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.
A Special Homotopy Continuation Method For A Class of Polynomial Systems
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Chongqing Inst. of Green and Intelligent Techn. Chinese Academy of Sciences（中国科学院重庆绿色智能技术研究院）
Galois groups of Schubert problems via homotopy computation
Institute for Mathematics；its Applications University of Minnesota Minneapolis MN \ 55455 USA；Department of Mathematics Texas A\&M University College Station Texas \ 77843 USA
Dipartimento di Matematica, Università di Trento（特伦托大学数学系）；Worcester State University（伍斯特州立大学）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame（圣母大学应用与计算数学及统计学系）；Inria Sophia Antipolis Méditerranée（索菲亚-安提波利斯地中海国家信息与自动化研究所）
