On the structure and properties of the quadrifocal tensor

Amnon Shashua, Lior Wolf

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

16 Scopus citations


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.

Original languageAmerican English
Title of host publicationComputer Vision - ECCV 2000 - 6th European Conference on Computer Vision, Proceedings
EditorsDavid Vernon
PublisherSpringer Verlag
Number of pages15
ISBN (Print)3540676856
StatePublished - 2000
Event6th European Conference on Computer Vision, ECCV 2000 - Dublin, Ireland
Duration: 26 Jun 20001 Jul 2000

Publication series

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


Conference6th European Conference on Computer Vision, ECCV 2000

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2000.


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