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1. 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 [3–6], as well as robotics (Kalman-Filter, Particle-Filter) [7, 8] and metrology [9]. In this work, we present a Bayesian framework for the localization of objects with constraints derived from interfaces (surfaces) or boundaries specific to a given localization problem.
Methods to find the locations of objects based on several projections are known from photogrammetry for a long time. These methods are also called triangulation in a multiple view geometry (e.g., see Hartley and Zisserman [1] for an overview of established methods). Introducing probabilities in order to deal with the Gaussian measurement noise is a well-known approach for estimating a 3-dimensional (3D) point based on two 2-dimensional (2D) projections, for example, see Hartley and Sturm [10]. In this context, Bedekar and Haralick [11] provided a Maximum A Posteriori (MAP) estimator by introducing prior knowledge independent of the observation for the estimation based on a set of 2D projections, which represents a starting point for this Paper but does not provide a more general perspective on estimation which is required to bring it to the level described in this paper. We are following this probabilistic interpretation of the inverse problem of triangulation and present explicit solutions in order to characterize the triangulation problem as well as embed it into the general framework of Bayesian estimation. In particular, we demonstrate the benefit...