Likelihood Functions for Errors-in-variables Models - Bias-free Local Estimation with Minimum Variance

Kai Krajsek, Christian Heinemann, Hanno Scharr

2014

Abstract

Parameter estimation in the presence of noisy measurements characterizes a wide range of computer vision problems. Thus, many of them can be formulated as errors-in-variables (EIV) problems. In this paper we provide a closed form likelihood function to EIV problems with arbitrary covariance structure. Previous approaches either do not offer a closed form, are restricted in the structure of the covariance matrix, or involve nuisance parameters. By using such a likelihood function, we provide a theoretical justification for well established estimators of EIV models. Furthermore we provide two maximum likelihood estimators for EIV parameters, a straight forward extension of a well known estimator and a novel, local estimator, as well as confidence bounds by means of the Cramer Rao Lower Bound. We show their performance by numerical experiments on optical flow estimation, as it is well explored and understood in literature. The straight forward extension turned out to have oscillating behavior, while the novel, local one performs favorably with respect to other methods. For small motions, it even performs better than an excellent global optical flow algorithm on the majority of pixel locations.

References

  1. Abatzoglou, T., Mendel, J., and Harada, G. (1991). The constrained total least squares technique and its applications to harmonic superresolution. Signal Processing, IEEE Transactions on, 39(5):1070 -1087.
  2. Andres, B., Kondermann, C., Kondermann, D., Kö the, U., Hamprecht, F. A., and Garbe, C. S. (2008). On errorsin-variables regression with arbitrary covariance and its application to optical flow estimation. In CVPR.
  3. Baker, S., Roth, S., Scharstein, D., Black, M. J., Lewis, J., and Szeliski, R. (2007). A database and evaluation methodology for optical flow. Computer Vision, IEEE International Conference on, 0:1-8.
  4. Barron, J. L., Fleet, D. J., and Beauchemin, S. S. (1994). Performance of optical flow techniques. Int. Journal of Computer Vision, 12:43-77.
  5. Chojnacki, W., Brooks, M. J., and Hengel, A. V. D. (2001). Rationalising the renormalisation method of kanatani. Journal of Mathematical Imaging and Vision, 14:21- 38. 10.1023/A:1008355213497.
  6. Clarke, T. A. and Fryer, J. G. (1998). The Development of Camera Calibration Methods and Models. The Photogrammetric Record, 16(91):51-66.
  7. Eldar, Y., Ben-Tal, A., and Nemirovski, A. (2005). Robust mean-squared error estimation in the presence of model uncertainties. Signal Processing, IEEE Transactions on, 53(1):168 - 181.
  8. Galvin, B., Mccane, B., Novins, K., Mason, D., and Mills, S. (1998). Recovering motion fields: An evaluation of eight optical flow algorithms. In British Machine Vision Conference, pages 195-204.
  9. Gleser, L. J. (1981). Estimation in a multivariate ”errors in variables” regression model: Large sample results. The Annals of Statistics, 9(1):24-44.
  10. Holme, A. (2010). Geometry: Our Cultural Heritage. Springer, 2 edition.
  11. Huffel, S. V. and Lemmerling, P. (2002). Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands.
  12. Kanatani, K. (2008). Statistical optimization for geometric fitting: Theoreticalaccuracy bound and high order error analysis. International Journal of Computer Vision, 80:167-188. 10.1007/s11263-007-0098-0.
  13. Kelley, C. (1995). Iterative Methods for Linear and Nonlinear Equations. Society for Industrial and Applied Mathematics, Philadelphia.
  14. Leedan, Y. and Meer, P. (2000). Heteroscedastic regression in computer vision: Problems with bilinear constraint. Int. J. of Computer Vision, 2:127-150.
  15. Lemmerling, P., De Moor, B., and Van Huffel, S. (1996). On the equivalence of constrained total least squares and structured total least squares. Signal Processing, IEEE Transactions on, 44(11):2908 -2911.
  16. Lucas, B. and Kanade, T. (August 1981). An iterative image registration technique with an application to stereo vision. In Proc. Seventh International Joint Conf. on Artificial Intelligence, pages 674-679, Vancouver, Canada.
  17. Markovsky, I. and Huffel, S. V. (2007). Overview of total least-squares methods. Signal Processing, 87(10):2283 - 2302. Special Section: Total Least Squares and Errors-in-Variables Modeling.
  18. Matei, B. C. and Meer, P. (2006). Estimation of nonlinear errors-in-variables models for computer vision applications. IEEE Trans. Pattern Anal. Mach. Intell., 28:1537-1552.
  19. Mü hlich, M. and Mester, R. (2004). Unbiased errorsin-variables estimation using generalized eigensystem analysis. In ECCV Workshop SMVP, pages 38-49.
  20. Nagel, H.-H. (1995). Optical flow estimation and the interaction between measurement errors at adjacent pixel positions. Intern. Journal of Computer Vision, 15:271-288.
  21. Nestares, O., Fleet, D. J., and Heeger, D. J. (2000). Likelihood functions and confidence bounds for Total Least Squares Problems. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition (CVPR'2000), Hilton Head.
  22. Scharr, H. (2000). Optimal Operators in Digital Image Processing. PhD thesis, Interdisciplinary Center for Scientific Computing, Univ. of Heidelberg.
  23. Simoncelli, E. P. (1993). Distributed Analysis and Representation of Visual Motion. PhD thesis, Massachusetts Institut of Technology, USA.
  24. Sun, D., Roth, S., and Black, M. J. (2010). Secrets of optical flow estimation and their principles. In CVPR, pages 2432-2439.
  25. Weber, J. and Malik, J. (1995). Robust computation of optical flow in a multi-scale differential framework. International Journal of Computer Vision, 14:67-81.
  26. Yeredor, A. (2000). The extended least squares criterion: minimization algorithms and applications. Signal Processing, IEEE Transactions on, 49(1):74 -86.
Download


Paper Citation


in Harvard Style

Krajsek K., Heinemann C. and Scharr H. (2014). Likelihood Functions for Errors-in-variables Models - Bias-free Local Estimation with Minimum Variance . In Proceedings of the 9th International Conference on Computer Vision Theory and Applications - Volume 3: VISAPP, (VISIGRAPP 2014) ISBN 978-989-758-009-3, pages 270-279. DOI: 10.5220/0004667402700279


in Bibtex Style

@conference{visapp14,
author={Kai Krajsek and Christian Heinemann and Hanno Scharr},
title={Likelihood Functions for Errors-in-variables Models - Bias-free Local Estimation with Minimum Variance},
booktitle={Proceedings of the 9th International Conference on Computer Vision Theory and Applications - Volume 3: VISAPP, (VISIGRAPP 2014)},
year={2014},
pages={270-279},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004667402700279},
isbn={978-989-758-009-3},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 9th International Conference on Computer Vision Theory and Applications - Volume 3: VISAPP, (VISIGRAPP 2014)
TI - Likelihood Functions for Errors-in-variables Models - Bias-free Local Estimation with Minimum Variance
SN - 978-989-758-009-3
AU - Krajsek K.
AU - Heinemann C.
AU - Scharr H.
PY - 2014
SP - 270
EP - 279
DO - 10.5220/0004667402700279