arxiv: 2605.02302 · v1 · submitted 2026-05-04 · 💻 cs.GR

Recognition: unknown

Structural MAT: Clean and Scalable Medial Axis Simplification via Explicit Surface Correspondence

Pengfei Wang , Shuangmin Chen , Dongming Yan , Ying He , Shiqing Xin , Changhe Tu , Wenping Wang

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Pith reviewed 2026-05-08 02:05 UTC · model grok-4.3

classification 💻 cs.GR

keywords medial axis transformmesh simplificationsurface correspondenceVoronoi diagramstructural alignmentedge collapsetriangle meshshape descriptor

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The pith

Explicit tracking of surface correspondences during MAT simplification produces structurally aligned medial axes for triangle meshes.

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

The paper seeks to simplify the medial axis transform of a triangle mesh while preserving its ability to reflect the shape's structural features and symmetries. It starts from a 3D Voronoi diagram of surface samples and then progressively collapses edges in a controlled way. The central mechanism is to keep explicit records of which surface region each medial vertex corresponds to at every step. These records are turned into priority rules that decide the order of collapses so the final triangles respect patch symmetries and fillet centers. A sympathetic reader would care because a simplified yet faithful MAT supports accurate reconstruction and feature extraction without requiring thousands of vertices even for complex inputs.

Core claim

The central claim is that explicitly tracking the correspondence between MAT vertices and surface regions throughout the progressive simplification process ensures that the resulting MAT triangles accurately reflect the intrinsic symmetries between surface patches. These geometric requirements are translated into priority control strategies that govern the sequencing of edge collapses on an initial 3D Voronoi structure. The outcome is a scalable MAT that remains highly expressive for articulated shapes and CAD models, capturing global structure with only a few hundred vertices while aligning with rolling-ball loci in fillet regions.

What carries the argument

explicit tracking of the correspondence between MAT vertices and surface regions, used to derive priority control strategies that sequence edge collapses in a QEM-like simplification of the Voronoi-initialized MAT

Load-bearing premise

The initial 3D Voronoi diagram of surface samples combined with the proposed priority control strategies will consistently produce MATs that align with rolling-ball loci in fillet regions without introducing topological errors on arbitrary triangle meshes.

What would settle it

Finding a triangle mesh on which the output MAT either fails to align its boundary with the locus of rolling ball centers in a fillet region or introduces a topological error such as incorrect connectivity would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.02302 by Changhe Tu, Dongming Yan, Pengfei Wang, Shiqing Xin, Shuangmin Chen, Wenping Wang, Ying He.

**Figure 1.** Figure 1: Representative medial axis results computed by our method on CAD models with sharp features and organic models with smooth surfaces. The view at source ↗

**Figure 2.** Figure 2: Medial axis computation on a filleted cube model. The true medial view at source ↗

**Figure 3.** Figure 3: The geometric correspondence between the boundary surface and view at source ↗

**Figure 4.** Figure 4: Pipeline overview. We begin by sampling the input surface and computing the 3D Voronoi diagram along with the Restricted Voronoi Diagram (RVD) view at source ↗

**Figure 5.** Figure 5: Medial face filtering near concave features. (a) Concave features on view at source ↗

**Figure 6.** Figure 6: Feature classification. Convex features and concave features on the view at source ↗

**Figure 7.** Figure 7: Comparison of feature vertices before (left) and after (right) being snapped to convex feature lines. By enforcing 𝑟 = 0, we prevent the medial boundary from drifting and ensure accurate capture of sharp CAD features. Feature Snapping. Prior to simplification, we perform a preprocessing step to align feature vertices with convex boundary features. We adapt the energy functional defined in Equation 5 to a … view at source ↗

**Figure 8.** Figure 8: Medial axis comparisons on representative CAD models with sharp features (top two rows) and regions with smooth transitions (bottom two rows). view at source ↗

**Figure 9.** Figure 9: Comparison with [Wang et al. 2025a] on CAD models. [Wang et al. 2025a] produces reasonable results when models can be cleanly segmented into patches (top models), but struggles or fails entirely when faced with models that resist straightforward patch decomposition (bottom models). Our method handles both categories uniformly without requiring segmentation preprocessing. 5.1 Comparisons on CAD Models We e… view at source ↗

**Figure 10.** Figure 10: Medial axis comparisons on four organic models, showing computation time (t), sphere count (#s), and the Hausdorff distance (HD) between the view at source ↗

**Figure 12.** Figure 12: Visualization of Atlas evolution across successive simplification view at source ↗

**Figure 11.** Figure 11: Runtime scaling with increasing sample count (fixed simplification view at source ↗

**Figure 14.** Figure 14: Medial axis results simplified to low vertex counts. Despite the view at source ↗

**Figure 16.** Figure 16: Ablation study on a CAD model. (a) Full method. (b) Without view at source ↗

**Figure 15.** Figure 15: Robustness to noise. We add random displacements with different view at source ↗

**Figure 19.** Figure 19: A representative failure case on a thin sheet-like model. With 15K view at source ↗

**Figure 18.** Figure 18: Effect of the stability threshold parameter view at source ↗

**Figure 21.** Figure 21: Surface extraction from an unsigned distance field on a garment view at source ↗

read the original abstract

The Medial Axis Transform (MAT) is a complete shape descriptor capable of reconstructing the geometry of the original domain. A high-quality MAT should not only facilitate high-fidelity reconstruction but also capture structural features -- for instance, by aligning the MAT boundary with the locus of rolling ball centers within fillet regions. However, computing such an ideal MAT remains a significant challenge, particularly when the input is a discrete triangle mesh. In this paper, we follow the established technical pipeline of initializing the MAT via a 3D Voronoi diagram of surface samples and subsequently simplifying the Voronoi structure through a QEM-like scheme. Our key insight is to explicitly track the correspondence between MAT vertices and surface regions throughout the progressive simplification process, ensuring that the resulting MAT triangles accurately reflect the intrinsic symmetries between surface patches. We translate these geometric requirements into a suite of priority control strategies that govern the sequencing of edge collapses. Through extensive evaluation against state-of-the-art MAT algorithms, we validate the strong performance of our approach regarding runtime efficiency, structural alignment, boundary regularity, triangle quality, and robustness to noise. Our resulting MATs remain highly expressive for both articulated shapes and CAD models, even under extreme simplification -- effectively capturing the global structure of complex geometries with only a few hundred vertices.

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.

Desk Editor's Note private letter to a colleague

The paper adds explicit MAT-to-surface correspondence tracking to a Voronoi-plus-QEM pipeline, which is a modest but concrete step for better structural fidelity if the initial diagram is already decent.

read the letter

The core addition is tracking which surface patches each MAT vertex links to during simplification, then using that to set collapse priorities so the result respects intrinsic symmetries and rolling-ball centers in fillets. This is presented as a direct way to make the simplified triangles align better than generic decimation. If the evaluations show clear gains in alignment and boundary regularity over prior methods, especially on CAD models and articulated shapes, that part lands as useful engineering for a descriptor that often needs cleanup afterward. The runtime and robustness claims also sound practical given the focus on extreme simplification down to a few hundred vertices. The evaluations against state-of-the-art methods are the main evidence offered, and they cover the right axes for this kind of work. The soft spot is the dependence on the starting 3D Voronoi diagram from surface samples. If that diagram already links wrong regions or drops branches on non-convex or noisy meshes, the correspondence tracking and priority rules only reorder what is already there; they do not add or remove connections. The paper would need to demonstrate that the priorities reliably avoid propagating those errors rather than assuming the initialization is close enough. Without detailed failure cases or ablation on how much the tracking actually corrects versus just preserves, the structural alignment claim stays harder to judge. This is for geometry processing groups that already use MATs in reconstruction or analysis pipelines and want a drop-in simplification step with less post-processing. A reader focused on practical shape descriptors would find the priority strategies worth trying. I would send it for peer review; the pipeline is standard enough that the new tracking mechanism and results can be checked directly.

Referee Report

1 major / 4 minor

Summary. The paper introduces Structural MAT, a method for clean and scalable simplification of the medial axis transform (MAT) from triangle meshes. It builds on the standard pipeline of computing a 3D Voronoi diagram from surface samples and then simplifying it via QEM-like edge collapses. The novel aspect is the explicit tracking of correspondences between MAT vertices and surface regions during simplification, which is used to define priority control strategies. These strategies aim to ensure that the resulting simplified MAT triangles align with intrinsic surface symmetries and the loci of rolling ball centers in fillet regions. Extensive experiments demonstrate advantages in runtime, structural alignment, boundary regularity, triangle quality, and robustness to noise, with the simplified MATs remaining effective for both articulated and CAD models even at high simplification ratios.

Significance. If the central claims hold, this provides a practical method for producing structurally faithful simplified MATs from discrete meshes, which could support downstream tasks in shape analysis, skeleton extraction, and geometric modeling. The explicit use of surface correspondence to guide simplification is a clear strength over purely geometric or heuristic approaches, and the reported evaluations across runtime, alignment, and robustness add concrete evidence of utility. The approach is algorithmic rather than data-driven, with priorities derived from geometric requirements.

major comments (1)

[§3.2] §3.2 (Initial Voronoi Construction and Correspondence Tracking): The central claim that explicit MAT-to-surface-region correspondence tracking during QEM-style collapses ensures alignment with rolling-ball loci and intrinsic patch symmetries rests on the assumption that the initial 3D Voronoi diagram of surface samples is already topologically close to the true medial axis. The description does not specify a mechanism (e.g., edge insertion, deletion, or reconnection) by which erroneous Voronoi connections between non-adjacent surface regions are corrected rather than merely reordered; if only reordering occurs, topological errors can propagate and undermine the alignment guarantee on non-convex or noisy inputs.

minor comments (4)

[Abstract] Abstract: The phrase 'extensive evaluation' would benefit from a brief parenthetical note on the number of models and categories tested to set expectations for the results section.
[§5.1] §5.1 (Evaluation Metrics): The structural alignment metric is presented without an equation or pseudocode definition; adding this would make the quantitative claims easier to reproduce and compare.
[Figure 4] Figure 4 caption: The color mapping for MAT triangles versus surface patches is not explained in the caption or legend, reducing clarity for readers examining the visual results.
[References] References: Several foundational works on medial axis extraction from meshes (pre-2020) are cited, but the list would be strengthened by including more recent Voronoi-based simplification papers from 2021-2023.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their constructive feedback and for recognizing the strengths of our approach. We provide a detailed response to the major comment below.

read point-by-point responses

Referee: [§3.2] §3.2 (Initial Voronoi Construction and Correspondence Tracking): The central claim that explicit MAT-to-surface-region correspondence tracking during QEM-style collapses ensures alignment with rolling-ball loci and intrinsic patch symmetries rests on the assumption that the initial 3D Voronoi diagram of surface samples is already topologically close to the true medial axis. The description does not specify a mechanism (e.g., edge insertion, deletion, or reconnection) by which erroneous Voronoi connections between non-adjacent surface regions are corrected rather than merely reordered; if only reordering occurs, topological errors can propagate and undermine the alignment guarantee on non-convex or noisy inputs.

Authors: We appreciate the referee pointing out this subtlety in our description. Our approach follows the standard pipeline of computing an initial 3D Voronoi diagram from surface samples, which we assume to be sufficiently dense to approximate the medial axis topology reasonably well. The explicit surface correspondence tracking is introduced to define priority functions for the QEM-style edge collapses, aiming to preserve alignments with surface symmetries and rolling ball loci during simplification. However, as the referee correctly notes, we do not provide mechanisms such as edge insertion, deletion, or reconnection to correct erroneous Voronoi connections between non-adjacent regions. The process primarily reorders the collapses based on our priority strategies. We acknowledge that on inputs with significant topological errors in the initial Voronoi (e.g., highly non-convex or extremely noisy meshes), these errors may persist or affect the final result. To address this, we will revise §3.2 to explicitly articulate the reliance on the quality of the initial Voronoi diagram and include a brief discussion of this limitation, along with suggestions for future work on topology correction. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic pipeline with independent geometric priorities

full rationale

The paper describes an algorithmic procedure: initialize MAT from 3D Voronoi diagram of surface samples, then simplify via QEM-like edge collapses while tracking explicit MAT-to-surface-region correspondences to derive priority strategies. No equations, fitted parameters, or self-referential definitions appear in the provided text; the priorities are stated as translations of geometric requirements (symmetries, rolling-ball loci) rather than outputs of the method itself. The initial Voronoi step is an external input, not derived from the simplification result. No self-citation chains or uniqueness theorems are invoked as load-bearing. The derivation chain remains self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. The method implicitly assumes standard Voronoi properties and mesh quality but does not introduce new postulated entities.

pith-pipeline@v0.9.0 · 5543 in / 1060 out tokens · 41898 ms · 2026-05-08T02:05:22.338636+00:00 · methodology

discussion (0)

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