Surface registration plays an important role for anatomical shape analysis in medical imaging. Existing surface registration methods often face a trade-off between efficiency and robustness. Local point matching methods are computationally efficient, but vulnerable to noise and initialisation. Methods designed for global point set alignment tend to incur a high computational cost. To address the challenge, here we present a fast surface registration method, which formulates surface meshes as probability measures and surface registration as a distributional optimisation problem. The discrepancy between two meshes is measured using an efficient sliced Wasserstein distance with log-linear computational complexity. We propose a novel optimisation method, AdamFlow, which generalises the well-known Adam optimisation method from the Euclidean space to the probability space for minimising the sliced Wasserstein distance. We theoretically analyse the asymptotic convergence of AdamFlow and empirically demonstrate its superior performance in both affine and non-rigid surface registration across various anatomical structures.

翻译：表面配准在医学影像的解剖形状分析中扮演着重要角色。现有表面配准方法常面临效率与鲁棒性之间的权衡。局部点匹配方法计算效率高，但易受噪声和初始化影响；而面向全局点集对齐的方法往往计算成本较高。为解决这一难题，本文提出一种快速表面配准方法，将表面网格形式化为概率测度，并将表面配准转化为分布优化问题。我们采用具有对数线性计算复杂度的高效切片Wasserstein距离来衡量两个网格间的差异，并提出一种新型优化方法AdamFlow，它将著名的Adam优化方法从欧几里得空间推广至概率空间，以最小化切片Wasserstein距离。我们在理论上分析了AdamFlow的渐近收敛性，并通过实验验证了其在多种解剖结构的仿射与非刚性表面配准中的优异性能。