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In [[computer vision]], the '''fundamental matrix''' <math> \mathbf{F} </math> is a 3×3 [[matrix (mathematics)|matrix]] which relates corresponding points in [[stereoscopy|stereo image]]s.  In [[epipolar geometry]], with [[Homogeneous_coordinates|homogeneous image coordinates]], '''x''' and '''x'''′, of corresponding points in a stereo image pair, '''Fx''' describes a line (an [[epipolar line]]) on which the corresponding point '''x'''′ on the other image must lie.  That means, for all pairs of corresponding points holds
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: <math> \mathbf{x}'^{\top}  \mathbf{F x} = 0. </math>
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Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone.
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The term "fundamental matrix" was coined by QT Luong in his influential PhD thesis. It is sometimes also referred to as the "'''bifocal tensor'''". As a tensor it is a [[two-point tensor]] in that it is a [[bilinear form]] relating points in distinct coordinate systems.
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The above relation which defines the fundamental matrix was published in 1992 by both Faugeras and Hartley. Although Longuet-Higgins' [[essential matrix]] satisfies a similar relationship, the essential matrix is a metric object pertaining to calibrated cameras, while the fundamental matrix describes the correspondence in more general and fundamental terms of projective geometry.
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This is captured mathematically by the relationship between a fundamental matrix
<math>\mathbf{F}</math> and its corresponding essential matrix <math>\mathbf{E}</math>,
which is
: <math> \mathbf{E} = \mathbf{K}'^{\top} \; \mathbf{F} \; \mathbf{K} </math>
<math>\mathbf{K}</math> and <math>\mathbf{K}'</math> being the intrinsic calibration
matrices of the two images involved.


== Introduction ==
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The fundamental matrix is a relationship between any two images of the same scene that constrains where the projection of points from the scene can occur in both images. Given the projection of a scene point into one of the images the corresponding point in the other image is constrained to a line, helping the search, and allowing for the detection of wrong correspondences. The relation between corresponding image points which the fundamental matrix represents is referred to as ''epipolar constraint'', ''matching constraint'', ''discrete matching constraint'', or ''incidence relation''.
 
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== Projective Reconstruction Theorem ==
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The fundamental matrix can be determined by a set of [[correspondence problem|point correspondences]].  Additionally, these corresponding image points may be ''triangulated'' to world points with the help of camera matrices derived directly from this fundamental matrix.  The scene composed of these world points is within a [[projective transformation]] of the true scene.<ref>Hartley 2003, p. 266-267</ref>


=== Proof ===
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Say that the image point correspondence <math>\mathbf{x} \leftrightarrow \mathbf{x'}</math> derives from the world point <math>\textbf{X}</math> under the camera matrices <math>\left ( \textbf{P}, \textbf{P}' \right )</math> as
 
:<math>
  <li>[http://www.observatoiredesreligions.fr/spip.php?article11 http://www.observatoiredesreligions.fr/spip.php?article11]</li>
\begin{align}
 
\mathbf{x} & = \textbf{P} \textbf{X} \\
  <li>[http://www.philatelie-france-russie.fr/spip.php?article51/ http://www.philatelie-france-russie.fr/spip.php?article51/]</li>
\mathbf{x'} & = \textbf{P}' \textbf{X}
 
\end{align}
  <li>[http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/ http://metransparent.nfrance.com/~k1001/spip.php?article8359&lang=ar&id_forum=8701/]</li>
</math>.
 
Say we transform space by a general [[homography (computer vision)|homography]] matrix <math>\textbf{H}_{4 \times 4}</math> such that <math>\textbf{X}_0 = \textbf{H} \textbf{X}</math>.
  <li>[http://aslm.org/aslm2012/index.php?option=com_kunena&func=view&catid=11&id=64929&Itemid=615#64929 http://aslm.org/aslm2012/index.php?option=com_kunena&func=view&catid=11&id=64929&Itemid=615#64929]</li>
 
</ul>


The cameras then transform as
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:<math>
\begin{align}
\textbf{P}_0 & = \textbf{P} \textbf{H}^{-1} \\
\textbf{P}_0' & = \textbf{P}' \textbf{H}^{-1}
\end{align}
</math>.


:<math>\textbf{P}_0 \textbf{X}_0 = \textbf{P} \textbf{H}^{-1} \textbf{H} \textbf{X} = \textbf{P} \textbf{X} = \mathbf{x}</math> and likewise with <math>\textbf{P}_0'</math> still get us the same image points.
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== Derivation of Fundamental matrix using coplanarity condition ==
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Fundamental matrix can be derived using the coplanarity condition. <ref>Jaehong Oh. [http://etd.ohiolink.edu/view.cgi?acc_num=osu1306250594 "Novel Approach to Epipolar Resampling of HRSI and Satellite Stereo Imagery-based Georeferencing of Aerial Images"], 2011,pp.22-29 accessed 2011-08-05.</ref>


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The fundamental matrix is of [[Rank (linear algebra)|rank]] 2. Its [[Kernel (matrix)|kernel]] defines the [[Epipolar geometry#Epipole|epipole]].
 
 
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==See also==
 
*[[Epipolar geometry]]
  <li>[http://204.13.164.121/spip.php?article4807&lang=nl/ http://204.13.164.121/spip.php?article4807&lang=nl/]</li>
*[[Essential matrix]]
 
 
  <li>[http://enseignement-lsf.com/spip.php?article64#forum24446803 http://enseignement-lsf.com/spip.php?article64#forum24446803]</li>
== Notes ==
 
{{reflist}}
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== References ==
</ul>
 
* {{cite conference |
title=What can be seen in three dimensions with an uncalibrated stereo rig? |
author=Olivier D. Faugeras |
booktitle=Proceedings of European Conference on Computer Vision |
year=1992
}}
 
* {{cite conference |
title=Camera self-calibration: Theory and experiments |
author=Olivier D. Faugeras |
coauthors=Quang-Tuan Luong and Steven Maybank |
booktitle=Proceedings of European Conference on Computer Vision |
year=1992
}}
 
*{{cite journal |
author=Q. T. Luong and Olivier D. Faugeras |
title=The Fundamental Matrix: Theory, Algorithms, and Stability Analysis  |
journal=International Journal of Computer Vision |
volume=17 |
pages=43–75 |
year=1996 |
doi=10.1007/BF00127818
| issue=1}}
 
*{{cite book |
author=Olivier Faugeras and Q. T. Luong |
title=The Geometry of Multiple Images |
publisher=MIT Press|
year=2001 |
isbn=0-262-06220-8}}
 
* {{cite conference |
title=Estimation of relative camera positions for uncalibrated cameras |
author=Richard I. Hartley |
booktitle=Proceedings of European Conference on Computer Vision |
year=1992
}}
 
*{{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}}
 
*{{cite journal |
doi=10.1109/34.601246 |
author=Richard I. Hartley |
title=In Defense of the Eight-Point Algorithm |
journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |
volume=19 |
pages=580–593 |
year=1997
| issue=6}}
 
* {{cite book |
title=Matrice fondamentale et auto-calibration en vision par ordinateur |
author=Q. T. Luong |
series=PhD Thesis, University of Paris, Orsay |
year=1992
}}
 
*{{cite book |
author=Yi Ma |
coauthors=Stefano Soatto, Jana Košecká and S. Shankar Sastry |
title=An Invitation to 3-D Vision |
publisher=Springer |
year=2004 |
isbn=0-387-00893-4}}
 
*{{cite journal |
author=Marc Pollefeys, Reinhard  Koch and Luc van Gool |
title=Self-Calibration and Metric Reconstruction in spite of Varying and Unknown Intrinsic Camera Parameters |
journal=International Journal of Computer Vision |
volume=32 |
pages=7–25 |
year=1999|
doi=10.1023/A:1008109111715
| issue=1}}
 
*{{cite journal |
author=Philip H. S. Torr|
title=The Development and Comparison of Robust Methods for Estimating the Fundamental Matrix |
journal=International Journal of Computer Vision |
volume=24 |
pages=271–300 |
year=1997 |
issue=3
|doi=10.1023/A:1007927408552}}
 
*{{cite journal |
author=Philip H. S. Torr and A. Zisserman |
title=MLESAC: A New Robust Estimator with Application to Estimating Image Geometry |
journal=Journal of Computer Vision and Image Understanding |
volume=78 |
pages=138–156|
year=2000 |
issue=1}}
 
*{{cite book |
author=Gang Xu and Zhengyou Zhang |
title=Epipolar geometry in Stereo, Motion and Object Recognition |
publisher=Kluwer Academic Publishers |
year=1996 |
isbn=0-7923-4199-6}}
 
*{{cite journal |
author=Zhengyou Zhang |
title=Determining the epipolar geometry and its uncertainty: A review |
journal=International Journal of Computer Vision |
volume=27 |
pages=161–195 |
year=1998 |
doi=10.1023/A:1007941100561
| issue=2}}
 
== Toolboxes ==
*[http://www.ics.forth.gr/%7elourakis/fundest/ fundest] is a [[GPL]] [[C (programming language)|C]]/[[C++]] library for [[robust statistics|robust]], non-linear (based on the [[Levenberg-Marquardt algorithm]]) fundamental matrix estimation from matched point pairs and various objective functions (Manolis Lourakis).
*[http://www.mathworks.com/matlabcentral/fileexchange/4576-structure-and-motion-toolkit-in-matlab Structure and Motion Toolkit in MATLAB (Philip H. S. Torr)]
* [http://eia.udg.es/%7Eqsalvi/recerca.html Fundamental Matrix Estimation Toolbox (Joaquim Salvi)]
* [http://egt.dii.unisi.it/ The Epipolar Geometry Toolbox (EGT)]
 
== External links ==
* [http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook2/HZepipolar.pdf Epipolar Geometry and the Fundamental Matrix (chapter from Hartley &amp; Zisserman)]
* [http://citeseer.ist.psu.edu/article/zhang96determining.html Determining the epipolar geometry and its uncertainty: A review (Zhengyou Zhang)]
* [http://www.informatik.hu-berlin.de/~blaschek/diplvortrag/learn_epi/EpipolarGeo.html Visualization of epipolar geometry] (originally by Sylvain Bougnoux of [[INRIA]] Robotvis, requires [[Java (programming language)|Java]])
* [http://danielwedge.com/fmatrix/ The Fundamental Matrix Song] Video demonstrating laws of epipolar geometry.
 
[[Category:Geometry in computer vision]]
 
{{Link FA|de}}

Latest revision as of 23:06, 10 December 2014

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