In this paper, we present a residual neural network-based method for point set registration that preserves the topological structure of the target point set. Similar to coherent point drift (CPD), the registration (alignment) problem is viewed as the movement of data points sampled from a target distribution along a regularized displacement vector field. While the coherence constraint in CPD is stated in terms of local motion coherence, the proposed regularization term relies on a global smoothness constraint as a proxy for preserving local topology. This makes CPD less flexible when the deformation is locally rigid but globally non-rigid as in the case of multiple objects and articulate pose registration. A Jacobian-based cost function and geometric-aware statistical distances are proposed to mitigate these issues. The latter allows for measuring misalignment between the target and the reference. The justification for the k-Nearest Neighbour(kNN) graph preservation of target data, when the Jacobian cost is used, is also provided. Further, to tackle the registration of high-dimensional point sets, a constant time stochastic approximation of the Jacobian cost is introduced. The proposed method is illustrated on several 2-dimensional toy examples and tested on high-dimensional flow Cytometry datasets where the task is to align two distributions of cells whilst preserving the kNN-graph in order to preserve the biological signal of the transformed data. The implementation of the proposed approach is available at https://github.com/MuhammadSaeedBatikh/kNN-Res_Demo/ under the MIT license.

翻译：本文提出一种基于残差神经网络的点集配准方法，该方法能够保持目标点集的拓扑结构。与相干点漂移（CPD）类似，配准（对齐）问题被视作从目标分布中采样的数据点沿正则化位移向量场的运动过程。CPD中的相干约束基于局部运动一致性，而本文提出的正则化项则依赖于全局平滑性约束作为保持局部拓扑的代理。这使得CPD在处理多物体和关节姿态配准等局部刚性但全局非刚性的形变时灵活性不足。为此，我们提出基于雅可比矩阵的代价函数和几何感知统计距离来缓解这些问题，其中后者可用于衡量目标与参考之间的未对齐程度。同时，本文论证了在采用雅可比代价函数时k最近邻（kNN）图对目标数据保持性的合理性。进一步地，为应对高维点集配准，我们引入了一种常数时间的随机近似雅可比代价计算方法。该方法在多个二维玩具示例上进行了验证，并在高维流式细胞术数据集上测试了性能——该数据集需要在保持kNN图以保留转换后数据生物信号的前提下对齐两个细胞分布。所提方法的实现代码已基于MIT许可证开源至https://github.com/MuhammadSaeedBatikh/kNN-Res_Demo/。