Uncertainty quantification and estimation of closed curves based on noisy data

Estimating closed curves based on noisy data has been a popular and yet a challenging problem in many fields of applications. Yet, uncertainty quantification of such estimation methods has received much less attention in the literature. The primary challenge stems from the fact that the parametrization of a closed curve is not generally unique and hence popular curve fitting methods (e.g., weighted least squares based on known parametrization) does not work well due to initialization instabilities leading to larger uncertainties. First, an initial set of cluster points are obtained by means of a constrained fuzzy c-means algorithm and an initial curve is constructed by fitting a B-spline curve based on the cluster centers. Second, a novel tuning parameter selection procedure is proposed to obtain optimal number of knots for the B-spline curve. Experimental results with simulated noisy data show that the proposed method works well for a variety of unknown closed curves with sharp changes of slopes and complex curvatures, even when moderate to large noises are added with heteroskedastic errors. Finally, a new curvature preserving uncertainty quantification method is proposed based on an adaptation of bootstrap method that provides confidence band around the fitted curve, an aspect that is rarely provided by popular curve fitting methods.