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

Wolfgang Hoegele,Barbara Dobler,Rainer Loeschel,Piotr Zygmanski,Oliver Koelbl

doi:10.1155/2013/960421

Wolfgang Hoegele, Barbara Dobler + Show 3 more

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https://doi.org/10.1155/2013/960421

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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.

Highlights

Due to their inherent ability to use prior knowledge, Bayesian approaches have gained interest in many fields in the recent years, such as in computer vision [1, 2] and medical applications [3,4,5,6], as well as robotics (Kalman-Filter, Particle-Filter) [7, 8] and metrology [9]
In order to show the practical differences, we present 5 different cases for the comparison of the estimators with 10,000 sample points x each, in 2D parallel beam and 3D cone beam geometry
For the standard parameters of case (A), in Figure 7 an example of the posterior Probability Density Function (PDF) in 2D for a single sample point x is presented. In this plot on the right side a profile of the likelihood and the posterior function is presented, which demonstrates the general behavior: the width of the posterior is reduced compared to the likelihood function by the cost of introducing a shift of the maximum position by prior knowledge

Summary

Introduction

Due to their inherent ability to use prior knowledge, Bayesian approaches have gained interest in many fields in the recent years, such as in computer vision [1, 2] and medical applications [3,4,5,6], as well as robotics (Kalman-Filter, Particle-Filter) [7, 8] and metrology [9]. The goal of estimation theory is to determine a concrete set of unknown real-valued parameters x ∈ ΩX ⊆ Rm (e.g., the 2D/3D position coordinates of the observed object in triangulation) from real-valued observations u ∈ ΩU ⊆ Rn (e.g., the detected or measured 1D/2D projections of the object), which are imperfectly or ambiguously related to each other (bold fonts of symbols are utilized to emphasize multidimensional vectors). Their ambiguous relation is due to the experimental uncertainties of the given problem, such as an imperfect Radon transform, or similar transforms. The probability that x takes a value lower than s is defined by s

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