Tensor embedding of the fundamental matrix

Shai Avidan, Amnon Shashua

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

7 Scopus citations


We revisit the bilinear matching constraint between two perspective views of a 3D scene. Our objective is to represent the constraint in the same manner and form as the trilinear constraint among three views. The motivation is to establish a common terminology that bridges between the fundamental matrix F (associated with the bilinear constraint) and the trifocal tensor Tjki (associated with the trilinearities). By achieving this goal we can unify both the properties and the techniques introduced in the past for working with multiple views for geometric applications. Doing that we introduce a 3×3×3 tensor Fjki, we call the bifocal tensor, that represents the bilinear constraint. The bifocal and trifocal tensors share the same form and share the same contraction properties. By close inspection of the contractions of the bifocal tensor into matrices we show that one can represent the family of rank-2 homography matrices by [δ]×F where δ is a free vector. We then discuss four applications of the new representation: (i) Quasi-metric viewing of projective data, (ii) triangulation, (iii) view synthesis, and (iv) recovery of camera ego-motion from a stream of views.

Original languageAmerican English
Title of host publication3D Structure from Multiple Images of Large-Scale Environments - European Workshop, SMILE 1998, Proceedings
EditorsReinhard Koch, Luc Van Gool
PublisherSpringer Verlag
Number of pages16
ISBN (Print)3540653104, 9783540653103
StatePublished - 1998
EventWorkshop on 3D Structure from Multiple Images of Large-Scale Environments, SMILE 1998 - Freiburg, Germany
Duration: 6 Jun 19987 Jun 1998

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


ConferenceWorkshop on 3D Structure from Multiple Images of Large-Scale Environments, SMILE 1998

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1998.


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