A Practical Solver for Scalar Data Topological Simplification

TL;DR

This work tackles the practical problem of topological simplification for 3D scalar fields by focusing on saddle-pair cancellation while preserving user-defined signal features. It introduces a practical solver built on persistence-optimization principles, augmented with two tailored accelerations: fast persistence updates via localized discrete Morse theory and fast assignment updates that reuse still-pairs across iterations. The method achieves substantial runtime reductions (average ~×64 speedup) and enables direct visualization of simplified data, improved filament extraction, and genus-defect repair in surfaces. By yielding a local minimum of the simplification energy with comparable output quality to baselines, it makes topological simplification feasible for real-life datasets and provides a reusable C++ implementation for reproducibility.

Abstract

This paper presents a practical approach for the optimization of topological simplification, a central pre-processing step for the analysis and visualization of scalar data. Given an input scalar field f and a set of "signal" persistence pairs to maintain, our approach produces an output field g that is close to f and which optimizes (i) the cancellation of "non-signal" pairs, while (ii) preserving the "signal" pairs. In contrast to pre-existing simplification algorithms, our approach is not restricted to persistence pairs involving extrema and can thus address a larger class of topological features, in particular saddle pairs in three-dimensional scalar data. Our approach leverages recent generic persistence optimization frameworks and extends them with tailored accelerations specific to the problem of topological simplification. Extensive experiments report substantial accelerations over these frameworks, thereby making topological simplification optimization practical for real-life datasets. Our approach enables a direct visualization and analysis of the topologically simplified data, e.g., via isosurfaces of simplified topology (fewer components and handles). We apply our approach to the extraction of prominent filament structures in three-dimensional data. Specifically, we show that our pre-simplification of the data leads to practical improvements over standard topological techniques for removing filament loops. We also show how our approach can be used to repair genus defects in surface processing. Finally, we provide a C++ implementation for reproducibility purposes.

Figures (9)

• Figure 1: Persistent diagrams for the lexicographic filtration of a clean (left) and a noisy (right) terrain example. Minimum-saddle persistence pairs are show with cyan bars in the birth-death space, while saddle-maximum pairs are shown with purple bars. Generators with infinite persistence are marked with an upward arrow. The persistence of each topological feature is given by the height of its bar. Critical simplices are shown in the data with spheres, with a radius proportional to their persistence.
• Figure 2: The Wasserstein distance $\pazocal{W}_{2}$ between $\pazocal{D}(f)$ (top) and $\pazocal{D}(g)$ (bottom) is computed by assignment optimization (\ref{['eq_wasserstein']}) in the 2D birth-death space (right). The optimal assignment $\phi^*$ (arrows) encodes a minimum cost transformation of $\pazocal{D}(f)$ into $\pazocal{D}(g)$, which displaces persistence pairs in the birth-death space or cancel them by sending them to the diagonal.
• Figure 3: Optimizing the simplification of an input scalar field $f = f_0$ into a field $g = f_{final}$ for the removal of a user selected saddle-maximum pair (cyan). At each iteration ($j < j' < j"$), given the point $p_i \in \pazocal{D}(f_j)$ to cancel, the data values of its birth and death vertices $v_{i_b}$ and $v_{i_d}$ (cyan spheres in the data) are modified to project $p_i$ to the diagonal. In this example, this results in a scalar field $g$ which is close to $f$, with the prescribed topology.
• Figure 4: Updated vertices (dark purple vertices, center insets) along the topological simplification optimization of a noisy terrain (non-signal pairs, to simplify, are shown in cyan). In this example, only 20% of the vertices are updated per iteration on average. The discrete gradient (at the core of a recent, fast persistence computation algorithm guillou_tvcg23) only needs to be recomputed for these, yielding a x2 speedup for persistence computation.
• Figure 5: Interactions between non-signal and signal pairs during the optimization. A multi-saddle vertex can be involved in both a non-signal pair $p$ (cyan bar in $\pazocal{D}(f)$) and a signal pair $p'$ (vertically aligned purple bar in $\pazocal{D}(f)$). At iteration $j$, the update of the non-signal pair $p$ unfolds the multi-saddle into multiple simple saddles of distinct values, effectively perturbing the birth of the signal pair $p'$ and making it non-still. In real-life data, especially in 3D, such configurations occur often, and cascade. In our experiments (\ref{['sec_results']}), at each iteration, $11\%$ of the signal pairs are perturbed this way by non-signal pairs (on average, and up to $32\%$). This is addressed by our loss (\ref{['sec_approach']}) which enforces signal pair preservation.