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This paper discusses the problem of recovering a planar polygon from its measured complex moments. These moments correspond to an indicator function defined over the polygon's support. Previous work on this problem gave necessary and sufficient conditions for such successful recovery process and focused mainly on the case of exact measurements being given. In this paper, we extend these results and treat the same problem in the case where a longer than necessary series of noise corrupted moments is given. Similar to methods found in array processing, system identification, and signal processing, we discuss a set of possible estimation procedures that are based on the Prony and the Pencil methods, relate them one to the other, and compare them through simulations. We then present an improvement over these methods based on the direct use of the maximum-likelihood estimator, exploiting the above methods as initialization. Finally, we show how regularization and, thus, maximum a posteriori probability estimator could be applied to reflect prior knowledge about the recovered polygon.