Shape space is an active research topic in computer vision and medical imaging fields. The distance defined in a shape space may provide a simple and refined index to represent a unique shape. This work studies the Wasserstein space and proposes a novel framework to compute the Wasserstein distance between general topological surfaces by integrating hyperbolic Ricci flow, hyperbolic harmonic map, and hyperbolic power Voronoi diagram algorithms. The resulting hyperbolic Wasserstein distance can intrinsically measure the similarity between general topological surfaces. Our proposed algorithms are theoretically rigorous and practically efficient. It has the potential to be a powerful tool for 3D shape indexing research. We tested our algorithm with human face classification and Alzheimer's disease (AD) progression tracking studies. Experimental results demonstrated that our work may provide a succinct and effective shape index.

中文翻译：

---

用于形状索引的双曲Wasserstein距离。

形状空间是计算机视觉和医学成像领域中一个活跃的研究主题。在形状空间中定义的距离可以提供简单且精确的索引来表示唯一形状。这项工作研究了Wasserstein空间，并提出了一个新颖的框架，通过集成双曲线Ricci流，双曲线调和图和双曲线功率Voronoi图算法来计算一般拓扑表面之间的Wasserstein距离。所得的双曲Wasserstein距离可以内在地度量一般拓扑表面之间的相似性。我们提出的算法在理论上严格且实用。它有可能成为3D形状索引研究的强大工具。我们使用人脸分类和阿尔茨海默氏病（AD）进展跟踪研究测试了我们的算法。