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Mathematical Problems in Engineering
Volume 2013, Article ID 960421, 17 pages
Research Article

Bayesian Estimation Applied to Stochastic Localization with Constraints due to Interfaces and Boundaries

1Department of Radiation Oncology, Regensburg University Medical Center, Franz-Josef-Strauss-Allee 11, 93053 Regensburg, Germany
2Department of Computer Science and Mathematics, University of Applied Sciences, Universitaetsstraße 31, 93053 Regensburg, Germany
3Department of Radiation Oncology, Brigham and Women’s Hospital and Harvard Medical School, 75 Francis Street, Boston, MA 02115, USA

Received 21 December 2012; Accepted 13 March 2013

Academic Editor: Marcelo Moreira Cavalcanti

Copyright © 2013 Wolfgang Hoegele et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Purpose. We present a systematic Bayesian formulation of the stochastic localization/triangulation problem close to constraining interfaces. Methods. For this purpose, the terminology of Bayesian estimation is summarized suitably for applied researchers including the presentation of Maximum Likelihood (ML), Maximum A Posteriori (MAP), and Minimum Mean Square Error (MMSE) estimation. Explicit estimators for triangulation are presented for the linear 2D parallel beam and the nonlinear 3D cone beam model. The priors in MAP and MMSE optionally incorporate (A) the hard constraints about the interface and (B) knowledge about the probability of the object with respect to the interface. All presented estimators are compared in several simulation studies for live acquisition scenarios with 10,000 samples each. Results. First, the presented application shows that MAP and MMSE perform considerably better, leading to lower Root Mean Square Errors (RMSEs) in the simulation studies compared to the ML approach by typically introducing a bias. Second, utilizing priors including (A) and (B) is very beneficial compared to just including (A). Third, typically MMSE leads to better results than MAP, by the cost of significantly higher computational effort. Conclusion. Depending on the specific application and prior knowledge, MAP and MMSE estimators strongly increase the estimation accuracy for localization close to interfaces.