Homotopy Continuation for Sensor Networks Self-Calibration

TL;DR

This paper applies homotopy continuation methods to solve complex polynomial systems in sensor network self-calibration, enabling accurate solutions for previously unsolvable configurations and demonstrating stability with noisy data.

Contribution

It introduces homotopy continuation techniques to solve challenging TDOA self-calibration problems, including minimal and subminimal configurations, with theoretical and practical advancements.

Findings

Successfully solved four previously unsolvable configurations in 2D TDOA calibration.

Proved the number of solutions for the minimal configuration is lower than prior estimates.

Demonstrated stability of the solvers with noisy data without nonlinear refinement.

Abstract

Given a sensor network, TDOA self-calibration aims at simultaneously estimating the positions of receivers and transmitters, and transmitters time offsets. This can be formulated as a system of polynomial equations. Due to the elevated number of unknowns and the nonlinearity of the problem, obtaining an accurate solution efficiently is nontrivial. Previous work has shown that iterative algorithms are sensitive to initialization and little noise can lead to failure in convergence. Hence, research has focused on algebraic techniques. Stable and efficient algebraic solvers have been proposed for some network configurations, but they do not work for smaller networks. In this paper, we use homotopy continuation to solve four previously unsolved configurations in 2D TDOA self-calibration, including a minimal one. As a theoretical contribution, we investigate the number of solutions of the new minimal configuration, showing this is much lower than previous estimates. As a more practical contribution, we also present new subminimal solvers, which can be used to achieve unique accurate solutions in previously unsolvable configurations. We demonstrate our solvers are stable both with clean and noisy data, even without nonlinear refinement afterwards. Moreover, we demonstrate the suitability of homotopy continuation for sensor network calibration problems, opening prospects to new applications.