Verifying Global Minima for L2 Minimization Problems in Multiple View Geometry
Author
Summary, in English
We consider the least-squares (L2) minimization
problems in multiple view geometry for triangulation, homography,
camera resectioning and structure-and-motion
with known rotatation, or known plane. Although optimal
algorithms have been given for these problems under an Linfinity
cost function, finding optimal least-squares solutions
to these problems is difficult, since the cost functions are not
convex, and in the worst case may have multiple minima.
Iterative methods can be used to find a good solution, but
this may be a local minimum. This paper provides a method
for verifying whether a local-minimum solution is globally
optimal, by providing a simple and rapid test involving the
Hessian of the cost function. The basic idea is that by showing
that the cost function is convex in a restricted but large
enough neighbourhood, a sufficient condition for global optimality
is obtained.
The method is tested on numerous problem instances of
real data sets. In the vast majority of cases we are able to
verify that the solutions are optimal, in particular, for small
to medium-scale problems.
problems in multiple view geometry for triangulation, homography,
camera resectioning and structure-and-motion
with known rotatation, or known plane. Although optimal
algorithms have been given for these problems under an Linfinity
cost function, finding optimal least-squares solutions
to these problems is difficult, since the cost functions are not
convex, and in the worst case may have multiple minima.
Iterative methods can be used to find a good solution, but
this may be a local minimum. This paper provides a method
for verifying whether a local-minimum solution is globally
optimal, by providing a simple and rapid test involving the
Hessian of the cost function. The basic idea is that by showing
that the cost function is convex in a restricted but large
enough neighbourhood, a sufficient condition for global optimality
is obtained.
The method is tested on numerous problem instances of
real data sets. In the vast majority of cases we are able to
verify that the solutions are optimal, in particular, for small
to medium-scale problems.
Department/s
- Mathematics (Faculty of Engineering)
- Mathematical Imaging Group
- ELLIIT: the Linköping-Lund initiative on IT and mobile communication
Publishing year
2012
Language
English
Pages
288-304
Publication/Series
International Journal of Computer Vision
Volume
101
Issue
2
Document type
Journal article
Publisher
Springer
Topic
- Computer Vision and Robotics (Autonomous Systems)
- Mathematics
Keywords
- reconstruction
- Geometric optimization
- convex programming
Status
Published
Research group
- Mathematical Imaging Group
ISBN/ISSN/Other
- ISSN: 1573-1405