Problems that require the parameterization of closed contours arise frequently in computer vision applications. This article introduces a new curve parameterization algorithm that is able to fit a closed curve to a set of points while being robust to the presence of outliers and occlusions in the data. This robustness property makes this algorithm applicable to computer vision applications where misclassification of features may lead to outliers. The algorithm starts by fitting ellipses to numerous five point subsets from the source data. The closed curve is parameterized by determining the median perimeter of the set of ellipses. The resulting curve is not an ellipse, allowing arbitrary closed contours to be parameterized. The use of the modal perimeter rather than the median perimeter is also explored. A detailed comparison is made between the proposed curve fitting algorithm and existing robust ellipse fitting algorithms. Finally, the utility of the algorithm for computer vision applications is demonstrated through the parameterization of the boundary of fuel droplets during combustion. The performance of the proposed algorithm and the performance of existing algorithms are compared to a ground truth segmentation of the fuel droplet images, which demonstrates improved performance for both area quantification and edge deviation.