The rank 4 constraint in multiple (≥ 3) view geometry

Amnon Shashua; Shai Avidan

doi:10.1007/3-540-61123-1_139

The rank 4 constraint in multiple (≥ 3) view geometry

Amnon Shashua, Shai Avidan

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

40 Scopus citations

Abstract

It has been established that certain trilinear froms of three perspective views give rise to a tensor of 27 intrinsic coefficients [8]. Further investigations have shown the existence of quadlinear forms across four views with the negative result that further views would not add any new constraints [3, 12, 5]. We show in this paper first, general results on any number of views. Rather than seeking new constraints (which we know now is not possible) we seek connections across trilinear tensors of triplets of views. Two main results are shown: (i) trilinear tensors across m > 3 views are embedded in a low dimensional linear subspace, (ii) given two views, all the induced homography matrices are embedded in a four-dimensional linear subspace. The two results, separately and combined, offer new possibilities of handling the consistency across multiple views in a linear manner (via factorization), some of which are further detailed in this paper.

Original language	English
Title of host publication	Computer Vision – ECCV 1996 - 4th European Conference on Computer Vision, Proceedings
Editors	Bernard Buxton, Roberto Cipolla
Publisher	Springer Verlag
Pages	196-206
Number of pages	11
ISBN (Print)	3540611231, 9783540611233
DOIs	https://doi.org/10.1007/3-540-61123-1_139
State	Published - 1996
Externally published	Yes
Event	4th European Conference on Computer Vision, ECCV 1996 - Cambridge, United Kingdom Duration: 15 Apr 1996 → 18 Apr 1996

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	1065
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	4th European Conference on Computer Vision, ECCV 1996
Country/Territory	United Kingdom
City	Cambridge
Period	15/04/96 → 18/04/96

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1996.

Keywords

3D recovery from 2D views
Algebraic and projective geometry
Matching constraints
Projective structure
Trilinearity

Access to Document

10.1007/3-540-61123-1_139

Cite this

Shashua, A., & Avidan, S. (1996). The rank 4 constraint in multiple (≥ 3) view geometry. In B. Buxton, & R. Cipolla (Eds.), Computer Vision – ECCV 1996 - 4th European Conference on Computer Vision, Proceedings (pp. 196-206). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1065). Springer Verlag. https://doi.org/10.1007/3-540-61123-1_139

Shashua, Amnon ; Avidan, Shai. / The rank 4 constraint in multiple (≥ 3) view geometry. Computer Vision – ECCV 1996 - 4th European Conference on Computer Vision, Proceedings. editor / Bernard Buxton ; Roberto Cipolla. Springer Verlag, 1996. pp. 196-206 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "It has been established that certain trilinear froms of three perspective views give rise to a tensor of 27 intrinsic coefficients [8]. Further investigations have shown the existence of quadlinear forms across four views with the negative result that further views would not add any new constraints [3, 12, 5]. We show in this paper first, general results on any number of views. Rather than seeking new constraints (which we know now is not possible) we seek connections across trilinear tensors of triplets of views. Two main results are shown: (i) trilinear tensors across m > 3 views are embedded in a low dimensional linear subspace, (ii) given two views, all the induced homography matrices are embedded in a four-dimensional linear subspace. The two results, separately and combined, offer new possibilities of handling the consistency across multiple views in a linear manner (via factorization), some of which are further detailed in this paper.",

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Shashua, A & Avidan, S 1996, The rank 4 constraint in multiple (≥ 3) view geometry. in B Buxton & R Cipolla (eds), Computer Vision – ECCV 1996 - 4th European Conference on Computer Vision, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1065, Springer Verlag, pp. 196-206, 4th European Conference on Computer Vision, ECCV 1996, Cambridge, United Kingdom, 15/04/96. https://doi.org/10.1007/3-540-61123-1_139

The rank 4 constraint in multiple (≥ 3) view geometry. / Shashua, Amnon; Avidan, Shai.
Computer Vision – ECCV 1996 - 4th European Conference on Computer Vision, Proceedings. ed. / Bernard Buxton; Roberto Cipolla. Springer Verlag, 1996. p. 196-206 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1065).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Shashua A, Avidan S. The rank 4 constraint in multiple (≥ 3) view geometry. In Buxton B, Cipolla R, editors, Computer Vision – ECCV 1996 - 4th European Conference on Computer Vision, Proceedings. Springer Verlag. 1996. p. 196-206. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-61123-1_139

The rank 4 constraint in multiple (≥ 3) view geometry

Abstract

Publication series

Conference

Bibliographical note

Keywords

Access to Document

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Cite this