TDOA Matrices: Algebraic Properties and their Application to Robust Denoising with Missing Data

TL;DR

This paper introduces the algebraic structure of TDOA matrices, enabling robust denoising and localization even with missing data and outliers, through a closed-form solution and low-rank estimation techniques.

Contribution

It characterizes TDOA matrices as rank-two with a unique SVD form, and develops a robust denoising method that handles noise, outliers, and missing data effectively.

Findings

Closed-form solution for denoising TDOA matrices with Gaussian noise

Effective handling of missing data in TDOA-based localization

Robust method resistant to outliers and data corruption

Abstract

Measuring the Time delay of Arrival (TDOA) between a set of sensors is the basic setup for many applications, such as localization or signal beamforming. This paper presents the set of TDOA matrices, which are built from noise-free TDOA measurements, not requiring knowledge of the sensor array geometry. We prove that TDOA matrices are rank-two and have a special SVD decomposition that leads to a compact linear parametric representation. Properties of TDOA matrices are applied in this paper to perform denoising, by finding the TDOA matrix closest to the matrix composed with noisy measurements. The paper shows that this problem admits a closed-form solution for TDOA measurements contaminated with Gaussian noise which extends to the case of having missing data. The paper also proposes a novel robust denoising method resistant to outliers, missing data and inspired in recent advances in robust low-rank estimation. Experiments in synthetic and real datasets show TDOA-based localization, both in terms of TDOA accuracy estimation and localization error.