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

Abstract

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.

Maximum a posteriori estimators as a limit of Bayes estimators

“…This estimator can be interpreted as an analog of maximum likelihood for Bayesian estimation, where the distribution has become a posterior. A number of sources ([19, §4.1.2] [8, Thm.2.4.3],[16, §7.6.5], [10], [6]) claim that maximum a posteriori estimators are limits of Bayes estimators, in the following sense. Consider the sequence of 0-1 loss functions, {L ν :…”