Multi-Tier Preservation of Discrete Morse Smale Complexes in Error-Bounded Lossy Compression

TL;DR

The paper addresses topology distortions caused by error-bounded lossy compression of scalar fields by preserving the discrete Morse-Smale complex (DMSC). It introduces DMTz, an iterative, edit-based workflow consisting of C-Loops and S-Loops that adjust vertex values to retain all critical cells (including saddles) and separatrices within a user-defined bound $\xi$, using a multitier preservation framework. Edits are quantized to reduce storage, with a theoretical guarantee of finite convergence and GPU-accelerated performance across multiple base compressors (SZ3, ZFP, TTHRESH). Empirical results show full MSC preservation across datasets and compressors, outperforming topology smoothing approaches in both accuracy and storage efficiency, while also highlighting limitations and directions for future work.

Abstract

We propose a novel method to preserve key topological structures (extrema, saddles, separatrices, and persistence diagrams) associated with Morse Smale complexes in error-bounded lossy compressed scalar fields. 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/3D discrete scalar fields by precisely preserving critical points (cells) and the separatrices that connect them. Our approach generates a series of (discrete) 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 cells and separatrices in alternating steps until convergence within finite iterations. Our approach addresses diverse application needs by offering users multitier 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.

Figures (13)

• Figure 1: Impacts of lossy compression (SZ3) on MSC under a relative error bound of $2 \times 10^{-3}$ with synthetic data. DMSC of (a) the original data and (b) SZ3's output. (c) and (d) Function Extension and Gradient Pairing.
• Figure 2: Illustration of the edit-based strategy: vertex $i$, originally a regular cell paired with edge $ij$, becomes an unpaired minimum after decompression. The height of each bar above a cell represents its current scalar value.
• Figure 3: An illustration of the workflow of our algorithm for preserving the full MSC. Our algorithm gets a series of edits during the compression time with two distinct loops: (1) C-loops, which iteratively fix all false critical cells, and (2) S-loops, which iteratively fix all false separatrices. The C-loops and S-loops alternate until no false critical cells or separatrices exist. The quantized edits are losslessly compressed and included with the compressed data in the compression output. The edits are applied to the decompressed data to correct MSC in the decompression stage.
• Figure 4: Fixing an FPmin/FNmin vertex $i$. (a) and (b) illustrate the process for fixing FPmin, while (c) and (d) demonstrate the process for FNmin. The height of the white cylinder above each vertex represents its lower bound ($f - \xi$), the height of the gray cylinder represents its current value, and the height of the translucent gray cylinder above the edge represents its current extended function value. The arrow from vertices to edges indicates the gradient pairing results.
• Figure 5: Fixing an FP1saddle that should be paired with a vertex: (a) and (b), or a triangle: (c) and (d), and an FN1saddle that incorrectly paired with a vertex: (e) and (f), or a triangle: (g) and (h).