Eight-point algorithm - Revision history

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← Previous revision		Revision as of 00:12, 25 May 2025
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	== Using fewer than eight points ==		== Using fewer than eight points ==

	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal= IEEE Transactions on Pattern Analysis and Machine Intelligence\|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite book\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|title=18th International Conference on Pattern Recognition (ICPR'06) \|chapter=Five-Point Motion Estimation Made Easy \|year=2006 \|chapter-url=https://ieeexplore.ieee.org/document/1698971 \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal= IEEE Transactions on Pattern Analysis and Machine Intelligence\|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 \|url-access=subscription }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite book\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|title=18th International Conference on Pattern Recognition (ICPR'06) \|chapter=Five-Point Motion Estimation Made Easy \|year=2006 \|chapter-url=https://ieeexplore.ieee.org/document/1698971 \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>

	== See also ==		== See also ==

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← Previous revision		Revision as of 23:27, 22 March 2024
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	== Using fewer than eight points ==		== Using fewer than eight points ==

	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite book\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|title=18th International Conference on Pattern Recognition (ICPR'06) \|chapter=Five-Point Motion Estimation Made Easy \|year=2006 \|chapter-url=https://ieeexplore.ieee.org/document/1698971 \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal= IEEE Transactions on Pattern Analysis and Machine Intelligence\|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite book\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|title=18th International Conference on Pattern Recognition (ICPR'06) \|chapter=Five-Point Motion Estimation Made Easy \|year=2006 \|chapter-url=https://ieeexplore.ieee.org/document/1698971 \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>

	== See also ==		== See also ==
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	\| title=In Defense of the Eight-Point Algorithm		\| title=In Defense of the Eight-Point Algorithm
	\|date=June 1997		\|date=June 1997
	\| journal=IEEE Transactions on Pattern ~~Recognition~~ and Machine Intelligence		\| journal= IEEE Transactions on Pattern Analysis and Machine Intelligence
	\| volume=19		\| volume=19
	\| issue=6		\| issue=6

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	== See also ==		== See also ==
	* [[Essential matrix]]		* [[Essential matrix]]
	** [[Essential matrix#Extracting rotation and translation]]		** {{section link\|Essential matrix\|Extracting rotation and translation}}
	* [[Fundamental matrix (computer vision)\|Fundamental matrix]]		* [[Fundamental matrix (computer vision)\|Fundamental matrix]]
	* [[Trifocal tensor]]		* [[Trifocal tensor]]

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← Previous revision		Revision as of 10:33, 25 July 2023
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	== Using fewer than eight points ==		== Using fewer than eight points ==

	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite ~~journal~~\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|~~year~~=~~2006~~ ~~\|url=https://ieeexplore.ieee.org/document/1698971~~ \|~~title~~=Five-Point Motion Estimation Made Easy \| ~~journal~~=~~18th~~ ~~International Conference on Pattern Recognition (ICPR'06)~~ \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite book\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|title=18th International Conference on Pattern Recognition (ICPR'06) \|chapter=Five-Point Motion Estimation Made Easy \|year=2006 \|chapter-url=https://ieeexplore.ieee.org/document/1698971 \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>

	== See also ==		== See also ==

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← Previous revision		Revision as of 08:03, 23 July 2023
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	== Using fewer than eight points ==		== Using fewer than eight points ==

	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite journal\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|year=2006 \|url=https://ieeexplore.ieee.org/document/1698971 \|title=Five-Point Motion Estimation Made Easy \| journal=18th International Conference on Pattern Recognition (ICPR'06) }}</ref>		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 \|pages=756–770 \|pmid=18579936 \|s2cid=886598 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite journal\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|year=2006 \|url=https://ieeexplore.ieee.org/document/1698971 \|title=Five-Point Motion Estimation Made Easy \| journal=18th International Conference on Pattern Recognition (ICPR'06) \|pages=630–633 \|isbn=0-7695-2521-0 \|s2cid=7745676 }}</ref>

	== See also ==		== See also ==
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	\| issue=6		\| issue=6
	\| pages=580–593		\| pages=580–593
	\| doi=10.1109/34.601246}}		\| doi=10.1109/34.601246\| s2cid=16919747
			}}

	* {{cite book		* {{cite book

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v2.05b - Bot T20 CW#61 - Fix errors for CW project (Reference before punctuation)

← Previous revision		Revision as of 14:39, 26 February 2023
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	== Using fewer than eight points ==		== Using fewer than eight points ==

	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 }}</ref>. Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm <ref>{{cite journal\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|year=2006 \|url=https://ieeexplore.ieee.org/document/1698971 \|title=Five-Point Motion Estimation Made Easy \| journal=18th International Conference on Pattern Recognition (ICPR'06) }}</ref>.		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm.<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 }}</ref> Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm.<ref>{{cite journal\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|year=2006 \|url=https://ieeexplore.ieee.org/document/1698971 \|title=Five-Point Motion Estimation Made Easy \| journal=18th International Conference on Pattern Recognition (ICPR'06) }}</ref>

	== See also ==		== See also ==

98.47.236.144: /* Using fewer than eight points */

2023-02-20T22:17:25Z

Using fewer than eight points

← Previous revision		Revision as of 22:17, 20 February 2023
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	== Using fewer than eight points ==		== Using fewer than eight points ==

	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. ~~Though~~ ~~possible~~ ~~from~~ a ~~theoretical~~ ~~point~~ of ~~view~~, the ~~practical~~ ~~implementation~~ of ~~this~~ is ~~not~~ ~~straightforward~~ and ~~relies~~ on ~~solving~~ ~~various~~ ~~non~~-~~linear~~ ~~equations~~.		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. David Nister proposed an efficient solution to estimate the essential matrix from set of five paired points, known as the five-point algorithm<ref>{{cite journal\| doi=10.1109/TPAMI.2004.17 \|last=Nister \|first=David \|year=2004 \|url=https://ieeexplore.ieee.org/document/1288525 \|title=An efficient solution to the five-point relative pose problem \| journal=IEEE Transactions on Pattern Analysis and Machine Intelligence \|volume=26 \|issue=6 }}</ref>. Hartley et. al. later proposed a modified and more stable five-point algorithm based on Nister's algorithm <ref>{{cite journal\| doi=10.1109/ICPR.2006.579 \|last=Li \|first=Hongdong \|year=2006 \|url=https://ieeexplore.ieee.org/document/1698971 \|title=Five-Point Motion Estimation Made Easy \| journal=18th International Conference on Pattern Recognition (ICPR'06) }}</ref>.

	Kaveh Fathian et al. proposed algorithms for five, six and seven points that circumvent the computation of the essential matrix by computing the rotation quaternion directly.<ref>{{cite journal\| doi=10.1109/LRA.2018.2792142 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1CBaHmTtslH662Fgqn2tim3NQup0Rh8L1 \|title=QuEst: A Quaternion-Based Approach for Camera Motion Estimation From Minimal Feature Points \|journal=IEEE Robotics and Automation Letters \|volume=3 \|issue=2 \|pages=857–864 \|arxiv=1704.02672 \|s2cid=51983009 }}</ref><ref>{{cite journal\| doi=10.1016/j.robot.2018.05.014 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1MhADXukLWHFAjxa1hyw0HgUrdv8lUq-Y \|title=Camera relative pose estimation for visual servoing using quaternions \|journal=Robotics and Autonomous Systems. \|volume=107 \|pages=45–62 \|s2cid=52011938 }}</ref>

	== See also ==		== See also ==

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2022-04-10T13:14:47Z

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← Previous revision		Revision as of 13:14, 10 April 2022
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	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. Though possible from a theoretical point of view, the practical implementation of this is not straightforward and relies on solving various non-linear equations.		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. Though possible from a theoretical point of view, the practical implementation of this is not straightforward and relies on solving various non-linear equations.

	Kaveh Fathian et al. proposed algorithms for five, six and seven points that circumvent the computation of the essential matrix by computing the rotation quaternion directly.<ref>{{cite journal\| doi=10.1109/LRA.2018.2792142 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1CBaHmTtslH662Fgqn2tim3NQup0Rh8L1 \|title=QuEst: A Quaternion-Based Approach for Camera Motion Estimation From Minimal Feature Points \|journal=IEEE Robotics and Automation Letters \|arxiv=1704.02672 }}</ref><ref>{{cite journal\| doi=10.1016/j.robot.2018.05.014 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1MhADXukLWHFAjxa1hyw0HgUrdv8lUq-Y \|title=Camera relative pose estimation for visual servoing using quaternions \|journal=Robotics and Autonomous Systems. }}</ref>		Kaveh Fathian et al. proposed algorithms for five, six and seven points that circumvent the computation of the essential matrix by computing the rotation quaternion directly.<ref>{{cite journal\| doi=10.1109/LRA.2018.2792142 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1CBaHmTtslH662Fgqn2tim3NQup0Rh8L1 \|title=QuEst: A Quaternion-Based Approach for Camera Motion Estimation From Minimal Feature Points \|journal=IEEE Robotics and Automation Letters \|volume=3 \|issue=2 \|pages=857–864 \|arxiv=1704.02672 \|s2cid=51983009 }}</ref><ref>{{cite journal\| doi=10.1016/j.robot.2018.05.014 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1MhADXukLWHFAjxa1hyw0HgUrdv8lUq-Y \|title=Camera relative pose estimation for visual servoing using quaternions \|journal=Robotics and Autonomous Systems. \|volume=107 \|pages=45–62 \|s2cid=52011938 }}</ref>

	== See also ==		== See also ==
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	\| issue=5828		\| issue=5828
	\| pages=133–135		\| pages=133–135
	\| doi=10.1038/293133a0}}		\| doi=10.1038/293133a0\| bibcode=1981Natur.293..133L
			\| s2cid=4327732
			}}

	[[Category:Geometry in computer vision]]		[[Category:Geometry in computer vision]]

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← Previous revision		Revision as of 12:51, 10 April 2022
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	Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. Though possible from a theoretical point of view, the practical implementation of this is not straightforward and relies on solving various non-linear equations.		Each point pair contributes with one constraining equation on the element in <math> \mathbf{E} </math>. Since <math> \mathbf{E} </math> has five degrees of freedom it should therefore be sufficient with only five point pairs to determine <math> \mathbf{E} </math>. Though possible from a theoretical point of view, the practical implementation of this is not straightforward and relies on solving various non-linear equations.

	Kaveh Fathian et al. proposed algorithms for five, six and seven points that circumvent the computation of the essential matrix by computing the rotation quaternion directly.<ref>{{cite journal\| doi=10.1109/LRA.2018.2792142 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1CBaHmTtslH662Fgqn2tim3NQup0Rh8L1 \|title=QuEst: A Quaternion-Based Approach for Camera Motion Estimation From Minimal Feature Points \|journal=IEEE Robotics and Automation Letters. \|arxiv=1704.02672 }}</ref><ref>{{cite journal\| doi=10.1016/j.robot.2018.05.014 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1MhADXukLWHFAjxa1hyw0HgUrdv8lUq-Y \|title=Camera relative pose estimation for visual servoing using quaternions \|journal=Robotics and Autonomous Systems. }}</ref>		Kaveh Fathian et al. proposed algorithms for five, six and seven points that circumvent the computation of the essential matrix by computing the rotation quaternion directly.<ref>{{cite journal\| doi=10.1109/LRA.2018.2792142 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1CBaHmTtslH662Fgqn2tim3NQup0Rh8L1 \|title=QuEst: A Quaternion-Based Approach for Camera Motion Estimation From Minimal Feature Points \|journal=IEEE Robotics and Automation Letters \|arxiv=1704.02672 }}</ref><ref>{{cite journal\| doi=10.1016/j.robot.2018.05.014 \|last=Fathian \|first=Kaveh \|year=2018 \|url=https://drive.google.com/uc?export=download&id=1MhADXukLWHFAjxa1hyw0HgUrdv8lUq-Y \|title=Camera relative pose estimation for visual servoing using quaternions \|journal=Robotics and Autonomous Systems. }}</ref>

	== See also ==		== See also ==

2601:47:4600:2930:0:0:0:231E at 21:21, 3 April 2022

2022-04-03T21:21:49Z

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	=== Step 2: Solving the equation ===		=== Step 2: Solving the equation ===

	A standard approach to solving this equation implies that <math> \mathbf{e} </math> is a [[Singular value decomposition\|~~left~~ singular vector]] of <math> \mathbf{Y} </math> corresponding to a [[Singular value decomposition\|singular value]] that equals zero. Provided that at least eight linearly independent vectors <math> \tilde{\mathbf{y}}_{k} </math> are used to construct <math> \mathbf{Y} </math> it follows that this singular vector is unique (disregarding scalar multiplication) and, consequently, <math> \mathbf{e} </math> and then <math> \mathbf{E} </math> can be determined.		A standard approach to solving this equation implies that <math> \mathbf{e} </math> is a [[Singular value decomposition\|right singular vector]] of <math> \mathbf{Y} </math> corresponding to a [[Singular value decomposition\|singular value]] that equals zero. Provided that at least eight linearly independent vectors <math> \tilde{\mathbf{y}}_{k} </math> are used to construct <math> \mathbf{Y} </math> it follows that this singular vector is unique (disregarding scalar multiplication) and, consequently, <math> \mathbf{e} </math> and then <math> \mathbf{E} </math> can be determined.

	In the case that more than eight corresponding points are used to construct <math> \mathbf{Y} </math> it is possible that it does not have any singular value equal to zero. This case occurs in practice when the image coordinates are affected by various types of noise. A common approach to deal with this situation is to describe it as a [[total least squares]] problem; find <math> \mathbf{e} </math> which minimizes		In the case that more than eight corresponding points are used to construct <math> \mathbf{Y} </math> it is possible that it does not have any singular value equal to zero. This case occurs in practice when the image coordinates are affected by various types of noise. A common approach to deal with this situation is to describe it as a [[total least squares]] problem; find <math> \mathbf{e} </math> which minimizes