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Publication# From Probability Graphical Models to Dynamic Networks — A Bayesian perspective on Smooth Best Estimate of Trajectory with applications in Geodetic Engineering

Abstract

Bayesian statistics is concerned with the integration of new information obtained through observations with prior knowledge, and accordingly, is often related to information theory (Jospin 2022). Recursive Bayesian estimation methods, such as Kalman Filters, have long been the de facto approach for Smooth Best Estimate of Trajectory (SBET) computations (Aslan 2021). Recently, the robotics and geodetic engineering communities have proposed to use an alternative approach referred to as Dynamic Networks (Rouzaud 2011), to globally determine the SBET solution. Global approaches are generally more complex to compute but are better at propagating additional information to all unobserved variables (Buntine 1994). In this presentation, we will outline the theoretical differences between recursive and global Bayesian models for SBET estimation from a probabilistic and information theoric perspective. We will show how prior knowledge can be integrated into the models and its effect on the accuracy of the SBET estimation in both cases. The theory will be illustrated with practical experimental results related to our recent work on line scanning Hyperspectral camera orientation. Line scanning sensor configuration is a good candidate to demonstrate the benefits of a Bayesian approach to sensor orientation, as each scan-line corresponds to a single pose of the platform, making them far less constrained as compared to pinhole cameras (Brell 2016). The traditional point estimate method for orientation estimation of pinhole cameras (i.e., Bundle Adjustment) is under-constrained in such a configuration. As such, the inclusion of a prior is necessary to make the solution unique.

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Ontological neighbourhood

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A Dynamic Bayesian Network (DBN) is a Bayesian network (BN) which relates variables to each other over adjacent time steps. A Dynamic Bayesian Network (DBN) is often called a Two-Timeslice BN (2TBN) because it says that at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1). DBNs were developed by Paul Dagum in the early 1990s at Stanford University's Section on Medical Informatics.

Kalman filter

For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

Bayesian statistics

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2023