Multiple Projective Views

We show how a special decomposition of general projection matrices, called canonic enables us to build geometric descriptions for a system of cameras which are invariant with respect to a given group

On the geometry and algebra of the point and line correspondences between N images

The formalism of the Grassmann-Cayley algebra is proposed to use as the simplest way to make both geometric and algebraic statements in a very synthetic and effective way (i.e. allowing actual computation if needed).

Reconstruction from Image Sequences by Means of Relative Depths

  • A. Heyden
  • Mathematics
    Proceedings of IEEE International Conference on Computer Vision
  • 1995
The paper shows how the problems of reconstructing the locations of n points in space from m different images without camera calibration can be put into a similar theoretical framework and a new concept, the reduced fundamental matrix, is introduced.

Tensorial Transfer: Representation of N > 3 Views of 3D scenes

The concatenation operation shows that N tensors are suucient for representing a set of N +2 views, i.e., the tensor of any triplet of views from the set can be generated in closed-form from an arbitrary collection of N tensor over this set.

The Dou8ble Algebra: An Effective Tool for Computing Invariants in Computer Vision

  • S. Carlsson
  • Mathematics
    Applications of Invariance in Computer Vision
  • 1993
This paper shows how to compute linear invariants of general configurations points and lines observed in two images and polyhedral configurations observed in one image without reconstructing individual points and Lines.

Algebraic Functions For Recognition

  • A. Shashua
  • Mathematics
    IEEE Trans. Pattern Anal. Mach. Intell.
  • 1995
The trilinearity result is shown to be of much practical use in visual recognition by alignment-yielding a direct reprojection method that cuts through the computations of camera transformation, scene structure and epipolar geometry.

Robust Recovery of the Epipolar Geometry for an Uncalibrated Stereo Rig

A robust correlation based approach that eliminates outliers is developed to produce a reliable set of corresponding high curvature points that are used to estimate the so-called Fundamental Matrix which is closely related to the epipolar geometry of the uncalibrated stereo rig.