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Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision

Byröd, Martin LU ; Josephson, Klas LU and Åström, Karl LU (2009) In International Journal of Computer Vision 84(3). p.237-256
Abstract
This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies... (More)
This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Polynomial equations, Gröbner basis, Minimal problems, Structure from motion
in
International Journal of Computer Vision
volume
84
issue
3
pages
237 - 256
publisher
Springer
external identifiers
  • wos:000267028600001
  • scopus:67349135634
ISSN
1573-1405
DOI
10.1007/s11263-009-0235-z
language
English
LU publication?
yes
id
3cb140a6-d814-4f55-b2ab-ac89d9f8be62 (old id 1389303)
date added to LUP
2009-05-15 14:40:18
date last changed
2017-10-01 03:41:40
@article{3cb140a6-d814-4f55-b2ab-ac89d9f8be62,
  abstract     = {This paper presents several new results on techniques for solving systems of polynomial equations in computer vision. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. In this paper we derive a generalization of the Gröbner basis method for polynomial equation solving, which improves overall numerical stability. We show how the action matrix can be computed in the general setting of an arbitrary linear basis for ℂ[x]/I. In particular, two improvements on the stability of the computations are made by studying how the linear basis for ℂ[x]/I should be selected. The first of these strategies utilizes QR factorization with column pivoting and the second is based on singular value decomposition (SVD). Moreover, it is shown how to improve stability further by an adaptive scheme for truncation of the Gröbner basis. These new techniques are studied on some of the latest reported uses of Gröbner basis methods in computer vision and we demonstrate dramatically improved numerical stability making it possible to solve a larger class of problems than previously possible.},
  author       = {Byröd, Martin and Josephson, Klas and Åström, Karl},
  issn         = {1573-1405},
  keyword      = {Polynomial equations,Gröbner basis,Minimal problems,Structure from motion},
  language     = {eng},
  number       = {3},
  pages        = {237--256},
  publisher    = {Springer},
  series       = {International Journal of Computer Vision},
  title        = {Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision},
  url          = {http://dx.doi.org/10.1007/s11263-009-0235-z},
  volume       = {84},
  year         = {2009},
}