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We discuss the recovery of a planar polygon from its measured complex moments. Previous work on this problem gave necessary and sufficient conditions for such successful recovery and focused mainly on the case of exact measurements. This paper extended these results by treating the case where a longer than necessary series of noise corrupted moments is given. Similar to methods found in array processing and system identification, a possible estimation procedure is discussed. We then present an improvement over these methods based on the direct use of the maximum likelihood estimator. Finally, we show how regularization and maximum a posteriori probability estimator could be applied to reflect prior knowledge about the recovered polygon.