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Critical configurations for projective reconstruction from multiple views

Hartley, Richard and Kahl, Fredrik LU (2007) In International Journal of Computer Vision 71(1). p.5-47
Abstract
This paper investigates a classical problem in computer vision: Given corresponding points in multiple images, when is there a unique projective reconstruction of the 3D geometry of the scene points and the camera positions? A set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points. For two views, it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface. We give a classification of all possible critical configurations for any number of points in three images, and show that in most cases, the ambiguity extends to any number of cameras. The underlying framework for deriving the critical sets is... (More)
This paper investigates a classical problem in computer vision: Given corresponding points in multiple images, when is there a unique projective reconstruction of the 3D geometry of the scene points and the camera positions? A set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points. For two views, it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface. We give a classification of all possible critical configurations for any number of points in three images, and show that in most cases, the ambiguity extends to any number of cameras. The underlying framework for deriving the critical sets is projective geometry. Using a generalization of Pascal's Theorem, we prove that any number of cameras and scene points on an elliptic quartic form a critical set. Another important class of critical configurations consists of cameras and points on rational quartics. The theoretical results are accompanied by many examples and illustrations. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
multiple view geometry, critical sets, degeneracy, projective geometry, structure from motion, geometry 3D reconstruction
in
International Journal of Computer Vision
volume
71
issue
1
pages
5 - 47
publisher
Springer
external identifiers
  • wos:000241228600001
  • scopus:33748430381
ISSN
1573-1405
DOI
10.1007/s11263-005-4796-1
language
English
LU publication?
yes
id
60aba156-0e38-4dd0-a175-c9d3bbbb0ae4 (old id 387983)
date added to LUP
2007-12-11 13:12:36
date last changed
2017-07-09 03:38:10
@article{60aba156-0e38-4dd0-a175-c9d3bbbb0ae4,
  abstract     = {This paper investigates a classical problem in computer vision: Given corresponding points in multiple images, when is there a unique projective reconstruction of the 3D geometry of the scene points and the camera positions? A set of points and cameras is said to be critical when there is more than one way of realizing the resulting image points. For two views, it has been known for almost a century that the critical configurations consist of points and camera lying on a ruled quadric surface. We give a classification of all possible critical configurations for any number of points in three images, and show that in most cases, the ambiguity extends to any number of cameras. The underlying framework for deriving the critical sets is projective geometry. Using a generalization of Pascal's Theorem, we prove that any number of cameras and scene points on an elliptic quartic form a critical set. Another important class of critical configurations consists of cameras and points on rational quartics. The theoretical results are accompanied by many examples and illustrations.},
  author       = {Hartley, Richard and Kahl, Fredrik},
  issn         = {1573-1405},
  keyword      = {multiple view geometry,critical sets,degeneracy,projective geometry,structure from motion,geometry 3D reconstruction},
  language     = {eng},
  number       = {1},
  pages        = {5--47},
  publisher    = {Springer},
  series       = {International Journal of Computer Vision},
  title        = {Critical configurations for projective reconstruction from multiple views},
  url          = {http://dx.doi.org/10.1007/s11263-005-4796-1},
  volume       = {71},
  year         = {2007},
}