Tutorial on Multiple View Geometry

000903: 9.00-13.30

In cunjunction with ICPR'00


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
  2. Tensor Calculus
  3. Modeling cameras
  4. Projective geometry
  5. Multiple view tensors
  6. Linear estimation of tensors
  7. Tensorial transfer
  8. Factorization and bundle adjustment
  9. Flexible calibration
  10. Conclusions
Lecture manuscript will be handed out during the course.

My home page

A paper about multiple view tensors:
Tensorial Properties of Multilinear Constraints
Mathematical Methods in the Applied Sciences, 23 (2000) 169-202.

Anders Heyden
Centre for Mathematical Sciences (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,