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In [[computer vision]], the '''trifocal tensor''' (also '''tritensor''') is a 3×3×3 array of numbers
(i.e., a [[tensor]]) that incorporates all [[projective geometry|projective]] geometric relationships
among three views. It relates the coordinates of corresponding points or lines in three views, being
independent of the scene structure and depending only on the relative motion (i.e., pose) among the
three views and their intrinsic calibration parameters. Hence, the trifocal tensor can be considered as
the generalization of the [[fundamental matrix (computer vision)|fundamental matrix]] in three views.
It is noted that despite that the tensor is made up of 27 elements, only 18 of them are actually independent.

== Correlation slices ==
The tensor can also be seen as a collection of three rank-two 3 x 3 matrices
<math>{\mathbf T}_1, \; {\mathbf T}_2, \; {\mathbf T}_3</math> known as its ''correlation slices''.
Assuming that the [[Camera matrix|projection matrices]] of three views are
<math>{\mathbf P}=[ {\mathbf I} \; | \; {\mathbf 0} ]</math>,
<math>{\mathbf P}^'=[ {\mathbf A} \; | \; {\mathbf a}_4 ]</math> and <math>{\mathbf P^{''}}=[{\mathbf B} \; | \; {\mathbf b}_4 ]</math>,
the correlation slices of the corresponding tensor can be expressed in closed form as
<math>{\mathbf T}_i={\mathbf a}_i {\mathbf b}_4^t - {\mathbf a}_4 {\mathbf b}_i^t, \; i=1 \ldots 3</math>,
where
<math>{\mathbf a}_i, \; {\mathbf b}_i</math>
are respectively the ''i''th columns of the camera matrices.
In practice, however, the tensor is estimated from point and line matches across the three views.

== Trilinear constraints ==
One of the most important properties of the trifocal tensor is that it gives rise to linear relationships between
lines and points in three images. More specifically, for triplets of corresponding points
<math>{\mathbf x} \; \leftrightarrow \; {\mathbf x}^{'} \; \leftrightarrow \;{\mathbf x}^{''}</math>
and any corresponding lines
<math>{\mathbf l} \; \leftrightarrow \; {\mathbf l}^{'} \; \leftrightarrow \;{\mathbf l}^{''}</math> through
them, the following ''trilinear constraints'' hold:
:<math>
({\mathbf l}^{'t} \left[{\mathbf T}_1, \; {\mathbf T}_2, \; {\mathbf T}_3 \right] {\mathbf l}^{''}) [{\mathbf l}]_{\times} = {\mathbf 0}^t
</math>

:<math>
{\mathbf l}^{'t} \left( \sum_i x_i {\mathbf T}_i \right) {\mathbf l}^{''} = 0
</math>

:<math>
{\mathbf l}^{'t} \left( \sum_i x_i {\mathbf T}_i \right) [{\mathbf x}^{''}]_{\times} = {\mathbf 0}^t
</math>

:<math>
[{\mathbf x}^']_{\times} \left( \sum_i x_i {\mathbf T}_i \right) {\mathbf l}^{''} = {\mathbf 0}
</math>

:<math>
[{\mathbf x}^']_{\times} \left( \sum_i x_i {\mathbf T}_i \right) [{\mathbf x}^{''}]_{\times} = {\mathbf 0}_{3 \times 3}
</math>
where <math> [\cdot]_{\times} </math> denotes the skew-symmetric [[Cross product#Conversion to matrix multiplication|cross product matrix]].

== Transfer ==
Given the trifocal tensor of three views and a pair of matched points in two views, it is possible to determine the
location of the point in the third view without any further information. This is known as ''point transfer'' and a
similar result holds for lines.

== References ==
*{{cite book |
author=Richard Hartley and Andrew Zisserman |
title=Multiple View Geometry in computer vision |
publisher=Cambridge University Press|
year=2003 |
isbn=0-521-54051-8}} Chapter on tensor is online [http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook2/HZtrifocal.pdf]
*{{cite journal |
author=Richard I. Hartley |
title=Lines and Points in Three Views and the Trifocal Tensor |
journal=International Journal of Computer Vision |
volume=22 |
pages=125–140 |
year=1997 |
doi=10.1023/A:1007936012022
| issue=2}}
*{{cite journal |
author=Philip Torr and Andrew Zisserman |
title=Robust Parameterization and Computation of the Trifocal Tensor |
journal=Image and Vision Computing |
volume=15 |
pages=591–607 |
year=1997 |
doi=10.1016/S0262-8856(97)00010-3
| issue=8}}

== External links ==
* [http://www.informatik.hu-berlin.de/~blaschek/diplvortrag/learn_epi/EpipolarGeo.html Visualization of trifocal geometry] (originally by Sylvain Bougnoux of [[INRIA]] Robotvis, requires [[Java (programming language)|Java]])

[[Category:Geometry in computer vision]]

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