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Fast Optimal Three View Triangulation

Byröd, Martin LU ; Josephson, Klas LU and Åström, Karl LU (2007) Asian Conference on Computer Vision (ACCV 2007), 2007 In Lecture Notes in Computer Science 4844. p.549-559
Abstract
We consider the problem of $L_2$-optimal triangulation from three separate views. Triangulation is an important part of numerous computer vision systems. Under gaussian noise, minimizing the $L_2$ norm of the reprojection error gives a statistically optimal estimate. This has been solved for two views. However, for three or more views, it is not clear how this should be done. A previously proposed, but computationally impractical, method draws on Gr{"o}bner basis techniques to solve for the complete set of stationary points of the cost function. We show how this method can be modified to become significantly more stable and hence given a fast implementation in standard IEEE double precision. We evaluate the precision and speed of the new... (More)
We consider the problem of $L_2$-optimal triangulation from three separate views. Triangulation is an important part of numerous computer vision systems. Under gaussian noise, minimizing the $L_2$ norm of the reprojection error gives a statistically optimal estimate. This has been solved for two views. However, for three or more views, it is not clear how this should be done. A previously proposed, but computationally impractical, method draws on Gr{"o}bner basis techniques to solve for the complete set of stationary points of the cost function. We show how this method can be modified to become significantly more stable and hence given a fast implementation in standard IEEE double precision. We evaluate the precision and speed of the new method on both synthetic and real data. The algorithm has been implemented in a freely available software package which can be downloaded from the Internet. (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Triangulation, Gröbner Basis, Optimal
in
Lecture Notes in Computer Science
editor
Yagi, Yasushi; Kweon, In So; Kang, Sing Bing and Zha, Hongbin
volume
4844
pages
549 - 559
publisher
Springer
conference name
Asian Conference on Computer Vision (ACCV 2007), 2007
external identifiers
  • WOS:000252603100054
  • Scopus:38149087651
ISBN
978-3-540-76389-5
DOI
10.1007/978-3-540-76390-1_54
language
English
LU publication?
yes
id
8b6dce46-a43c-469b-b8a8-478c4fb90706 (old id 787775)
date added to LUP
2008-01-08 16:30:15
date last changed
2016-10-13 04:44:02
@misc{8b6dce46-a43c-469b-b8a8-478c4fb90706,
  abstract     = {We consider the problem of $L_2$-optimal triangulation from three separate views. Triangulation is an important part of numerous computer vision systems. Under gaussian noise, minimizing the $L_2$ norm of the reprojection error gives a statistically optimal estimate. This has been solved for two views. However, for three or more views, it is not clear how this should be done. A previously proposed, but computationally impractical, method draws on Gr{"o}bner basis techniques to solve for the complete set of stationary points of the cost function. We show how this method can be modified to become significantly more stable and hence given a fast implementation in standard IEEE double precision. We evaluate the precision and speed of the new method on both synthetic and real data. The algorithm has been implemented in a freely available software package which can be downloaded from the Internet.},
  author       = {Byröd, Martin and Josephson, Klas and Åström, Karl},
  editor       = {Yagi, Yasushi and Kweon, In So and Kang, Sing Bing and Zha, Hongbin},
  isbn         = {978-3-540-76389-5},
  keyword      = {Triangulation,Gröbner Basis,Optimal},
  language     = {eng},
  pages        = {549--559},
  publisher    = {ARRAY(0x8dfe7c0)},
  series       = {Lecture Notes in Computer Science},
  title        = {Fast Optimal Three View Triangulation},
  url          = {http://dx.doi.org/10.1007/978-3-540-76390-1_54},
  volume       = {4844},
  year         = {2007},
}