An Angular Approach for Range-Based Approximate Maximum Likelihood Source Localization Through Convex Relaxation

Abstract—This work considers the problem of locating a single source from noisy range measurements to a set of nodes in a wireless sensor network. We propose two new techniques that we designate as Source Localization with Nuclear Norm (SLNN) and Source Localization with 1-norm (SL-1), which extend to arbitrary real dimensions our prior work on 2D source localization formulated in the complex plane. Our approach is based on for-mulating a Maximum-Likelihood (ML) estimation problem, and then using convex relaxation techniques to obtain a semidefinite program (SDP) that can be globally and efficiently solved. SLNN directly approximates the Gaussian ML solution, and the relax-ation is shown to be tighter than in other methods in the same class. We present an analysis of the convexity properties of the con-straint set for the 2D complex version of SLNN (SLCP) to justify the observed tightness of the relaxation. We propose the SL-1 algorithm to address the Laplacian noise case, which models the presence of outliers in range measurements. We overcome the non-differentiability of the Laplacian likelihood function by rewriting the ML problem as an exact weighted version of the Gaussian case. In terms of accuracy of localization, the proposed algorithms globally outperform state-of-the-art optimization-based methods in different noise scenarios, while exhibiting moderate computa-tional complexity. Index Terms—Centralized method, convex hull, convex relax-ation, range-based source localization, semidefinite programming. I