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.

通过最优传输理论，我们提出了一种基于最优部分传输问题的完整的非刚性注册方法集，并通过分层切片的方式扩展了算法以获得显著的计算效率，从而实现了快速且稳健的非刚性注册算法。