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.

基于噪声数据估计闭合曲线一直是许多应用领域中一个流行但具有挑战性的问题。然而，这种估计方法的不确定性量化在文献中受到的关注要少得多。主要挑战源于这样一个事实，即闭合曲线的参数化通常不是唯一的，因此流行的曲线拟合方法（例如，基于已知参数化的加权最小二乘法）由于初始化不稳定性导致更大的不确定性而不能很好地工作。首先，通过约束模糊c-means算法获得一组初始聚类点，并通过基于聚类中心拟合B样条曲线构建初始曲线。其次，提出了一种新的调谐参数选择程序，以获得 B 样条曲线的最佳节点数。模拟噪声数据的实验结果表明，该方法适用于各种斜率急剧变化和复杂曲率的未知闭合曲线，即使在中到大的噪声中加入异方差误差也是如此。最后，基于自举方法的适应性提出了一种新的曲率保持不确定性量化方法，该方法在拟合曲线周围提供置信带，这是流行的曲线拟合方法很少提供的方面。