Estimation of a circular arc center and its radius
Computer Vision, Graphics, and Image Processing
Direct least-squares fitting of algebraic surfaces
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Direct Least Square Fitting of Ellipses
IEEE Transactions on Pattern Analysis and Machine Intelligence
Convex Optimization
IEEE Transactions on Signal Processing
Iterative Minimization Schemes for Solving the Single Source Localization Problem
SIAM Journal on Optimization
Exact and Approximate Solutions of Source Localization Problems
IEEE Transactions on Signal Processing
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We consider the problem of locating a user's position from a set of noisy pseudoranges to a group of satellites. We consider both the nonlinear least squares formulation of the problem, which is nonconvex and nonsmooth, and the nonlinear squared least squares variant, in which the objective function is smooth, but still nonconvex. We show that the squared least squares problem can be reformulated as a generalized trust region subproblem and as such can be solved efficiently. Conditions for attainment of the optimal solutions of both problems are derived. The nonlinear least squares problem is shown to have tight connections to the well-known geometric circle fitting and orthogonal regression problems. Finally, a fixed point method for the nonlinear least squares formulation is derived and analyzed.