We propose a new topological tool for computer vision - Scalar Function Topology Divergence (SFTD), which measures the dissimilarity of multi-scale topology between sublevel sets of two functions having a common domain. Functions can be defined on an undirected graph or Euclidean space of any dimensionality. Most of the existing methods for comparing topology are based on Wasserstein distance between persistence barcodes and they don't take into account the localization of topological features. The minimization of SFTD ensures that the corresponding topological features of scalar functions are located in the same places. The proposed tool provides useful visualizations depicting areas where functions have topological dissimilarities. We provide applications of the proposed method to 3D computer vision. In particular, experiments demonstrate that SFTD as an additional loss improves the reconstruction of cellular 3D shapes from 2D fluorescence microscopy images, and helps to identify topological errors in 3D segmentation. Additionally, we show that SFTD outperforms Betti matching loss in 2D segmentation problems.

翻译：我们提出了一种用于计算机视觉的新型拓扑工具——标量函数拓扑散度（SFTD），用于度量定义在共同域上的两个函数其子水平集之间多尺度拓扑结构的差异性。函数可定义于无向图或任意维度的欧几里得空间。现有拓扑比较方法大多基于持久条形码的Wasserstein距离，未考虑拓扑特征的局部化特性。SFTD的最小化能确保标量函数的对应拓扑特征定位于相同空间区域。该工具可生成直观的可视化结果，清晰展示函数间存在拓扑差异的区域。我们提供了该方法在三维计算机视觉中的具体应用：实验表明，将SFTD作为附加损失函数可提升基于二维荧光显微镜图像的细胞三维形状重建质量，并有效识别三维分割中的拓扑错误。此外，在二维分割问题中，SFTD的表现优于Betti匹配损失函数。