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We formulate parametric curve fitting as a concave cost minimization problem. Our formulation is general encompassing any parametric curve where parameters can be free or constrained. The proposed concave cost opens the future possibility of applying several available concave programming algorithms in curve fitting. In this paper, we propose a fast local minimization of the concave cost and utilize it to a specific application- automated detection of ellipse-shaped leukocytes (white blood cells) from microscopy images. We illustrate that our solution can cope well with outliers compared to other competitive methods of ellipse fitting and leukocyte detection.