A comprehensive analysis of the geometry of TDOA maps in localisation problems

Abstract

In this manuscript we consider the well-established problem of TDOA-based source localization and propose a comprehensive analysis of its solutions for arbitrary sensor measurements and placements. More specifically, we define the TDOA map from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space. We then study the identifiability of the model, giving a complete analytical characterization of the image of this map and its invertibility. This analysis has been conducted in a completely mathematical fashion, using many different tools which make it valid for every sensor configuration. These results are the first step towards the solution of more general problems involving, for example, a larger number of sensors, uncertainty in their placement, or lack of synchronization.

51 pages (3 appendices of 12 pages), 12 figures

References in corpus (2)

• A Hilbert Scheme in Computer Vision
• Positioning systems in Minkowski space-time: Bifurcation problem and observational data

Cited by in corpus (5)

• A Geometrical-Statistical approach to outlier removal for TDOA measuments
• TDOA Matrices: Algebraic Properties and their Application to Robust Denoising with Missing Data
• Source localization and denoising: a perspective from the TDOA space
• Geolocation with FDOA Measurements via Polynomial Systems and RANSAC
• The algebro-geometric study of range maps