Tutorial on Tensorial Description of Multiple View Geometry

990921: 8.30-12.00

In cunjunction with ICCV'99


Lecture Notes available now.


The tensorial description of multiple view geometry has become increasingly popular during the last few years. The fundamental matrix (or bifocal tensor) has been known for quite a while. The trifocal tensor has been discovered and used succesfully lately. The quadrifocal tensor has recently been discovered and its full potential has not yet been exploited. These tensors encodes the multiple view geometry for two, three and four views respectively. They can be used both in RANSAC algorithms to find correct point matches and in linear reconstruction algorithms. Another application is to transfer features seen in two or three images to another image, which has applications in view synthesis. The tensorial description has many advantages compared to using camera matrices to parameterize the geometry. Firstly, a minimal parameterization without internal gauge freedoms is obtained. Secondly, many transfer formulas can easily be understood and remembered from the tensor formulas. Thirdly, a very elegant and compact description are obtained. Moreover, both point and line features can be handled uniformly and changes of coordinates in the images are easily transfered to the tensor components. Finally, the tensor components along with the epipoles contain all projective invariants of the viewing geometry.


  1. To give a theoretical understanding of multiple view geometry and the tensorial description.
  2. To give working knowledge of how tensors can be used to solve the structure and motion problem.
  3. To show how multiple view tensors can be used in some different applications:

Intended Audience

The target audience is anyone interested in knowing about the principles and applications of multiple view geometry and its latest developments. The tutorial is directed towards both researchers in closely related areas, application-oriented peaple and PhD-students. The course is self-contained in the sense that no background about tensor calculus is needed. Some knowledge about multiple view geometry might be helpful but is not essential.


This tutorial focuses on the understanding and use of multiple view tensors in computer vision. We will cover the following topics:
  1. Introduction (camera model, matrix algebra, manifolds).
  2. What is a tensor?
  3. Matrix formulation of multiple view geometry.
  4. Definition of multiple view tensors.
  5. Linear estimation of tensors.
  6. Linear reconstruction techniques.
  7. Properties of multiple view tensors.
  8. Transfer equations.
  9. Conclusions.
Lecture manuscript will be handed out during the course.

My home page

My ECCV'98 paper about multiple view tensors:
A Common Framework for Multiple-View Tensors

Anders Heyden
Department of Mathematics (LTH)
Lund Institute of Technology / Lund University
P.O. Box 118, S-221 00 LUND

Room: 453A
Direct Phone: +46 46 22 204 91
Dept. Phone: +46 46 22 285 37
Fax: +46 46 22 240 10
e-mail: Anders.Heyden@math.lth.se

Last edited, 1999-03-05,