pith. sign in

arxiv: 2605.25921 · v1 · pith:77DCM4ZSnew · submitted 2026-05-25 · 💻 cs.GR · cs.CV

Curve Skeletonization in Continuous domain for Meshes and Point Clouds

Jai Bardhan , Ramya Hebbalaguppe , Aravind Udupa This is my paper

Pith reviewed 2026-06-29 19:21 UTC · model grok-4.3

classification 💻 cs.GR cs.CV
keywords curve skeletonizationcontinuous domainintrinsic triangulationpoint cloudsmesheslocal separatorstufted Laplaciansshape analysis
0
0 comments X

The pith

CSCD generalizes discrete local separators to continuous manifolds for curve skeletonization on meshes and point clouds.

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

The paper introduces CSCD to fix representation inaccuracies that arise when skeletonization methods operate only on discrete graphs. It does so by lifting the local separators approach to continuous manifolds, with one version for meshes that uses intrinsic triangulation and another for point clouds that uses tufted Laplacians. CSCD-M is presented as the first intrinsic curve-skeletonization method and is shown to match LS performance on diverse meshes while outperforming it on the Thingi10k dataset; CSCD-PC is reported to give qualitatively better results than CoverageAxis++ and EPCS. The work also shows the extracted skeletons support downstream tasks including object classification, shape segmentation, and detection of handles, tunnels, and constrictions.

Core claim

CSCD is a framework that generalizes Skeletonization via Local Separators to the continuous domain on manifolds. CSCD-M achieves this for meshes by leveraging intrinsic triangulation to gain resilience to noise and improved topological preservation. CSCD-PC applies the same idea to point clouds through tufted Laplacians. The resulting method matches LS across varied meshes, outperforms LS on Thingi10k, and produces qualitatively superior skeletons for point clouds compared with CoverageAxis++ and EPCS.

What carries the argument

The CSCD framework that generalizes the discrete local separators approach to continuous manifolds via intrinsic triangulation for meshes and tufted Laplacians for point clouds.

If this is right

  • CSCD-M matches or exceeds LS performance on standard mesh benchmarks and on Thingi10k.
  • CSCD-PC produces qualitatively better curve skeletons from point clouds than CoverageAxis++ or EPCS.
  • The extracted skeletons improve accuracy in downstream tasks such as object classification, shape segmentation, and identification of handles, tunnels, and constrictions.
  • The continuous formulation reduces discretization errors while retaining the computational advantages of the original local-separators method.

Where Pith is reading between the lines

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

  • The same lifting strategy could be applied to other discrete graph algorithms in geometry processing to reduce discretization artifacts.
  • Direct operation on manifold data may enable skeletonization pipelines that avoid an intermediate meshing step for raw scans.
  • Improved topological fidelity on noisy inputs could benefit applications that rely on consistent medial-axis or curve representations across varying sampling densities.

Load-bearing premise

Intrinsic triangulation and tufted Laplacians successfully extend the local separators property to continuous representations while preserving its efficiency and topological guarantees.

What would settle it

A mesh or point cloud where CSCD-M or CSCD-PC produces a skeleton with incorrect topology or connectivity that LS or the compared methods do not produce would falsify the claim of improved preservation without new inaccuracies.

Figures

Figures reproduced from arXiv: 2605.25921 by Aravind Udupa, Jai Bardhan, Ramya Hebbalaguppe.

Figure 1
Figure 1. Figure 1: Representative results of the proposed CSCD on diverse 3D shapes from various benchmark datasets - Meshes (left) and Point Clouds (right): (a) We show the result of our method on the neptune mesh – The inset illustrates the excellent skeletal quality for both the hand and the trident; (b) We show a comparison of CSCD to a contemporary method (LS [2]) for Copper-key – CSCD reconstructs the holes of the shap… view at source ↗
Figure 2
Figure 2. Figure 2: CSCD Framework Overview: Our framework gen￾eralizes LS [2], such that a graph-based realization results in an algorithm similar to LS (Appendix K.1), mesh-based realization leads to CSCD-M, and a point cloud realization leads to CSCD-PC (Sec. 3). Method Graph Mesh Point Cloud LS [2] ✓ ✗ ✗ MSLS [3] ✓ ✗ ✗ ROSA [45] ✗ ✗ ✓ MCF [46] ✗ ✓ ✗ EPCS [23] ✗ ✗ ✓ CA++ [12, 53] ✗ ✓ ✓ CSCD (Ours) ✓ (App. K.1) ✓CSCD-M ✓CSC… view at source ↗
Figure 3
Figure 3. Figure 3: [Schematic of CSCD for Curve Skeletonization]. The entire Curve Skeletonization framework can be divided into two stages. In Stage 1, we calculate the various local separators given the 3D object as input(input can be a mesh or a point cloud). Once we have a sampled point, we calculate the geodesic distance to all other points and find the cut loci of the point to identify the target cut locus. Then an app… view at source ↗
Figure 4
Figure 4. Figure 4: [Illustration of a local separator.] The green ver￾tex is the source s, the purple vertex is the target cut locus t (Sec. 3.1.2). The two paths ˆl (red and lime) are the approximate paths (Sec. 3.1.4), and the cyan curve l is the final local separator (Sec. 3.1.5). See Sec. M for Terminology/definitions. where, f(·) denotes functions specific to each step. 2.2. Stage 1: Local Separator Construction An idea… view at source ↗
Figure 5
Figure 5. Figure 5: Constraint region for the loop optimization procedure. The green sphere shows the Eu￾clidean sphere constraint. Simply shortening the curve yields a local geodesic loop that can drift significantly from the initial path. For ex￾ample, a loop drawn at the bottom of a cone may slide upward to￾ward the tip, which is undesirable for curve skeletonization. Two observations guide our constraint: (1) the target c… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Comparison of our (right) estimated cut loci ( [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cliques in the armadillo hand and their removal. (a) Original hand; (b) With cliques; (c) Cliques replaced by central star-like nodes. 3.2.5. Constructing the Graph The centroids of the regions form the nodes of the curve skeleton graph. Nodes corresponding to neighboring re￾gions are connected. When a region neighbors more than two others, resulting cliques are simplified by converting them to a star form… view at source ↗
Figure 8
Figure 8. Figure 8: Qualitative results of our method: ROSA [45] struggles to capture mesh details, while MCF [46] produces overly smooth skeletons. CSCD-M, LS [2] and MSLS [3] yield comparable results on meshes; however, our method correctly captures details, as seen on the copper key. In fertility, our approach results in smoother skeletons compared to LS and MSLS. ThingiID: 37358 ThingiID: 90736 ThingiID : 44395 ThingiID: … view at source ↗
Figure 9
Figure 9. Figure 9: Qualitative results of CSCD-M on Thingi10k. CSCD￾M performs well even on poorly triangulated meshes. Performance on poorly triangulated meshes: Our intrinsic triangulation scheme, based on the Integer coordinate sys￾tem [16], uses intrinsic edge flips to enforce Delaunay con￾ditions. As shown in [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: CSCD-M and MSLS on a torus with holes. Left panel, CSCD-M local separators on the torus – highlighting how the separators go around the holes. Right panel, comparison of the curve skeleton obtained by MSLS and Ours (CSCD-M). Performance on meshes with holes: We evaluate on a torus mesh with three holes—two partial and one fully discon￾necting the shape—as a controlled test case. Our method is the only one… view at source ↗
Figure 11
Figure 11. Figure 11: CSCD-PC vs ROSA [45] vs CA++ [Eurographics’24][53]: CA++ fails to generate a valid skeleton (since it’s a MAT inspired algorithm). Our method captures object details better, yielding more nodes and centered skeletons compared to ROSA. EPCS Ours [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: CSCD-PC vs EPCS [CAG’23]: Our method captures object details better, compared to EPCS [23]. In hand (left), our skeleton is centered in the palm and shows skeleton consistent with square shape. In deer (middle), we capture the snout and the tail. In dino (right), EPCS fails to capture the curvature of the arm completely while our skeleton follows the arm. 6. Acknowledgments We would like to thank Rahul Na… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of our estimated cut loci (right) and the original code (left) [ [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Our adaptation for cut locus identification on point clouds on various point clouds [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: CSCD-M vs LS on complex shapes: Our method results in skeletons that are comparable to LS. Specifically on the xyzrgb-dragon object, our skeleton appears to be smoother and contains fewer noisy branches in the the body of the dragon. to use only a single thread of the CPU. Why is this in the appendix? There are a few reasons as to why we decided to put this analysis within the appendix: 1. LS, MSLS and ou… view at source ↗
Figure 16
Figure 16. Figure 16: Smoothened xyzrgb-dragon for CSCD-M. We see that the spurious branches have reduced after smoothing the bod MSLS Ours [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: CSCD-M and MSLS on a torus with holes. Left panel, CSCD-M local separators on the torus – highlighting how the separators go around the holes. Right panel, comparison of the curve skeleton obtained by MSLS and Ours (CSCD-M). armadillo fertility [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Results of CSCD-M on noisy meshes armadillo and fertility: Here we show the curve skeleton generated at progres￾sively noisier vertex positions. N [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Results on a low mesh resolution: Both LS and MSLS lead to distorted skeletons at reduced mesh resolution while our method still obtains well-centered skeletons. Here the mesh resolution was reduced by 2× the original mesh through mesh decimation [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: CSCD-PC vs ROSA: Comparative results of our method vs ROSA. We are able to capture the details of the object while ROSA [45] struggles to do so. Specifically, we see that ROSA results in fewer nodes and coarser skeletons compared to CSCD-PC [PITH_FULL_IMAGE:figures/full_fig_p024_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: [Downstream Application: Shape Segmentation]: Our method works as a strong skeletonization technique for the shape diameter function (SDF) based unsupervised segmentation. The obtained segmentation is robust to pose variations and works very well. 13 [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: [Downstream Application: Handle/Tunnel/Constricting Loop Detection]: With minor changes, our local separator identifi￾cation algorithm works effectively to identify handles/tunnels and constricting loops on the shape. These loops form useful basis for other tasks such as surface cutting, etc. 14 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Geodesic Constraint vs Euclid Constraint for calculating the local separators: We see that for many objects the resultant skeleton looks pretty acceptable. However, for certain meshes like gorilla, the geodesic constraint would lead to poor results. 15 [PITH_FULL_IMAGE:figures/full_fig_p026_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Number of local separators vs reconstruction Error for armadillo: Increasing the number of local separators reduces the error drastically, but the decrease in error varies based on the complexity of the object. The reconstruction errors are evaluated for 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 and 8192 number of local separators. 16 [PITH_FULL_IMAGE:figures/full_fig_p027_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Number of local separators vs reconstruction Error for fertility: Increasing the number of local separators reduces the error drastically, but the decrease in error varies based on the complexity of the object. The reconstruction errors are evaluated for 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 and 8192 number of local separators. 17 [PITH_FULL_IMAGE:figures/full_fig_p028_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Number of local separators vs reconstruction Error for TID:133568: Increasing the number of local separators reduces the error drastically, but the decrease in error varies based on the complexity of the object. The reconstruction errors are evaluated for 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096 and 8192 number of local separators. 18 [PITH_FULL_IMAGE:figures/full_fig_p029_26.png] view at source ↗
read the original abstract

Advancements in 3D curve skeletonization are accelerating progress across a wide range of applications. However, developing robust skeletonization algorithms that capture intricate object details remains challenging. Skeletonization via Local Separators (LS) offers an efficient graph-based approach but suffers from representation inaccuracies due to its discrete nature. To address this, we introduce CSCD, a novel framework for Curve Skeletonization in the Continuous Domain, generalizing LS to manifolds. Specifically, we present two realizations: CSCD-M for meshes and CSCD-PC for point clouds. CSCD-M leverages the intrinsic triangulation of a mesh for resilience to noise and improved topological preservation, while CSCD-PC employs tufted Laplacians for enhanced robustness. To our knowledge, CSCD-M is the first intrinsic method for curve skeletonization. Our results show CSCD-M matches LS performance across diverse meshes and outperforms LS (TOG'21) on benchmarks like Thingi10k dataset. CSCD-PC qualitatively outperforms CoverageAxis++ (Eurographics'24) and EPCS (CAG'23). Finally, we demonstrate the efficacy of CSCD in a few downstream tasks: object classification, shape segmentation, identifying handles, tunnels, and constrictions in objects. Project Website: https://cscd-skel.pages.dev

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

0 major / 2 minor

Summary. The paper introduces CSCD, a framework for curve skeletonization in the continuous domain that generalizes the discrete Local Separators (LS) approach. CSCD-M realizes this for meshes via intrinsic triangulation, claimed to be the first intrinsic method, while CSCD-PC uses tufted Laplacians for point clouds. The authors state that CSCD-M matches LS performance across diverse meshes and outperforms LS on Thingi10k, CSCD-PC qualitatively outperforms CoverageAxis++ and EPCS, and both support downstream tasks including classification, segmentation, and detection of handles/tunnels/constrictions.

Significance. If the central generalization holds, the work supplies a continuous-domain extension of LS that preserves topological guarantees and efficiency while adding noise resilience through intrinsic triangulation and tufted Laplacians. The derivations appear parameter-free and internally consistent, with experiments supporting the efficiency and topological claims. This could strengthen skeletonization tools for noisy or manifold-based 3D data in graphics applications.

minor comments (2)
  1. Abstract: the outperformance claim on Thingi10k would be clearer if a parenthetical reference to the specific quantitative metric (e.g., Hausdorff distance or coverage score) were added, even though the full results section presumably contains the supporting data.
  2. §3 (method): the transition from discrete local separators to the continuous intrinsic operator could include one additional sentence contrasting the two formulations for readers unfamiliar with intrinsic triangulations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and the recommendation of minor revision. We are pleased that the significance of the continuous-domain generalization of Local Separators, along with the topological and efficiency claims, is recognized.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper generalizes the prior discrete Local Separators (LS) construction to continuous domain via intrinsic triangulation (CSCD-M) and tufted Laplacians (CSCD-PC). No step reduces a claimed prediction or first-principles result to its own inputs by definition, fitted parameter, or self-citation chain. The topological guarantees and performance claims rest on the new continuous realizations and experiments rather than renaming or smuggling prior ansatzes. The derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all claims rest on the generalization from discrete LS without further specification.

pith-pipeline@v0.9.1-grok · 5765 in / 1062 out tokens · 46154 ms · 2026-06-29T19:21:23.648074+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 5 canonical work pages · 3 internal anchors

  1. [1]

    Skeleton extraction by mesh contraction.ACM Trans

    Oscar Kin-Chung Au, Chiew-Lan Tai, Hung-Kuo Chu, Daniel Cohen-Or, and Tong-Yee Lee. Skeleton extraction by mesh contraction.ACM Trans. Graph., 27(3):1–10, 2008. 2, 20

    2008

  2. [2]

    Skeletonization via local separators.ACM Trans

    Andreas Bærentzen and Eva Rotenberg. Skeletonization via local separators.ACM Trans. Graph., 40(5), 2021. 1, 2, 5, 6, 7, 8, 10, 20

    2021

  3. [3]

    Andreas Bærentzen, Rasmus Emil Christensen, Emil Toftegaard Gæde, and Eva Rotenberg

    J. Andreas Bærentzen, Rasmus Emil Christensen, Emil Toftegaard Gæde, and Eva Rotenberg. Multilevel skele- tonization using local separators. InProceedings of the 39th International Symposium on Computational Geometry (SoCG 2023), 2023. 2, 6, 7, 8, 5, 10, 20

    2023

  4. [4]

    A robust algorithm for voxel-to-polygon mesh phantom con- version.Brain and Human Body Modeling: Computational Human Modeling at EMBC 2018, pages 317–327, 2019

    Justin L Brown, Takuya Furuta, and Wesley E Bolch. A robust algorithm for voxel-to-polygon mesh phantom con- version.Brain and Human Body Modeling: Computational Human Modeling at EMBC 2018, pages 317–327, 2019. 20

    2018

  5. [5]

    Skeletonization via dual of shape segmentation.Computer Aided Geometric Design, 80:101856, 2020

    Jingliang Cheng, Xinyu Zheng, Shuangmin Chen, Guozhu Liu, Shiqing Xin, Lin Lu, Yuanfeng Zhou, and Changhe Tu. Skeletonization via dual of shape segmentation.Computer Aided Geometric Design, 80:101856, 2020. 2, 20

    2020

  6. [6]

    Y . Chu, W. Wang, and L. Li. Robustly extracting concise 3d curve skeletons by enhancing the capture of prominent features.IEEE Trans Vis Comput Graph, 29(8):3472–3488,

  7. [7]

    Epub 2023 Jun 29, PMID: 35324442. 2

    2023

  8. [8]

    A simple discrete calculus for digital surfaces

    David Coeurjolly and Jacques-Olivier Lachaud. A simple discrete calculus for digital surfaces. InDiscrete Geome- try and Mathematical Morphology, pages 341–353, Cham,

  9. [9]

    Springer International Publishing. 12

  10. [10]

    Curve- skeleton applications

    Nicu D Cornea, Deborah Silver, and Patrick Min. Curve- skeleton applications. InVIS 05. IEEE Visualization, 2005., pages 95–102. IEEE, 2005. 2, 20

    2005

  11. [11]

    The heat method for distance computation.Commun

    Keenan Crane, Clarisse Weischedel, and Max Wardetzky. The heat method for distance computation.Commun. ACM, 60(11):90–99, 2017. 4, 2, 19

    2017

  12. [12]

    A survey of algorithms for geodesic paths and distances

    Keenan Crane, Marco Livesu, Enrico Puppo, and Yipeng Qin. A survey of algorithms for geodesic paths and distances. ArXiv, abs/2007.10430, 2020. 19

    work page arXiv 2007

  13. [13]

    Coverage axis: In- ner point selection for 3d shape skeletonization.Computer Graphics Forum, 41(2):419–432, 2022

    Zhiyang Dou, Cheng Lin, Rui Xu, Lei Yang, Shiqing Xin, Taku Komura, and Wenping Wang. Coverage axis: In- ner point selection for 3d shape skeletonization.Computer Graphics Forum, 41(2):419–432, 2022. 2, 20

    2022

  14. [14]

    Coverage axis: In- ner point selection for 3d shape skeletonization

    Zhiyang Dou, Cheng Lin, Rui Xu, Lei Yang, Shiqing Xin, Taku Komura, and Wenping Wang. Coverage axis: In- ner point selection for 3d shape skeletonization. InCom- puter Graphics Forum, pages 419–432. Wiley Online Li- brary, 2022. 2, 20

    2022

  15. [15]

    Towards skeleton based recon- struction: From projective skeletonization to canal surface estimation

    Bastein Durix, G ´eraldine Morin, Sylvie Chambon, C ´eline Roudet, and Lionel Garnier. Towards skeleton based recon- struction: From projective skeletonization to canal surface estimation. In2015 International Conference on 3D Vision, pages 545–553, 2015. 2

    2015

  16. [16]

    Shachar Fleishman, Daniel Cohen-Or, and Cl ´audio T. Silva. Robust moving least-squares fitting with sharp features. ACM SIGGRAPH 2005 Papers, 2005. 4

    2005

  17. [17]

    NeRF: Neural Radiance Field in 3D Vision: A Comprehensive Review (Updated Post-Gaussian Splatting)

    Kyle Gao, Yina Gao, Hongjie He, Dening Lu, Linlin Xu, and Jonathan Li. Nerf: Neural radiance field in 3d vision, a comprehensive review.arXiv preprint arXiv:2210.00379,

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    Integer coordinates for intrinsic geometry processing.arXiv preprint arXiv:2106.00220, 2021

    Mark Gillespie, Nicholas Sharp, and Keenan Crane. Integer coordinates for intrinsic geometry processing.arXiv preprint arXiv:2106.00220, 2021. 7

    work page arXiv 2021

  19. [19]

    Strategies for shape matching using skele- tons.Computer Vision and Image Understanding, 110(3): 326–345, 2008

    Wooi-Boon Goh. Strategies for shape matching using skele- tons.Computer Vision and Image Understanding, 110(3): 326–345, 2008. Similarity Matching in Computer Vision and Multimedia. 2

    2008

  20. [20]

    L1-medial skeleton of point cloud.ACM Trans

    Hui Huang, Shihao Wu, Daniel Cohen-Or, Minglun Gong, Hao Zhang, Guiqing Li, and Baoquan Chen. L1-medial skeleton of point cloud.ACM Trans. Graph., 32(4), 2013. 2, 20

    2013

  21. [21]

    libigl: A simple C++ geometry processing library, 2018

    Alec Jacobson, Daniele Panozzo, et al. libigl: A simple C++ geometry processing library, 2018. https://libigl.github.io/. 2

    2018

  22. [22]

    Jones, J.A

    M.W. Jones, J.A. Baerentzen, and M. Sramek. 3d dis- tance fields: a survey of techniques and applications.IEEE Transactions on Visualization and Computer Graphics, 12 (4):581–599, 2006. 1

    2006

  23. [23]

    Computing geodesic paths on manifolds.Proceedings of the national academy of Sciences, 95(15):8431–8435, 1998

    Ron Kimmel and James A Sethian. Computing geodesic paths on manifolds.Proceedings of the national academy of Sciences, 95(15):8431–8435, 1998. 2

    1998

  24. [24]

    New Greedy Heuristics For Set Cover and Set Packing

    David Kordalewski. New greedy heuristics for set cover and set packing.ArXiv, abs/1305.3584, 2013. 4

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  25. [25]

    Epcs: Endpoint-based part-aware curve skeleton extraction for low-quality point clouds.Com- puters & Graphics, 117:209–221, 2023

    Chunhui Li, Mingquan Zhou, Guohua Geng, Yifei Xie, Yuhe Zhang, and Yangyang Liu. Epcs: Endpoint-based part-aware curve skeleton extraction for low-quality point clouds.Com- puters & Graphics, 117:209–221, 2023. 2, 8, 20

    2023

  26. [26]

    Q-mat: Computing medial axis trans- form by quadratic error minimization.ACM Transactions on Graphics (TOG), 35(1):1–16, 2015

    Pan Li, Bin Wang, Feng Sun, Xiaohu Guo, Caiming Zhang, and Wenping Wang. Q-mat: Computing medial axis trans- form by quadratic error minimization.ACM Transactions on Graphics (TOG), 35(1):1–16, 2015. 2

    2015

  27. [27]

    Point2skeleton: Learning skeletal representations from point clouds.2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 4275–4284, 2020

    Chu-Hsing Lin, Changjian Li, Yuan Liu, Nenglun Chen, Yi- King Choi, and Wenping Wang. Point2skeleton: Learning skeletal representations from point clouds.2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 4275–4284, 2020. 2, 20

    2021

  28. [28]

    An optimization-driven approach for computing geodesic paths on triangle meshes.Computer- Aided Design, 90:105–112, 2017

    Bangquan Liu, Shuangmin Chen, Shi-Qing Xin, Ying He, Zhen Liu, and Jieyu Zhao. An optimization-driven approach for computing geodesic paths on triangle meshes.Computer- Aided Design, 90:105–112, 2017. SI:SPM2017. 20

    2017

  29. [29]

    Re- constructing the curve-skeletons of 3d shapes using the vi- sual hull.IEEE Transactions on Visualization and Computer Graphics, 18:1891–1901, 2012

    Marco Livesu, Fabio Guggeri, and Riccardo Scateni. Re- constructing the curve-skeletons of 3d shapes using the vi- sual hull.IEEE Transactions on Visualization and Computer Graphics, 18:1891–1901, 2012. 2, 20

    1901

  30. [30]

    V oxel structure- based mesh reconstruction from a 3d point cloud.IEEE Transactions on Multimedia, 24:1815–1829, 2022

    Chenlei Lv, Weisi Lin, and Baoquan Zhao. V oxel structure- based mesh reconstruction from a 3d point cloud.IEEE Transactions on Multimedia, 24:1815–1829, 2022. 20

    2022

  31. [31]

    Transfer4d: A framework for frugal motion capture and deformation transfer

    Shubh Maheshwari, Rahul Narain, and Ramya Hebbal- aguppe. Transfer4d: A framework for frugal motion capture and deformation transfer. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 12836–12846, 2023. 2

    2023

  32. [32]

    Mancinelli, M

    C. Mancinelli, M. Livesu, and E. Puppo. Practical computa- tion of the cut locus on discrete surfaces.Computer Graphics Forum, 40(5):261–273, 2021. 4, 6, 2, 5 9

    2021

  33. [33]

    Carvalho

    Dimas Mart ´ınez, Luiz Velho, and Paulo C. Carvalho. Com- puting geodesics on triangular meshes.Computers and Graphics, 29(5):667–675, 2005. 20

    2005

  34. [34]

    Srinivasan, Matthew Tancik, Jonathan T

    Ben Mildenhall, Pratul P. Srinivasan, Matthew Tancik, Jonathan T. Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view syn- thesis.Commun. ACM, 65:99–106, 2020. 1

    2020

  35. [35]

    Curve-skeleton properties, applications and algorithms.IEEE Transactions on Visual- ization and Computer Graphics, 13(3):530–548, 2007

    D Nicu, C Silver, and D Silver. Curve-skeleton properties, applications and algorithms.IEEE Transactions on Visual- ization and Computer Graphics, 13(3):530–548, 2007. 2, 20

    2007

  36. [36]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V . Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V . Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in Python.Journal of Machine Learning Research, 12:2825–2830, 2011. 2

    2011

  37. [37]

    Dennie Reniers, Jarke van Wijk, and Alexandru Telea. Com- puting multiscale curve and surface skeletons of genus 0 shapes using a global importance measure.IEEE Transac- tions on Visualization and Computer Graphics, 14(2):355– 368, 2008. 2, 20

    2008

  38. [38]

    Saha, Gunilla Borgefors, and Gabriella Sanniti di Baja

    Punam K. Saha, Gunilla Borgefors, and Gabriella Sanniti di Baja. A survey on skeletonization algorithms and their appli- cations.Pattern Recognition Letters, 76:3–12, 2016. Special Issue on Skeletonization and its Application. 2, 20

    2016

  39. [39]

    Consis- tent mesh partitioning and skeletonisation using the shape diameter function.The Visual Computer, 24:249–259, 2008

    Lior Shapira, Ariel Shamir, and Daniel Cohen-Or. Consis- tent mesh partitioning and skeletonisation using the shape diameter function.The Visual Computer, 24:249–259, 2008. 8

    2008

  40. [40]

    A laplacian for nonman- ifold triangle meshes.Computer Graphics Forum, 39(5):69– 80, 2020

    Nicholas Sharp and Keenan Crane. A laplacian for nonman- ifold triangle meshes.Computer Graphics Forum, 39(5):69– 80, 2020. 12

    2020

  41. [41]

    You can find geodesic paths in triangle meshes by just flipping edges.ACM Trans

    Nicholas Sharp and Keenan Crane. You can find geodesic paths in triangle meshes by just flipping edges.ACM Trans. Graph., 39(6), 2020. 5, 2, 4, 20

    2020

  42. [42]

    Geometrycentral: A modern c++ library of data structures and algorithms for ge- ometry processing.github, 2019

    Nicholas Sharp, Keenan Crane, et al. Geometrycentral: A modern c++ library of data structures and algorithms for ge- ometry processing.github, 2019. 2

    2019

  43. [43]

    The princeton shape benchmark

    Philip Shilane, Patrick Min, Michael Kazhdan, and Thomas Funkhouser. The princeton shape benchmark. InProceed- ings Shape Modeling Applications, 2004., pages 167–178. IEEE, 2004. 8

    2004

  44. [44]

    PhD thesis, CO- MUE Universit´e Cˆote d’Azur (2015-2019), 2019

    Alvaro Javier Fuentes Suarez.Modeling shapes with skele- tons: scaffolds & anisotropic convolution. PhD thesis, CO- MUE Universit´e Cˆote d’Azur (2015-2019), 2019. 7, 3

    2015

  45. [45]

    Sundar, D

    H. Sundar, D. Silver, N. Gagvani, and S. Dickinson. Skeleton based shape matching and retrieval. In2003 Shape Modeling International., pages 130–139, 2003. 2

    2003

  46. [46]

    Fast exact and approx- imate geodesics on meshes.ACM transactions on graphics (TOG), 24(3):553–560, 2005

    Vitaly Surazhsky, Tatiana Surazhsky, Danil Kirsanov, Steven J Gortler, and Hugues Hoppe. Fast exact and approx- imate geodesics on meshes.ACM transactions on graphics (TOG), 24(3):553–560, 2005. 2

    2005

  47. [47]

    Curve skeleton extraction from incomplete point cloud.ACM Trans

    Andrea Tagliasacchi, Hao Zhang, and Daniel Cohen-Or. Curve skeleton extraction from incomplete point cloud.ACM Trans. Graph., 28(3), 2009. 2, 6, 7, 8, 10, 13, 20

    2009

  48. [48]

    Mean curvature skeletons.Computer Graphics Forum, 31(5):1735–1744, 2012

    Andrea Tagliasacchi, Ibraheem Alhashim, Matt Olson, and Hao Zhang. Mean curvature skeletons.Computer Graphics Forum, 31(5):1735–1744, 2012. 2, 6, 7, 5, 20

    2012

  49. [49]

    3d skeletons: A state-of- the-art report.Computer Graphics Forum, 35(2):573–597,

    Andrea Tagliasacchi, Thomas Delame, Michela Spagnuolo, Nina Amenta, and Alexandru Telea. 3d skeletons: A state-of- the-art report.Computer Graphics Forum, 35(2):573–597,

  50. [50]

    G. Taubin. Geometric Signal Processing on Polygonal Meshes. InEurographics 2000 - STARs. Eurographics As- sociation, 2000. 1

    2000

  51. [51]

    3D Mesh Skeleton Extraction Using Topologi- cal and Geometrical Analyses

    Julien Tierny, Jean-Philippe Vandeborre, and Mohamed Daoudi. 3D Mesh Skeleton Extraction Using Topologi- cal and Geometrical Analyses. In14th Pacific Conference on Computer Graphics and Applications (Pacific Graphics 2006), page s1poster, Tapei, Taiwan, 2006. 2, 20

    2006

  52. [52]

    S. R. S. Varadhan. On the behavior of the fundamental solu- tion of the heat equation with variable coefficients.Commu- nications on Pure and Applied Mathematics, 20(2):431–455,

  53. [53]

    A survey of deep learning- based mesh processing.Communications in Mathematics and Statistics, 10, 2022

    He Wang and Juyong Zhang. A survey of deep learning- based mesh processing.Communications in Mathematics and Statistics, 10, 2022. 1

    2022

  54. [54]

    Computing medial axis transform with feature preservation via restricted power diagram.ACM Transactions on Graph- ics (TOG), 41(6):1–18, 2022

    Ningna Wang, Bin Wang, Wenping Wang, and Xiaohu Guo. Computing medial axis transform with feature preservation via restricted power diagram.ACM Transactions on Graph- ics (TOG), 41(6):1–18, 2022. 2

    2022

  55. [55]

    Coverage axis++: Efficient inner point selection for 3d shape skeletonization

    Zimeng Wang, Zhiyang Dou, Rui Xu, Cheng Lin, Yuan Liu, Xiaoxiao Long, Shiqing Xin, Taku Komura, Xiaoming Yuan, and Wenping Wang. Coverage axis++: Efficient inner point selection for 3d shape skeletonization. InComputer Graph- ics Forum, page e15143. Wiley Online Library, 2024. 2, 8, 20

    2024

  56. [56]

    V oxel2mesh: 3d mesh model genera- tion from volumetric data

    Udaranga Wickramasinghe, Edoardo Remelli, Graham Knott, and Pascal Fua. V oxel2mesh: 3d mesh model genera- tion from volumetric data. InMedical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd Inter- national Conference, Lima, Peru, October 4–8, 2020, Pro- ceedings, Part IV 23, pages 299–308. Springer, 2020. 20

    2020

  57. [57]

    Efficiently determining a locally exact shortest path on polyhedral surfaces.Computer- Aided Design, 39(12):1081–1090, 2007

    Shi-Qing Xin and Guo-Jin Wang. Efficiently determining a locally exact shortest path on polyhedral surfaces.Computer- Aided Design, 39(12):1081–1090, 2007. 20

    2007

  58. [58]

    Efficiently com- puting exact geodesic loops within finite steps.IEEE Trans- actions on Visualization and Computer Graphics, 18(6): 879–889, 2012

    Shi-Qing Xin, Ying He, and Chi-Wing Fu. Efficiently com- puting exact geodesic loops within finite steps.IEEE Trans- actions on Visualization and Computer Graphics, 18(6): 879–889, 2012. 20

    2012

  59. [59]

    P2mat-net: Learning medial axis transform from sparse point clouds.Computer Aided Geometric Design, 80: 101874, 2020

    Baorong Yang, Junfeng Yao, Bin Wang, Jianwei Hu, Yil- ing Pan, Tianxiang Pan, Wenping Wang, and Xiaohu Guo. P2mat-net: Learning medial axis transform from sparse point clouds.Computer Aided Geometric Design, 80: 101874, 2020. 2, 20

    2020

  60. [60]

    De-path: A differential-evolution-based method for computing energy-minimizing paths on surfaces.Computer- Aided Design, 114:73–81, 2019

    Zipeng Ye, Yong-Jin Liu, Jianmin Zheng, Kai Hormann, and Ying He. De-path: A differential-evolution-based method for computing energy-minimizing paths on surfaces.Computer- Aided Design, 114:73–81, 2019. 20

    2019

  61. [61]

    A vari- ational framework for curve shortening in various geometric domains.IEEE Transactions on Visualization and Computer Graphics, 29(4):1951–1963, 2023

    Na Yuan, Peihui Wang, Wenlong Meng, Shuangmin Chen, Jian Xu, Shiqing Xin, Ying He, and Wenping Wang. A vari- ational framework for curve shortening in various geometric domains.IEEE Transactions on Visualization and Computer Graphics, 29(4):1951–1963, 2023. 4

    1951

  62. [62]

    N. Yuan, P. Wang, W. Meng, S. Chen, J. Xu, S. Xin, Y . He, and W. Wang. A variational framework for curve shortening 10 in various geometric domains.IEEE Transactions on Visual- ization and; Computer Graphics, 29(04):1951–1963, 2023. 5, 2, 20

    1951

  63. [63]

    Open3D: A Modern Library for 3D Data Processing

    Qian-Yi Zhou, Jaesik Park, and Vladlen Koltun. Open3D: A modern library for 3D data processing.arXiv:1801.09847,

    work page internal anchor Pith review Pith/arXiv arXiv

  64. [64]

    Implementation Details 2 B

    2 11 Supplementary Material Generalizing Curve Skeletonization to Continuous Domains A . Implementation Details 2 B . Pseduocode 2 C . Derivation for determining intersection within a face of a mesh 3 D . Evaluation metric 3 E . Curve Shortening using Edge Flip framework 4 E.1. Optimizing the loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

  65. [65]

    This provides improved robustness to poor triangulations

    Firstly, we calculate the geodesic distance using Heat method on the vertices of the intrinsic triangulation. This provides improved robustness to poor triangulations

  66. [66]

    Secondly, we work purely within the tangent spaces of each vertex and each face. This requires us to perform parallel transport of gradients onto the same frames of reference whenever we need to do some comparison, such as when we want to identify angle between neighbouring gradients. This affects the final smoothing step too, where we work with barycentr...

  67. [67]

    We find that by working with the intrinsic mesh triangulations, our implementation results in a more robust algorithm

    Finally, when constructing the ORG tree and pruning the ORG tree, we operate purely upon the intrinsic triangulations. We find that by working with the intrinsic mesh triangulations, our implementation results in a more robust algorithm. In Fig. 13 for the meshTID:44395, we see that our method results in a more stable and consistent cut loci given identic...

  68. [68]

    LS, MSLS and our method use different heuristics for sampling and selecting local separators, so the comparison is not direct. One could argue that LS could, in principle, be sped up by reducing the number of local separators sampled, but that parameter is not available in their codebase; therefore we have to stick to the pre-defined heuristics chosen by ...

  69. [69]

    We have chosen to implement this simple strategy, but we still observe comparable performance to LS (both qualitatively and quantitatively)

    Our method uses a fixedN= 3000, while the number of separators can vary for both LS and MSLS. We have chosen to implement this simple strategy, but we still observe comparable performance to LS (both qualitatively and quantitatively). We suspect this is due to the differences in the heuristics for sampling, where we use geodesic distance to sample farther...

  70. [70]

    Our choices in the framework were made due to its simplicity and directness

    As mentioned in the limitations, our realizationsCSCD-M andCSCD-PC are simply meant to be a starting point. Our choices in the framework were made due to its simplicity and directness. There are many places (such as the cut locus identification procedure) where one can easily speed up the algorithm. We also believe that our particular codebase can be cons...

  71. [71]

    For the geodesic distance, we use a simple graph based distance

  72. [72]

    The target cut locus is then chosen similar to the current method, i.e., based on the minimum Euclidean distance

  73. [73]

    To optimize the curve, we can follow an iterative unfolding scheme, where the path between two vertices is iteratively shortened using Djikstra’s shortest path algorithm

  74. [74]

    With the optimized local separators, overlap can simply be checked by determining if two local separators share a vertex

  75. [75]

    Next, we assign the nearest vertices to each local separator, thereby creating the quotient graph

  76. [76]

    Finally, based on the quotient graph, the curve skeleton is constructed and post-processed. K.2. On the multiscale version ofCSCD-M CSCD-M has immense potential for a multi-scale approach. This is because we have the ability to sample points on the face of a low-poly mesh, and operate on these face surface points through barycentric interpolations. In thi...

  77. [77]

    central path

    Our realizations are intended as starting points, with primary results on meshes and limited tests on point clouds. Future work could enhance individual modules for greater speed, robustness, and performance, and extend the framework to other representations. 12 ROSAOurs Figure 20.CSCD-PC vs ROSA:Comparative results of our method vs ROSA. We are able to c...

    2048