Tutorial on Multiple View Geometry
In cunjunction with ICPR'00
NEW: 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.
To give a theoretical understanding of multiple view geometry and the
To give working knowledge of how tensors can be used to solve
the structure and motion problem.
To show how multiple view tensors can be used in some different applications:
- Structure and motion
- View synthesis
- The correspondance problem
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:
Lecture manuscript will be handed out during the course.
- Tensor Calculus
- Modeling cameras
- Projective geometry
- Multiple view tensors
- Linear estimation of tensors
- Tensorial transfer
- Factorization and bundle adjustment
- Flexible calibration
My home page
A paper about multiple view tensors:
Tensorial Properties of Multilinear Constraints
Mathematical Methods in the Applied Sciences, 23 (2000) 169-202.
Centre for Mathematical Sciences (LTH)
Lund Institute of Technology / Lund University
P.O. Box 118, S-221 00 LUND
Direct Phone: +46 46 22 204 91
Dept. Phone: +46 46 22 285 37
Fax: +46 46 22 240 10
Last edited, 1999-03-05,