This paper presents a novel non-rigid point set registration method that is
inspired by unsupervised clustering analysis. Unlike previous approaches that
treat the source and target point sets as separate entities, we develop a
holistic framework where they are formulated as clustering centroids and
clustering members, separately. We then adopt Tikhonov regularization with an
$\ell_1$-induced Laplacian kernel instead of the commonly used Gaussian kernel
to ensure smooth and more robust displacement fields. Our formulation delivers
closed-form solutions, theoretical guarantees, independence from dimensions,
and the ability to handle large deformations. Subsequently, we introduce a
clustering-improved Nystr\"om method to effectively reduce the computational
complexity and storage of the Gram matrix to linear, while providing a rigorous
bound for the low-rank approximation. Our method achieves high accuracy results
across various scenarios and surpasses competitors by a significant margin,
particularly on shapes with substantial deformations. Additionally, we
demonstrate the versatility of our method in challenging tasks such as shape
transfer and medical registration.

该论文介绍了一种受无监督聚类分析启发的新型非刚性点集配准方法，通过聚类中心和聚类成员分别将源点集和目标点集表述为一个整体框架，并采用带有 Ι₁- 诱导 Laplacian 核的 Tikhonov 正则化来确保平滑而更强健的位移场，在保证封闭形式解、理论保证、与维度无关以及处理大形变的能力的同时，引入了改进的聚类 Nyström 方法来有效减少计算复杂度和 Gram 矩阵的存储，同时为低秩逼近提供了严格的界限。我们的方法在各种场景中实现了高精度的结果，并在具有显著形变的形状上明显超过竞争对手，在形状转换和医学配准等具有挑战性的任务中展现了方法的多功能性。