Advertisement

Recursive Estimation with Implicit Constraints

  • Richard Steffen
  • Christian Beder
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4713)

Abstract

Recursive estimation or Kalman filtering usually relies on explicit model functions, that directly and explicitly describe the effect of the parameters on the observations. However, many problems in computer vision, including all those resulting in homogeneous equation systems, are easier described using implicit constraints between the observations and the parameters. By implicit we mean, that the constraints are given by equations, that are not easily solvable for the observation vector.

We present a framework, that allows to incorporate such implicit constraints as measurement equations into a Kalman filter. The algorithm may be used as a black-box, simplifying the process of specifying suitable measurement equations for many problems. As a byproduct, the possibility of specifying model equations non-explicitly, some non-linearities may be avoided and better results can be achieved for certain problems.

Keywords

Explicit Function Bundle Adjustment Recursive Estimation Unscented Transformation Implicit Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beder, C., Steffen, R.: Determining an initial image pair for fixing the scale of a 3d reconstruction from an image sequence. In: Franke, K., Müller, K.-R., Nickolay, B., Schäfer, R. (eds.) Pattern Recognition. LNCS, vol. 4174, pp. 657–666. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Davison, A.J.: Real-Time Simultaneous Localisation and Mapping with a Single Camera. In: Proceeding of the 9th International Conference on Computer Vision, pp. 674–679 (2003)Google Scholar
  3. 3.
    Engels, C., Stewenius, H., Nister, D.: Bundle Adjustment Rules. Photogrammetric Computer Vision (PCV) (September 2006)Google Scholar
  4. 4.
    Förstner, W., Wrobel, B.: Mathematical Concepts in Photogrammetry. In: McGlome, J.C., Mikhail, E.M., Bethel, J. (eds.) Manual of Photogrammetry, pp. 15–180, ASPRS (2004)Google Scholar
  5. 5.
    Grün, A.: An Optimum Algorithm for On-Line Triangulation. International Society for Photogrammetry and Remote Sensing 24 - III/2, 131–151 (1982)Google Scholar
  6. 6.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  7. 7.
    Heuel, S.: Points, Lines and Planes and their Optimal Estimation. In: Radig, B., Florczyk, S. (eds.) Pattern Recognition. LNCS, vol. 2191, pp. 92–99. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Heuel, S.: Uncertain Projective Geometry - Statistical Reasoning for Polyhedral Object Reconstruction. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  9. 9.
    Julier, S., Uhlmann, J.: A new extension of the Kalman filter to nonlinear systems. In: Int. Symp. Aerospace/Defense Sensing, Simul. and Controls, Orlando, FL (1997)Google Scholar
  10. 10.
    Kalman, R.E.: A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 35–45 (1960)Google Scholar
  11. 11.
    Koch, K.R.: Parameter Estimation and Hypothesis Testing in Linear Models, 2nd edn. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  12. 12.
    Montiel, J., Civera, J., Davison, A.: Unified Inverse Depth Parametrization for Monocular SLAM. In: Proceedings of Robotics: Science and Systems, Philadelphia, USA (2006)Google Scholar
  13. 13.
    Perwass, C., Gebken, C., Sommer, G.: Estimation of geometric entities and operators from uncertain data. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) Pattern Recognition. LNCS, vol. 3663, pp. 459–467. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Triggs, B., McLauchlan, P., Hartley, R., Fitzgibbon, A.: Bundle Adjustment – A Modern Synthesis. In: Triggs, B., Zisserman, A., Szeliski, R. (eds.) Vision Algorithms: Theory and Practice. LNCS, vol. 1883, pp. 298–375. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Welch, G., Bishop, G.: An Introduction to the Kalman Filter. Technical Report, University of North Carolina at Chapel Hill (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Richard Steffen
    • 1
  • Christian Beder
    • 2
  1. 1.Department of Photogrammetry, Bonn UniversityGermany
  2. 2.Computer Science Department, Kiel UniversityGermany

Personalised recommendations