On the structure and properties of the quadrifocal tensor

Amnon Shashua; Lior Wolf

doi:10.1007/3-540-45054-8_46

On the structure and properties of the quadrifocal tensor

Amnon Shashua, Lior Wolf

The Rachel and Selim Benin School of Engineering and Computer Science

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

16 Scopus citations

Abstract

The quadrifocal tensor which connects image measurements along 4 views is not yet well understood as its counterparts the fundamental matrix and the trifocal tensor. This paper establishes the structure of the tensor as an “epipole-homography” pairing Q^ijkl = υ^′jH^ikl − υ^″kH^ijl + υ^‴lH^ijk where υ′, υ″, υ‴ are the epipoles in views 2,3,4, H is the “homography tensor” the 3-view analogue of the homography matrix, and the indi- ces i; j; k; l are attached to views 1,2,3,4 respectively - i.e., H^ikl is the homography tensor of views 1,3,4. In the course of deriving the structure Q^ijkl we show that Linear Line Complex (LLC) mappings are the basic building block in the process. We also introduce a complete break-down of the tensor slices: 3×3×3 slices are homography tensors, and 3×3 slices are LLC mappings. Furthermore, we present a closed-form formula of the quadrifocal tensor described by the trifocal tensor and fundamental matrix, and also show how to recover projection matrices from the quadrifocal tensor. We also describe the form of the 51 non-linear constraints a quadrifocal tensor must adhere to.

Original language	English
Title of host publication	Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings
Editors	David Vernon
Publisher	Springer Verlag
Pages	710-724
Number of pages	15
ISBN (Print)	3540676856
DOIs	https://doi.org/10.1007/3-540-45054-8_46
State	Published - 2000
Event	6th European Conference on Computer Vision, ECCV 2000 - Dublin, Ireland Duration: 26 Jun 2000 → 1 Jul 2000

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	1842
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	6th European Conference on Computer Vision, ECCV 2000
Country/Territory	Ireland
City	Dublin
Period	26/06/00 → 1/07/00

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2000.

Access to Document

10.1007/3-540-45054-8_46

Cite this

Shashua, A., & Wolf, L. (2000). On the structure and properties of the quadrifocal tensor. In D. Vernon (Ed.), Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings (pp. 710-724). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1842). Springer Verlag. https://doi.org/10.1007/3-540-45054-8_46

Shashua, Amnon ; Wolf, Lior. / On the structure and properties of the quadrifocal tensor. Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings. editor / David Vernon. Springer Verlag, 2000. pp. 710-724 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{988f41a4781e486694b35ba704f988bf,

title = "On the structure and properties of the quadrifocal tensor",

abstract = "The quadrifocal tensor which connects image measurements along 4 views is not yet well understood as its counterparts the fundamental matrix and the trifocal tensor. This paper establishes the structure of the tensor as an “epipole-homography” pairing Qijkl = υ′jHikl − υ″kHijl + υ‴lHijk where υ′, υ″, υ‴ are the epipoles in views 2,3,4, H is the “homography tensor” the 3-view analogue of the homography matrix, and the indi- ces i; j; k; l are attached to views 1,2,3,4 respectively - i.e., Hikl is the homography tensor of views 1,3,4. In the course of deriving the structure Qijkl we show that Linear Line Complex (LLC) mappings are the basic building block in the process. We also introduce a complete break-down of the tensor slices: 3×3×3 slices are homography tensors, and 3×3 slices are LLC mappings. Furthermore, we present a closed-form formula of the quadrifocal tensor described by the trifocal tensor and fundamental matrix, and also show how to recover projection matrices from the quadrifocal tensor. We also describe the form of the 51 non-linear constraints a quadrifocal tensor must adhere to.",

author = "Amnon Shashua and Lior Wolf",

note = "Publisher Copyright: {\textcopyright} Springer-Verlag Berlin Heidelberg 2000.; 6th European Conference on Computer Vision, ECCV 2000 ; Conference date: 26-06-2000 Through 01-07-2000",

year = "2000",

doi = "10.1007/3-540-45054-8_46",

language = "אנגלית",

isbn = "3540676856",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "710--724",

editor = "David Vernon",

booktitle = "Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings",

address = "גרמניה",

}

Shashua, A & Wolf, L 2000, On the structure and properties of the quadrifocal tensor. in D Vernon (ed.), Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1842, Springer Verlag, pp. 710-724, 6th European Conference on Computer Vision, ECCV 2000, Dublin, Ireland, 26/06/00. https://doi.org/10.1007/3-540-45054-8_46

On the structure and properties of the quadrifocal tensor. / Shashua, Amnon; Wolf, Lior.
Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings. ed. / David Vernon. Springer Verlag, 2000. p. 710-724 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1842).

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

TY - GEN

T1 - On the structure and properties of the quadrifocal tensor

AU - Shashua, Amnon

AU - Wolf, Lior

PY - 2000

Y1 - 2000

N2 - The quadrifocal tensor which connects image measurements along 4 views is not yet well understood as its counterparts the fundamental matrix and the trifocal tensor. This paper establishes the structure of the tensor as an “epipole-homography” pairing Qijkl = υ′jHikl − υ″kHijl + υ‴lHijk where υ′, υ″, υ‴ are the epipoles in views 2,3,4, H is the “homography tensor” the 3-view analogue of the homography matrix, and the indi- ces i; j; k; l are attached to views 1,2,3,4 respectively - i.e., Hikl is the homography tensor of views 1,3,4. In the course of deriving the structure Qijkl we show that Linear Line Complex (LLC) mappings are the basic building block in the process. We also introduce a complete break-down of the tensor slices: 3×3×3 slices are homography tensors, and 3×3 slices are LLC mappings. Furthermore, we present a closed-form formula of the quadrifocal tensor described by the trifocal tensor and fundamental matrix, and also show how to recover projection matrices from the quadrifocal tensor. We also describe the form of the 51 non-linear constraints a quadrifocal tensor must adhere to.

AB - The quadrifocal tensor which connects image measurements along 4 views is not yet well understood as its counterparts the fundamental matrix and the trifocal tensor. This paper establishes the structure of the tensor as an “epipole-homography” pairing Qijkl = υ′jHikl − υ″kHijl + υ‴lHijk where υ′, υ″, υ‴ are the epipoles in views 2,3,4, H is the “homography tensor” the 3-view analogue of the homography matrix, and the indi- ces i; j; k; l are attached to views 1,2,3,4 respectively - i.e., Hikl is the homography tensor of views 1,3,4. In the course of deriving the structure Qijkl we show that Linear Line Complex (LLC) mappings are the basic building block in the process. We also introduce a complete break-down of the tensor slices: 3×3×3 slices are homography tensors, and 3×3 slices are LLC mappings. Furthermore, we present a closed-form formula of the quadrifocal tensor described by the trifocal tensor and fundamental matrix, and also show how to recover projection matrices from the quadrifocal tensor. We also describe the form of the 51 non-linear constraints a quadrifocal tensor must adhere to.

UR - http://www.scopus.com/inward/record.url?scp=84944228823&partnerID=8YFLogxK

U2 - 10.1007/3-540-45054-8_46

DO - 10.1007/3-540-45054-8_46

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.conference???

AN - SCOPUS:84944228823

SN - 3540676856

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 710

EP - 724

BT - Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings

A2 - Vernon, David

PB - Springer Verlag

T2 - 6th European Conference on Computer Vision, ECCV 2000

Y2 - 26 June 2000 through 1 July 2000

ER -

Shashua A, Wolf L. On the structure and properties of the quadrifocal tensor. In Vernon D, editor, Computer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings. Springer Verlag. 2000. p. 710-724. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-45054-8_46

On the structure and properties of the quadrifocal tensor

Abstract

Publication series

Conference

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this