Point cloud registration plays a crucial role in various fields, including robotics, computer graphics, and medical imaging. This process involves determining spatial relationships between different sets of points, typically within a 3D space. In real-world scenarios, complexities arise from non-rigid movements and partial visibility, such as occlusions or sensor noise, making non-rigid registration a challenging problem. Classic non-rigid registration methods are often computationally demanding, suffer from unstable performance, and, importantly, have limited theoretical guarantees. The optimal transport problem and its unbalanced variations (e.g., the optimal partial transport problem) have emerged as powerful tools for point-cloud registration, establishing a strong benchmark in this field. These methods view point clouds as empirical measures and provide a mathematically rigorous way to quantify the `correspondence' between (the transformed) source and target points. In this paper, we approach the point-cloud registration problem through the lens of optimal transport theory and first propose a comprehensive set of non-rigid registration methods based on the optimal partial transportation problem. Subsequently, leveraging the emerging work on efficient solutions to the one-dimensional optimal partial transport problem, we extend our proposed algorithms via slicing to gain significant computational efficiency, resulting in fast and robust non-rigid registration algorithms. We demonstrate the effectiveness of our proposed methods and compare them against baselines on various 3D and 2D non-rigid registration problems where the source and target point clouds are corrupted by random noise.

翻译：点云配准在机器人学、计算机图形学和医学成像等多个领域中发挥着关键作用。该过程涉及确定不同点集之间的空间关系（通常是在三维空间中）。在实际场景中，非刚性运动和部分可见性（如遮挡或传感器噪声）带来的复杂性使得非刚性配准成为一个具有挑战性的问题。经典的非刚性配准方法通常计算成本高、性能不稳定，且重要的是，其理论保证有限。最优传输问题及其非平衡变体（例如，最优部分传输问题）已成为点云配准的强有力工具，在该领域建立了坚实的基准。这些方法将点云视为经验测度，并提供了一种数学上严谨的方式来量化（变换后的）源点与目标点之间的“对应关系”。在本文中，我们从最优传输理论的视角来研究点云配准问题，并首次提出了一套基于最优部分传输问题的全面非刚性配准方法。随后，利用一维最优部分传输问题高效求解的新兴研究成果，我们通过切片方法扩展了所提算法，以显著提升计算效率，从而得到快速且鲁棒的非刚性配准算法。我们展示了所提方法的有效性，并在源点云和目标点云受随机噪声污染的各种三维和二维非刚性配准问题中，将其与基线方法进行了比较。