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Tensor Embedding of the Fundamental Matrix

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3D Structure from Multiple Images of Large-Scale Environments (SMILE 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1506))

Abstract

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 T jk i (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 F jk i , 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.

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Author information

Authors and Affiliations

  1. Institute of Computer Science, The Hebrew University, 91904, Jerusalem, Israel

    Shai Avidan & Amnon Shashua

Authors
  1. Shai Avidan
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  2. Amnon Shashua
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Editor information

Editors and Affiliations

  1. ESAT-PSI, Katholieke Universiteit Leuven, Kardinaal Mercilaan 9, B-3001, Leuven, Belgium

    Reinhard Koch  & Luc Van Gool  & 

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© 1998 Springer-Verlag Berlin Heidelberg

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Avidan, S., Shashua, A. (1998). Tensor Embedding of the Fundamental Matrix. In: Koch, R., Van Gool, L. (eds) 3D Structure from Multiple Images of Large-Scale Environments. SMILE 1998. Lecture Notes in Computer Science, vol 1506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49437-5_4

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  • DOI: https://doi.org/10.1007/3-540-49437-5_4

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65310-3

  • Online ISBN: 978-3-540-49437-9

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