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arxiv: 2606.08258 · v1 · pith:5GC5W5Q3new · submitted 2026-06-06 · 💻 cs.GR · cs.CV· cs.LG

MS-COOT: Comparing Morse-Smale Complexes with Co-Optimal Transport

Guangyu Meng , Mingzhe Li , Erin Wolf Chambers This is my paper

Pith reviewed 2026-06-27 18:46 UTC · model grok-4.3

classification 💻 cs.GR cs.CVcs.LG
keywords Morse-Smale complexco-optimal transporthypergraphtopological distancescalar field comparisonregion matchingfeature analysisvisualization
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The pith

MS-COOT computes distances between Morse-Smale complexes by matching both critical points and their induced regions via co-optimal transport.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes representing Morse-Smale complexes as hypergraphs to preserve region-level information that graph methods lose. It then defines a co-optimal transport distance that finds joint correspondences between points and regions. This allows the method to detect events like region splitting and merging during comparison. The approach is tested on datasets from simulations and meshes, showing it captures changes missed by existing distances while supporting tasks like classifying different structures. If correct, it offers a more complete way to compare topological features in scalar fields for visualization applications.

Core claim

The central discovery is that by formulating the comparison of Morse-Smale complexes as a co-optimal transport problem on their hypergraph representations, one can jointly optimize correspondences for both critical points and regions, thereby obtaining a distance that reflects region-level structural changes such as splits and merges.

What carries the argument

The co-optimal transport formulation on a hypergraph where nodes are critical points and hyperedges are regions, instantiated with a hypernetwork for relationships, persistence-based measures, and attribute-based sample costs.

If this is right

  • It identifies region splitting and merging events during comparison.
  • It achieves strong results in classifying different scalar field structures.
  • It distinguishes data at different resolutions more effectively than graph-based methods.
  • It works on 2D simulations, 3D meshes, and volumetric data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach could be used to track topological changes in time-varying scalar fields.
  • Similar techniques might improve comparisons in other areas of topological data analysis.
  • The hypergraph transport could generalize to matching other types of topological structures.

Load-bearing premise

That the hypergraph representation of regions and the co-optimal transport with the specified components yields correspondences and distances that accurately capture the topological structure and events in the scalar field.

What would settle it

A test where two complexes differ only by a known region split or merge, but the computed distance does not increase accordingly or fails to outperform graph distances in detecting the change.

Figures

Figures reproduced from arXiv: 2606.08258 by Erin Wolf Chambers, Guangyu Meng, Mingzhe Li.

Figure 1
Figure 1. Figure 1: MS-COOT produces region-to-region correspondence between 3D Morse-Smale complexes. (a) 3D Morse-Smale complexes extracted by TTK [52] from a Viscous Finger simulation [34] at t=51 (9 regions) and t=52 (5 regions). (b) Exploded view revealing all regions individually, labeled S1–S9 and T1–T5. (c) Row-normalized region coupling ξ for t=51→52: each bar shows how a source region distributes mass across target … view at source ↗
Figure 2
Figure 2. Figure 2: MS-COOT simultaneously matches critical points and regions between two Morse-Smale complexes. (a, b) Clean and noisy sinusoidal scalar fields. (c, d) Morse-Smale complexes extracted by TTK, showing critical points (CPs: minima, saddles, maxima) and separatrices. (e, f) Region coupling ξ: each region of the noisy field in (f) inherits the color and ID of its best-matched region in the clean field (e). When … view at source ↗
Figure 3
Figure 3. Figure 3: MS-COOT workflow. Given two scalar fields f and g in (a), we compute their Morse-Smale complexes in (b) and represent each as a measure hypernetwork H. Two inputs are constructed for the MS-COOT solver: the hypernetwork function ω in (c), encoding CP-to-region proximity via shortest-path distances dG on an augmented graph with virtual region centers (VCs), and persistence-based probability measures µ, ν in… view at source ↗
Figure 4
Figure 4. Figure 4: Shortest-path example on the augmented graph G, con￾structed from the clean sinusoidal dataset ( [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Vortex Street [44]: periodic dynamics and robustness to local perturbations (157 timesteps). (a) Pairwise distance matrices for WD, GWD, FGW, and MS-COOT. Periodicity of the vortex shedding appears as off-diagonal banding (dark blocks indicating high similarity), most clearly captured by MS-COOT. Baselines exhibit noisier patterns with weaker temporal consistency. (b) Zoom into timesteps 107–110: WD, GWD, … view at source ↗
Figure 6
Figure 6. Figure 6: Ionization Front [60]: MS-COOT detects a region merge that is less visible to graph-based baselines. (a) Pairwise distance matrices for four methods; red arrows highlight the timestep 3→4 transition. Zoom panels show the highlighted neighborhood: FGW exhibits little contrast, while MS-COOT shows a clear increase in distance. (b) Terrains representing the scalar fields at timesteps 3 and 4; cyan circles ind… view at source ↗
Figure 7
Figure 7. Figure 7: Heated Cylinder [52]: phase transition around timestep 56–57. (a) All-pairs distance matrices for WD, GWD, FGW, and MS-COOT. All methods reveal a transition region around timesteps 56–57 (red arrows), but MS-COOT and GWD show a sharper boundary and more consistent block structure after the transition, while WD and FGW exhibit more diffuse patterns. (b) MS complexes overlaid on the velocity magnitude field … view at source ↗
Figure 8
Figure 8. Figure 8: Region count evolution for the Heated Cylinder dataset. Number of regions per timestep. A clear shift occurs at timestep 56→57 (dashed line), separating two regimes with lower and higher region counts, respectively. Cat Centaur David Dog Gorilla Horse Michael Victoria Wolf Min Max [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: TOSCA dataset: one representative mesh per category col￾ored by the average geodesic distance (AGD) scalar field, normalized to [0, 1]. Extremities (paws, hands, tails) have high AGD (red); torso regions have low AGD (blue). The 9 categories span quadrupeds (cat, dog, horse, wolf), humanoids (David, Michael, Victoria), and mixed forms (centaur, gorilla). tion between regimes. Overall, MS-COOT captures both… view at source ↗
Figure 10
Figure 10. Figure 10: Parameter sensitivity of MS-COOT on TOSCA. Overall micro recall (i.e., accuracy) using k-NN (k=1/3/5). (a) Sample cost C (Eq. (1)): binary type penalty (Eq. (2)), scalar cost (Eq. (8)), and their combination. (b) Balance weight α, controlling the trade-off between structure and sample cost (default α=0.5, dashed line). (c) PI bandwidth σ in the node measure µ (Eq. (6)) (default σ=0.3, dashed line). (d) Si… view at source ↗
read the original abstract

Understanding and comparing structures in scalar fields is a central challenge in scientific visualization, with applications ranging from feature analysis to temporal and structural comparison. The Morse-Smale (MS) complex provides a natural representation by decomposing a scalar field into regions induced by gradient flow. However, existing approaches typically rely on graph-based representations, capturing relationships between critical points while discarding region-level structure. In this work, we represent the MS complex as a hypergraph, where critical points form nodes and regions define hyperedges. We introduce MS-COOT, a co-optimal transport distance that jointly computes correspondences between critical points and regions. This formulation enables explicit region-to-region matching within a distance-based framework, allowing identification of region-level events such as splitting and merging. We instantiate this framework with domain-specific components, including a hypernetwork function encoding critical point-region relationships, persistence-based probability measures that emphasize topologically significant features, and a sample cost term that incorporates critical point attributes. We evaluate MS-COOT on five datasets spanning 2D simulations, 3D surface meshes, and volumetric data. Our results show that MS-COOT captures region-level structural changes that are not reflected by graph-based distances, while achieving strong performance in downstream tasks such as classification and resolution discrimination.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces MS-COOT, a co-optimal transport distance for comparing Morse-Smale complexes represented as hypergraphs (critical points as nodes, regions as hyperedges). It jointly computes correspondences between critical points and regions via a hypernetwork encoding, persistence-based probability measures, and sample costs incorporating critical-point attributes. The central claim is that this enables explicit region-to-region matching to identify topological events such as splitting and merging, which graph-based distances miss; evaluation on five datasets (2D simulations, 3D meshes, volumetric data) shows improved capture of region-level changes and strong performance on classification and resolution discrimination tasks.

Significance. If the region correspondences are shown to align with actual topological events, the work would provide a meaningful advance in structural comparison for scientific visualization by moving beyond critical-point graphs to region-level hypergraph matching. The integration of co-optimal transport with persistence weighting and hypernetwork encoding is a technically interesting synthesis. The multi-dataset evaluation spanning dimensions is a positive aspect.

major comments (1)
  1. [Abstract] Abstract and evaluation description: the claim that MS-COOT 'captures region-level structural changes that are not reflected by graph-based distances' and enables identification of splitting/merging rests on the assumption that the hypergraph co-optimal transport produces correspondences that track these events. However, only downstream classification and resolution discrimination results are reported; no direct quantitative validation (e.g., precision/recall of detected split/merge events against ground-truth region correspondences or comparison to known topological changes) is described. This is load-bearing for the central contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting the need for stronger evidence supporting the central claims regarding region-level event detection. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and evaluation description: the claim that MS-COOT 'captures region-level structural changes that are not reflected by graph-based distances' and enables identification of splitting/merging rests on the assumption that the hypergraph co-optimal transport produces correspondences that track these events. However, only downstream classification and resolution discrimination results are reported; no direct quantitative validation (e.g., precision/recall of detected split/merge events against ground-truth region correspondences or comparison to known topological changes) is described. This is load-bearing for the central contribution.

    Authors: We agree that the manuscript's evidence for explicit region-to-region matching tracking split/merge events is indirect, relying on improved downstream task performance across the five datasets and qualitative visualizations of correspondences in the results. No direct quantitative validation (such as precision/recall against ground-truth event labels on controlled data) is presented. This is a substantive point. We will add a new evaluation subsection using synthetic scalar fields with programmatically induced split and merge events, reporting quantitative alignment metrics between computed region correspondences and the known topological changes. This will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: MS-COOT is a constructed distance with independent evaluation

full rationale

The paper defines MS-COOT as a co-optimal transport distance on a hypergraph representation of the Morse-Smale complex, with explicit components (hypernetwork encoding, persistence measures, sample costs) chosen by the authors. No equation or claim reduces a 'prediction' to a fitted parameter by construction, no self-citation chain supports a uniqueness theorem, and no ansatz is smuggled in. Downstream classification and resolution results on five datasets serve as external checks rather than tautological outputs. The central claim that region-level events are captured beyond graph distances is a methodological assertion, not a definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to identify specific free parameters, axioms, or invented entities; review is limited to abstract only.

pith-pipeline@v0.9.1-grok · 5758 in / 1024 out tokens · 27352 ms · 2026-06-27T18:46:20.850094+00:00 · methodology

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