Skip to main content
Top
Published in:

07-07-2017

Distortion Varieties

Authors: Joe Kileel, Zuzana Kukelova, Tomas Pajdla, Bernd Sturmfels

Published in: Foundations of Computational Mathematics | Issue 4/2018

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions. These are based on Chow polytopes and Gröbner bases. Multi-parameter distortions are studied using tropical geometry. The motivation for distortion varieties comes from multi-view geometry in computer vision. Our theory furnishes a new framework for formulating and solving minimal problems for camera models with image distortion.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
3.
go back to reference M. Bujnak, Algebraic solutions to absolute pose problems, Doctoral Thesis, Czech Technical University in Prague, 2012. M. Bujnak, Algebraic solutions to absolute pose problems, Doctoral Thesis, Czech Technical University in Prague, 2012.
4.
go back to reference M. Bujnak, Z. Kukelova, and T. Pajdla, 3D reconstruction from image collections with a single known focal length, in Proceedings of the 2009 IEEE International Conference on Computer Vision (ICCV 2009), pp. 351–358. M. Bujnak, Z. Kukelova, and T. Pajdla, 3D reconstruction from image collections with a single known focal length, in Proceedings of the 2009 IEEE International Conference on Computer Vision (ICCV 2009), pp. 351–358.
5.
go back to reference M. Bujnak, Z. Kukelova and T. Pajdla, Making minimal solvers fast, in Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2012), pp. 1506–1513. M. Bujnak, Z. Kukelova and T. Pajdla, Making minimal solvers fast, in Proceedings of the 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2012), pp. 1506–1513.
6.
go back to reference M. Byrod, Z. Kukelova, K. Josephson, T. Pajdla and K. Åström, Fast and robust numerical solutions to minimal problems for cameras with radial distortion, in Proceedings of the 2008 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008), pp. 1–8. M. Byrod, Z. Kukelova, K. Josephson, T. Pajdla and K. Åström, Fast and robust numerical solutions to minimal problems for cameras with radial distortion, in Proceedings of the 2008 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008), pp. 1–8.
7.
go back to reference E. Cattani, M. A. Cueto, A. Dickenstein, S. Di Rocco and B. Sturmfels, Mixed discriminants, Math. Z. 274 (2013) 761–778.MathSciNetCrossRefMATH E. Cattani, M. A. Cueto, A. Dickenstein, S. Di Rocco and B. Sturmfels, Mixed discriminants, Math. Z. 274 (2013) 761–778.MathSciNetCrossRefMATH
8.
go back to reference D. Cox, J. Little and H. Schenck, Toric varieties, Graduate Studies in Mathematics 124, American Mathematical Society, Providence, 2011. D. Cox, J. Little and H. Schenck, Toric varieties, Graduate Studies in Mathematics 124, American Mathematical Society, Providence, 2011.
9.
go back to reference J. Dalbec and B. Sturmfels, Introduction to Chow forms, in Invariant methods in discrete and computational geometry (N. White, ed.), Springer, New York, 1995, pp. 37–58. J. Dalbec and B. Sturmfels, Introduction to Chow forms, in Invariant methods in discrete and computational geometry (N. White, ed.), Springer, New York, 1995, pp. 37–58.
10.
go back to reference M. Démazure, Sur deux problèmes de reconstruction, Technical Report 882, INRIA, Rocquencourt, 1988. M. Démazure, Sur deux problèmes de reconstruction, Technical Report 882, INRIA, Rocquencourt, 1988.
11.
go back to reference S. Di Rocco, Linear toric fibrations, in Combinatorial algebraic geometry, Lecture Notes in Mathematics 2108, Springer, Cham, 2014, pp. 119–147. S. Di Rocco, Linear toric fibrations, in Combinatorial algebraic geometry, Lecture Notes in Mathematics 2108, Springer, Cham, 2014, pp. 119–147.
12.
go back to reference D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account), in Algebraic geometry, Bowdoin 1985, Part 1, Proceedings of Symposia in Pure Mathematics 46 (S.J. Bloch, ed.), American Mathematics Society, Providence, 1987, pp. 3–13. D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account), in Algebraic geometry, Bowdoin 1985, Part 1, Proceedings of Symposia in Pure Mathematics 46 (S.J. Bloch, ed.), American Mathematics Society, Providence, 1987, pp. 3–13.
13.
14.
go back to reference M. Fischler and R. Bolles, Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography, Commun. ACM 24 (1981) 381–395.MathSciNetCrossRef M. Fischler and R. Bolles, Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography, Commun. ACM 24 (1981) 381–395.MathSciNetCrossRef
15.
go back to reference A. Fitzgibbon, Simultaneous linear estimation of multiple view geometry and lens distortion, in Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2001), pp. 125–132. A. Fitzgibbon, Simultaneous linear estimation of multiple view geometry and lens distortion, in Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2001), pp. 125–132.
16.
go back to reference G. Fløystad, J. Kileel and G. Ottaviani, The Chow form of the essential variety in computer vision, J. Symbolic Comput., to appear. G. Fløystad, J. Kileel and G. Ottaviani, The Chow form of the essential variety in computer vision, J. Symbolic Comput., to appear.
17.
go back to reference I.M. Gel’fand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, 1994.CrossRefMATH I.M. Gel’fand, M.M. Kapranov and A.V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, 1994.CrossRefMATH
19.
go back to reference J. Harris, Algebraic geometry: a first course, Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992.CrossRef J. Harris, Algebraic geometry: a first course, Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992.CrossRef
20.
go back to reference R. Hartley and A. Zisserman, Multiple view geometry in computer vision, Cambridge University Press, 2nd ed., Cambridge, 2003. R. Hartley and A. Zisserman, Multiple view geometry in computer vision, Cambridge University Press, 2nd ed., Cambridge, 2003.
22.
go back to reference F. Jiang, Y. Kuang, J.E. Solem and K. Åström, A minimal solution to relative pose with unknown focal length and radial distortion, in Proceedings of the 2014 Asian Conference on Computer Vision (ACCV 2014), pp. 443–456. F. Jiang, Y. Kuang, J.E. Solem and K. Åström, A minimal solution to relative pose with unknown focal length and radial distortion, in Proceedings of the 2014 Asian Conference on Computer Vision (ACCV 2014), pp. 443–456.
23.
24.
go back to reference Y. Kuang, J.E. Solem, F. Kahl and K. Åström, Minimal solvers for relative pose with a single unknown radial distortion, in Proceedings of 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2014), pp. 33–40. Y. Kuang, J.E. Solem, F. Kahl and K. Åström, Minimal solvers for relative pose with a single unknown radial distortion, in Proceedings of 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2014), pp. 33–40.
25.
go back to reference Z. Kukelova, Algebraic Methods in Computer Vision, Doctoral Thesis, Czech Technical University in Prague, 2013. Z. Kukelova, Algebraic Methods in Computer Vision, Doctoral Thesis, Czech Technical University in Prague, 2013.
26.
go back to reference Z. Kukelova, M. Bujnak and T. Pajdla, Automatic Generator of Minimal Problem Solvers, in Proceedings of the 2008 European Conference on Computer Vision (ECCV 2008), Lecture Notes in Computer Science 5304, Springer 2008, pp. 302–315. Z. Kukelova, M. Bujnak and T. Pajdla, Automatic Generator of Minimal Problem Solvers, in Proceedings of the 2008 European Conference on Computer Vision (ECCV 2008), Lecture Notes in Computer Science 5304, Springer 2008, pp. 302–315.
27.
go back to reference D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, 2015. D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics 161, American Mathematical Society, 2015.
28.
go back to reference B. Micusik and T. Pajdla, Structure from motion with wide circular field of view cameras, IEEE T. Pattern Anal. 28 (2006) 1135–1149.CrossRef B. Micusik and T. Pajdla, Structure from motion with wide circular field of view cameras, IEEE T. Pattern Anal. 28 (2006) 1135–1149.CrossRef
29.
go back to reference G. Newsam, D. Q. Huynh, M. Brooks and H. P. Pan, Recovering unknown focal lengths in self-calibration: an essentially linear algorithm and degenerate configurations. ISPRS J. Photogramm., vol. XXXI-B3 (1996), 575–580. G. Newsam, D. Q. Huynh, M. Brooks and H. P. Pan, Recovering unknown focal lengths in self-calibration: an essentially linear algorithm and degenerate configurations. ISPRS J. Photogramm., vol. XXXI-B3 (1996), 575–580.
30.
go back to reference D. Nistér, An efficient solution to the five-point relative pose problem, IEEE T. Pattern Anal. 26 (2004) 756–770.CrossRef D. Nistér, An efficient solution to the five-point relative pose problem, IEEE T. Pattern Anal. 26 (2004) 756–770.CrossRef
31.
32.
go back to reference H. Stewenius, D. Nistér, F. Kahl and F. Schaffalitzky, A minimal solution for relative pose with unknown focal length, in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), pp. 789–794. H. Stewenius, D. Nistér, F. Kahl and F. Schaffalitzky, A minimal solution for relative pose with unknown focal length, in Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2005), pp. 789–794.
33.
go back to reference P. Sturm, On focal length calibration from two views, in Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2001), pp. 145–150. P. Sturm, On focal length calibration from two views, in Proceedings of the 2001 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2001), pp. 145–150.
34.
go back to reference B. Sturmfels, Gröbner bases and convex polytopes, American Mathematical Society, University Lectures Series 8, Providence, 1996. B. Sturmfels, Gröbner bases and convex polytopes, American Mathematical Society, University Lectures Series 8, Providence, 1996.
35.
go back to reference B. Sturmfels, Solving systems of polynomial equations, American Mathematical Society, CBMS Regional Conferences Series 97, Providence, 2002. B. Sturmfels, Solving systems of polynomial equations, American Mathematical Society, CBMS Regional Conferences Series 97, Providence, 2002.
36.
go back to reference C. Traverso, Gröbner trace algorithms, in Proceedings of the 1988 International Symposium of Symbolic and Algebraic Computation (ISSAC 1988), Lecture Notes in Computer Science 358 (P. Gianni, ed.), Springer-Verlag, Berlin Heidelberg, 1989, pp. 125–138. C. Traverso, Gröbner trace algorithms, in Proceedings of the 1988 International Symposium of Symbolic and Algebraic Computation (ISSAC 1988), Lecture Notes in Computer Science 358 (P. Gianni, ed.), Springer-Verlag, Berlin Heidelberg, 1989, pp. 125–138.
Metadata
Title
Distortion Varieties
Authors
Joe Kileel
Zuzana Kukelova
Tomas Pajdla
Bernd Sturmfels
Publication date
07-07-2017
Publisher
Springer US
Published in
Foundations of Computational Mathematics / Issue 4/2018
Print ISSN: 1615-3375
Electronic ISSN: 1615-3383
DOI
https://doi.org/10.1007/s10208-017-9361-0

Premium Partner