We propose a multi-tier paradigm to preserve various components of Morse-Smale complexes in lossy compressed scalar fields, including extrema, saddles, separatrices, and persistence diagrams. Existing error-bounded lossy compressors rarely consider preserving topological structures such as discrete Morse-Smale complexes, leading to significant inaccuracies in data interpretation and potentially resulting in incorrect scientific conclusions. This paper mainly focuses on preserving the Morse-Smale complexes in 2D or 3D discrete scalar fields by precisely preserving critical simplices and the separatrices that connect them. Our approach generates a series of edits during compression time, which are applied to the decompressed data to accurately reconstruct the complexes while maintaining the error within prescribed bounds. We design a workflow that iteratively fixes critical simplices and separatrices in alternating steps until convergence within finite iterations. Our approach addresses diverse application needs by offering users flexible options to balance compression efficiency and feature preservation. To enable effective integration with lossy compressors, we use GPU parallelism to enhance the performance of each workflow component. We conduct experiments on various datasets to demonstrate the effectiveness of our method in accurately preserving Morse-Smale complexes.

翻译：本文提出一种多层级范式，用于在有损压缩标量场中保留Morse-Smale复形的各类组分，包括极值点、鞍点、分界线与持续性图。现有误差有界有损压缩器鲜少考虑保留离散Morse-Smale复形等拓扑结构，导致数据解译严重失准，并可能引发错误科学结论。本文主要聚焦于通过精确保留临界单纯形及其连接分界线，在二维或三维离散标量场中保持Morse-Smale复形结构。我们的方法在压缩阶段生成一系列修正操作，将其应用于解压数据以精确重建复形，同时将误差维持在预设界内。我们设计了一种工作流程，通过交替步骤迭代修正临界单纯形与分界线，直至在有限迭代内收敛。该方法通过为用户提供平衡压缩效率与特征保留的灵活选项，满足多样化的应用需求。为实现与有损压缩器的有效集成，我们采用GPU并行技术提升各工作流组件的性能。我们在多组数据集上进行实验，验证了本方法在精确保留Morse-Smale复形方面的有效性。