  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.
