Scalar Function Topology Divergence: Comparing Topology of 3D Objects
This addresses the need for more accurate topological comparisons in computer vision, particularly for 3D shape analysis and segmentation, though it is incremental as it builds on existing persistence barcode methods.
The paper tackles the problem of comparing the topology of 3D objects by proposing Scalar Function Topology Divergence (SFTD), a new tool that measures dissimilarity in multi-scale topology and accounts for feature localization, resulting in improved 3D shape reconstruction from 2D images and better identification of topological errors in segmentation.
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.