IEEE Trans Pattern Anal Mach Intell. 2020 Jun;42(6):1362-1376. doi: 10.1109/TPAMI.2019.2898400. Epub 2019 Feb 8.
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.
形状空间是计算机视觉和医学成像领域的一个活跃研究课题。形状空间中的距离可以提供一个简单而精细的指标来表示独特的形状。本工作研究了 Wasserstein 空间，并提出了一种新的框架，通过整合双曲 Ricci 流、双曲调和映射和双曲幂 Voronoi 图算法，来计算一般拓扑曲面之间的 Wasserstein 距离。所得到的双曲 Wasserstein 距离可以内在地度量一般拓扑曲面之间的相似性。我们提出的算法在理论上是严谨的，在实践中是高效的。它有可能成为 3D 形状索引研究的有力工具。我们用人脸分类和阿尔茨海默病（AD）进展跟踪研究来测试我们的算法。实验结果表明，我们的工作可能提供了一个简洁而有效的形状指标。
