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Geometry and Critical Configurations of Multiple Views

Kahl, Fredrik LU (2001) In Doctoral Theses in Mathematical Sciences 2001:4.
Abstract
This thesis is concerned with one of the core problems in computer vision, namely to reconstruct a real world scene from several images of it. The interplay between the geometry of the scene, the cameras and the images is analyzed. The framework is based on projective geometry, which is the natural language for describing the geometry of multiple views. The affine and Euclidean geometries are regarded as special cases of projective geometry.



First, the projection of several different geometric primitives in multiple views is described and analyzed. The analysis includes points, lines, quadrics and curved surfaces. The cameras are assumed to be uncalibrated and both the perspective/projective and the affine camera model... (More)
This thesis is concerned with one of the core problems in computer vision, namely to reconstruct a real world scene from several images of it. The interplay between the geometry of the scene, the cameras and the images is analyzed. The framework is based on projective geometry, which is the natural language for describing the geometry of multiple views. The affine and Euclidean geometries are regarded as special cases of projective geometry.



First, the projection of several different geometric primitives in multiple views is described and analyzed. The analysis includes points, lines, quadrics and curved surfaces. The cameras are assumed to be uncalibrated and both the perspective/projective and the affine camera model are considered. Several new reconstruction methods are developed. Some features of these methods include the possibility of handling: (i) missing data, (ii) several different primitives simultaneously and (iii) minimal cases.



Then, focus is turned to the process of obtaining a Euclidean reconstruction of the scene from uncalibrated images. This problem is known as auto-calibration. A reconstruction method which imposes regularity constraints on the camera motion is introduced which makes the auto-calibration problem more stable.



The last part of the thesis is devoted to a theoretical study of necessary and sufficient conditions for obtaining a unique scene reconstruction. One classical result is that for two images of a 3D scene this happens if and only if the scene points and the camera centres do not lie on a ruled quadric. This is generalized to any number of views. Furthermore, analogous critical configurations for the 1D camera are derived. In auto-calibration, it is shown that the critical configurations depend only on the camera motion. Complete classifications of such critical motions which lead to ambiguous reconstructions are given under different settings. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Professor Zisserman, Andrew, Oxford University
organization
publishing date
type
Thesis
publication status
published
subject
keywords
algebra, algebraic geometry, field theory, Number Theory, Matematik, Mathematics, reconstruction, image sequence, absolute conic, critical motions, critical surfaces, perspective projection, affine geometry, Euclidean geometry, multiple view geometry, projective geometry, group theory, Talteori, fältteori, algebraisk geometri, gruppteori, Mathematical logic, set theory, combinatories, Matematisk logik, mängdlära, kombinatorik
in
Doctoral Theses in Mathematical Sciences
volume
2001:4
pages
172 pages
publisher
Fredrik Kahl, Centre for Mathematical Sciences, P.O. Box 118, SE-221 00 Lund, Sweden,
defense location
MH:C, Matematikcentrum
defense date
2001-09-21 13:15
ISSN
1404-0034
ISBN
91-628-4935-2
language
English
LU publication?
yes
id
7813c2c3-48e5-4b0d-b0c3-f3ddf977624c (old id 20228)
date added to LUP
2007-05-28 08:34:57
date last changed
2016-09-19 08:44:52
@phdthesis{7813c2c3-48e5-4b0d-b0c3-f3ddf977624c,
  abstract     = {This thesis is concerned with one of the core problems in computer vision, namely to reconstruct a real world scene from several images of it. The interplay between the geometry of the scene, the cameras and the images is analyzed. The framework is based on projective geometry, which is the natural language for describing the geometry of multiple views. The affine and Euclidean geometries are regarded as special cases of projective geometry.<br/><br>
<br/><br>
First, the projection of several different geometric primitives in multiple views is described and analyzed. The analysis includes points, lines, quadrics and curved surfaces. The cameras are assumed to be uncalibrated and both the perspective/projective and the affine camera model are considered. Several new reconstruction methods are developed. Some features of these methods include the possibility of handling: (i) missing data, (ii) several different primitives simultaneously and (iii) minimal cases.<br/><br>
<br/><br>
Then, focus is turned to the process of obtaining a Euclidean reconstruction of the scene from uncalibrated images. This problem is known as auto-calibration. A reconstruction method which imposes regularity constraints on the camera motion is introduced which makes the auto-calibration problem more stable.<br/><br>
<br/><br>
The last part of the thesis is devoted to a theoretical study of necessary and sufficient conditions for obtaining a unique scene reconstruction. One classical result is that for two images of a 3D scene this happens if and only if the scene points and the camera centres do not lie on a ruled quadric. This is generalized to any number of views. Furthermore, analogous critical configurations for the 1D camera are derived. In auto-calibration, it is shown that the critical configurations depend only on the camera motion. Complete classifications of such critical motions which lead to ambiguous reconstructions are given under different settings.},
  author       = {Kahl, Fredrik},
  isbn         = {91-628-4935-2},
  issn         = {1404-0034},
  keyword      = {algebra,algebraic geometry,field theory,Number Theory,Matematik,Mathematics,reconstruction,image sequence,absolute conic,critical motions,critical surfaces,perspective projection,affine geometry,Euclidean geometry,multiple view geometry,projective geometry,group theory,Talteori,fältteori,algebraisk geometri,gruppteori,Mathematical logic,set theory,combinatories,Matematisk logik,mängdlära,kombinatorik},
  language     = {eng},
  pages        = {172},
  publisher    = {Fredrik Kahl, Centre for Mathematical Sciences, P.O. Box 118, SE-221 00 Lund, Sweden,},
  school       = {Lund University},
  series       = {Doctoral Theses in Mathematical Sciences},
  title        = {Geometry and Critical Configurations of Multiple Views},
  volume       = {2001:4},
  year         = {2001},
}