The Key to Three-View Geometry.
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| Title: | The Key to Three-View Geometry. |
|---|---|
| Authors: | Nordberg, Klas1 klas@isy.liu.se |
| Source: | International Journal of Computer Vision. Sep2011, Vol. 94 Issue 3, p282-294. 13p. 1 Diagram. |
| Subjects: | Contact transformations, Geometric analysis, Tensor algebra, Cameras, Matrices (Mathematics), Three-dimensional imaging |
| Abstract: | In this article we describe a set of canonical transformations of the image spaces that make the description of three-view geometry very simple. The transformations depend on the three-view geometry and the canonically transformed trifocal tensor $\mathcal {T}'$ takes the form of a sparse array where 17 elements in well-defined positions are zero, it has a linear relation to the camera matrices and to two of the fundamental matrices, a third order relation to the third fundamental matrix, a second order relation to the other two trifocal tensors, and first order relations to the 10 three-view all-point matching constraints. In this canonical form, it is also simple to determine if the corresponding camera configuration is degenerate or co-linear. An important property of the three canonical transformations of the images spaces is that they are in SO(3). The 9 parameters needed to determine these transformations and the 9 parameters that determine the elements of $\mathcal {T}'$ together provide a minimal parameterization of the tensor. It does not have problems with multiple maps or multiple solutions that other parameterizations have, and is therefore simple to use. It also provides an implicit representation of the trifocal internal constraints: the sparse canonical representation of the trifocal tensor can be determined if and only if it is consistent with its internal constraints. In the non-ideal case, the canonical transformation can be determined by solving a minimization problem and a simple algorithm for determining the solution is provided. This allows us to extend the standard linear method for estimation of the trifocal tensor to include a constraint enforcement as a final step, similar to the constraint enforcement of the fundamental matrix. Experimental evaluation of this extended linear estimation method shows that it significantly reduces the geometric error of the resulting tensor, but on average the algebraic estimation method is even better. For a small percentage of cases, however, the extended linear method gives a smaller geometric error, implying that it can be used as a complement to the algebraic method for these cases. [ABSTRACT FROM AUTHOR] |
| Copyright of International Journal of Computer Vision is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 61141738 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The Key to Three-View Geometry. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Nordberg%2C+Klas%22">Nordberg, Klas</searchLink><relatesTo>1</relatesTo><i> klas@isy.liu.se</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Computer+Vision%22">International Journal of Computer Vision</searchLink>. Sep2011, Vol. 94 Issue 3, p282-294. 13p. 1 Diagram. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Contact+transformations%22">Contact transformations</searchLink><br /><searchLink fieldCode="DE" term="%22Geometric+analysis%22">Geometric analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Tensor+algebra%22">Tensor algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Cameras%22">Cameras</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Three-dimensional+imaging%22">Three-dimensional imaging</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In this article we describe a set of canonical transformations of the image spaces that make the description of three-view geometry very simple. The transformations depend on the three-view geometry and the canonically transformed trifocal tensor $\mathcal {T}'$ takes the form of a sparse array where 17 elements in well-defined positions are zero, it has a linear relation to the camera matrices and to two of the fundamental matrices, a third order relation to the third fundamental matrix, a second order relation to the other two trifocal tensors, and first order relations to the 10 three-view all-point matching constraints. In this canonical form, it is also simple to determine if the corresponding camera configuration is degenerate or co-linear. An important property of the three canonical transformations of the images spaces is that they are in SO(3). The 9 parameters needed to determine these transformations and the 9 parameters that determine the elements of $\mathcal {T}'$ together provide a minimal parameterization of the tensor. It does not have problems with multiple maps or multiple solutions that other parameterizations have, and is therefore simple to use. It also provides an implicit representation of the trifocal internal constraints: the sparse canonical representation of the trifocal tensor can be determined if and only if it is consistent with its internal constraints. In the non-ideal case, the canonical transformation can be determined by solving a minimization problem and a simple algorithm for determining the solution is provided. This allows us to extend the standard linear method for estimation of the trifocal tensor to include a constraint enforcement as a final step, similar to the constraint enforcement of the fundamental matrix. Experimental evaluation of this extended linear estimation method shows that it significantly reduces the geometric error of the resulting tensor, but on average the algebraic estimation method is even better. For a small percentage of cases, however, the extended linear method gives a smaller geometric error, implying that it can be used as a complement to the algebraic method for these cases. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of International Journal of Computer Vision is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s11263-011-0428-0 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 13 StartPage: 282 Subjects: – SubjectFull: Contact transformations Type: general – SubjectFull: Geometric analysis Type: general – SubjectFull: Tensor algebra Type: general – SubjectFull: Cameras Type: general – SubjectFull: Matrices (Mathematics) Type: general – SubjectFull: Three-dimensional imaging Type: general Titles: – TitleFull: The Key to Three-View Geometry. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Nordberg, Klas IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 09 Text: Sep2011 Type: published Y: 2011 Identifiers: – Type: issn-print Value: 09205691 Numbering: – Type: volume Value: 94 – Type: issue Value: 3 Titles: – TitleFull: International Journal of Computer Vision Type: main |
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