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  In this paper we discuss the problem of recovering the vertices of a planar polygon from its measured complex moments.Because the given, measured moments can be noisy, the recovered vertices are only estimates of the true ones. The literature offers many algorithms for solving such an estimation problem. We will restrict our discussion to the Total Least Squares (TLS) data fitting models HTLS and STLS and the matrix pencil method GPOF. We show the close link between the HTLS and the GPOF method. We use the HTLS method to compute starting values for the STLS method. We compare the accuracy of these three methods on simulated data. 2005 Elsevier B.V. All rights reserved.