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arxiv: 1907.00523 · v1 · pith:ZUJ2YIPTnew · submitted 2019-07-01 · 💻 cs.GR · cs.CG

Geodesic Centroidal Voronoi Tessellations: Theories, Algorithms and Applications

Zipeng Ye , Ran Yi , Minjing Yu , Yong-jin Liu , Ying He This is my paper

Pith reviewed 2026-05-25 11:50 UTC · model grok-4.3

classification 💻 cs.GR cs.CG
keywords Geodesic Centroidal Voronoi TessellationsManifold MeshesComputer VisionComputer GraphicsIntrinsic Geometric StructuresCombinatorial StructuresTime and Space ComplexityVoronoi Diagrams
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The pith

Geodesic centroidal Voronoi tessellations on manifold meshes enable efficient search, location and indexing for computer vision and graphics tasks.

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

The paper summarizes recent work defining geodesic centroidal Voronoi tessellations as intrinsic geometric structures on manifold meshes embedded in high-dimensional feature space. It claims these structures support a wide range of applications in computer vision and graphics because of built-in efficiency for search, location, and indexing operations on digital media such as images, videos, and 3D models. The authors also lay out open theoretical and algorithmic problems around constructing the combinatorial structures of GCVTs and determining their time and space complexities.

Core claim

Geodesic centroidal Voronoi tessellations (GCVTs) are intrinsic geometric structures on manifold meshes. They support a wide range of applications in computer vision and graphics due to the efficiency of search, location and indexing inherent in these structures. The work reviews theories and algorithms for GCVTs while framing the construction of their combinatorial structures and the analysis of their time and space complexities as challenging open issues.

What carries the argument

Geodesic centroidal Voronoi tessellations (GCVTs), intrinsic partitions of manifold meshes that group points by geodesic distance to centroids and thereby enable direct geometric indexing.

If this is right

  • GCVTs can be applied directly to big data tasks involving images, videos, and 3D graphical models.
  • The intrinsic nature of GCVTs allows search and indexing operations without embedding into ambient Euclidean space.
  • Theoretical results on GCVT structures provide foundations for new algorithms with explicit complexity bounds.
  • The same structures support multiple distinct applications across computer vision and computer graphics.

Where Pith is reading between the lines

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

  • Similar geodesic partitioning ideas could be tested on non-manifold or dynamic meshes where standard Voronoi methods fail.
  • If complexity bounds turn out favorable, GCVTs might serve as a drop-in primitive for manifold-based machine learning pipelines.
  • The open construction problem suggests room for approximation schemes that trade exact geodesic distance for faster combinatorial output.
  • Connections between GCVT centroids and existing mesh simplification or remeshing pipelines remain to be explored.

Load-bearing premise

That the combinatorial structures of GCVTs can be constructed and analyzed for time and space complexity in a manner that supports the claimed applications.

What would settle it

A concrete example of a manifold mesh where building the GCVT combinatorial structure requires superlinear time or space relative to the number of vertices, such that the claimed efficiency for vision or graphics tasks no longer holds.

Figures

Figures reproduced from arXiv: 1907.00523 by Minjing Yu, Ran Yi, Ying He, Yong-jin Liu, Zipeng Ye.

Figure 1
Figure 1. Figure 1: (b) shows a constrained Voronoi tessellation of 30 point generators on the 2-manifold (a) and its Voronoi cell [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A geodesic Voronoi tessellation on a genus-2 manifold whose cells are multiply connected. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The map Φ that stretches an image I ⊂ R 2 to an image manifold M2 ⊂ R 5 . (b) A uniform tessellation on M2 such as GCVT naturally induces content-sensitive superpixels in I via Φ −1; for an easy illustration, a gray image is used for a mapping in R 3 . tessellation in Euclidean space. Therefore, the cells in CCVT may be disconnected or multiply-connected2 ; see Figures 1 and 2 for an illustration. GCVT… view at source ↗
Figure 4
Figure 4. Figure 4: Some examples of two content-sensitive superpixels (MSLIC [ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The stretching map Φ : Υ → M3 ⊂ R 6 maps a video Υ ⊂ R 3 into a 3-manifold M3 ⊂ R 6 . (3) Compactness: cells in GCVT are regular and uniform in non-feature regions; (4) Feature preservation: the feature regions (such as object boundary) have a large color variation and therefore lead to a large stretching/area on M2. The larger the area in M2, the higher possibility that a cell boundary passes through it; … view at source ↗
Figure 6
Figure 6. Figure 6: Examples of content-sensitive supervoxels [ [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) GCVT with a small number of generators [21]. (b) Its dual intrinsic Delaunay triangulation (IDT) [22]. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Geodesic paths and circles on curved manifolds. There are two different geodesic circles passing through [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A visibility wedge (VW) on the bottom mesh edge. [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Each iso-contour of the distance field on a closed [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) If all vertices do not have the same geodesic distance to a given pair of source points, their bisector [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: If a saddle vertex vs lies on the bisector of two points p and q, i.e., dg(p, vs) = dg(q, vs), this bisector contains a 2D regions (shaded area) [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The geodesic Voronoi tessellation of the generator set [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A real example of geodesic Voronoi tessellation of 12 point generators on a 2-manifold mesh with 2022 [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Some examples of geodesic Voronoi tessellations. [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: MDE pipeline. 6.1 MDE Global Method MDE is a stochastic global optimization method, which extends the classic differential evolution [27] to the manifold setting. Classic differential evolution applies agents that have operations of addition, subtraction and scala multiplication, all defined in a vector space. To use these operations on a manifold, MDE assigned an order to the generators in an agent for e… view at source ↗
Figure 17
Figure 17. Figure 17: The regular mesh in (a) left and the irregular mesh in (b) left represent the same geometry. By applying the [PITH_FULL_IMAGE:figures/full_fig_p012_17.png] view at source ↗
read the original abstract

Nowadays, big data of digital media (including images, videos and 3D graphical models) are frequently modeled as low-dimensional manifold meshes embedded in a high-dimensional feature space. In this paper, we summarized our recent work on geodesic centroidal Voronoi tessellations(GCVTs), which are intrinsic geometric structures on manifold meshes. We show that GCVT can find a widely range of interesting applications in computer vision and graphics, due to the efficiency of search, location and indexing inherent in these intrinsic geometric structures. Then we present the challenging issues of how to build the combinatorial structures of GCVTs and establish their time and space complexities, including both theoretical and algorithmic results.

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 / 1 minor

Summary. The manuscript summarizes the authors' recent work on geodesic centroidal Voronoi tessellations (GCVTs) as intrinsic geometric structures on manifold meshes. It claims that GCVTs enable a wide range of applications in computer vision and graphics due to efficiencies in search, location, and indexing. The paper also presents challenging issues regarding the construction of combinatorial structures of GCVTs and the establishment of their time and space complexities, including theoretical and algorithmic results.

Significance. If the presented theoretical and algorithmic results on GCVT construction and complexities hold and demonstrate the claimed efficiencies, the work could provide a foundation for broader use of these structures in graphics and vision applications. However, the abstract frames the core construction and complexity analysis as open challenging issues rather than resolved results, which limits the immediate significance for the application claims.

major comments (1)
  1. [Abstract] Abstract: The central claim that GCVTs enable applications 'due to the efficiency of search, location and indexing inherent in these intrinsic geometric structures' is load-bearing for the paper's contribution but is directly undercut by the subsequent framing of 'the challenging issues of how to build the combinatorial structures of GCVTs and establish their time and space complexities' as topics the paper presents (rather than resolved results). This internal tension means the application significance rests on unshown prior work without new evidence of settled complexities.
minor comments (1)
  1. [Abstract] Abstract: 'a widely range of interesting applications' should be corrected to 'a wide range of interesting applications'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that GCVTs enable applications 'due to the efficiency of search, location and indexing inherent in these intrinsic geometric structures' is load-bearing for the paper's contribution but is directly undercut by the subsequent framing of 'the challenging issues of how to build the combinatorial structures of GCVTs and establish their time and space complexities' as topics the paper presents (rather than resolved results). This internal tension means the application significance rests on unshown prior work without new evidence of settled complexities.

    Authors: We acknowledge the referee's observation of tension in the abstract phrasing. This manuscript is a survey summarizing our prior published results on GCVT constructions and their demonstrated use in applications (where efficiencies in search, location, and indexing were shown via concrete implementations on manifold meshes). The 'challenging issues' section discusses remaining open problems in fully general combinatorial structures and complexity bounds, which were not fully resolved in our prior work. We agree the abstract should be revised for precision to explicitly attribute the efficiency claims to the summarized prior results rather than implying new resolutions here. We will update the abstract accordingly in the revised version. revision: yes

Circularity Check

0 steps flagged

No derivation chain or equations present; paper is an overview summary.

full rationale

The provided abstract and description characterize the document as a summary of the authors' recent work on GCVTs, explicitly framing construction of combinatorial structures and their complexities as 'challenging issues' rather than presenting any new derivations, equations, predictions, or load-bearing steps. No self-definitional reductions, fitted inputs called predictions, or uniqueness claims via self-citation are exhibited because no mathematical content or derivation chain appears. The application claims rest on stated efficiencies of intrinsic structures but are not derived within this text, so no circularity can be identified by the required standards of quoting specific reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities are described in the abstract.

pith-pipeline@v0.9.0 · 5649 in / 1038 out tokens · 23500 ms · 2026-05-25T11:50:05.384465+00:00 · methodology

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Spatial Tessellations: Concept and Applications of Voronoi Diagrams

    Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, and Sung Nok Chiu. Spatial Tessellations: Concept and Applications of Voronoi Diagrams. Wiley, 2000

    work page 2000

  2. [2]

    On the construction of the voronoi mesh on a sphere

    Jeffrey M Augenbaum and Charles S Peskin. On the construction of the voronoi mesh on a sphere. Journal of Computational Physics, 59(2):177–192, 1985

    work page 1985

  3. [3]

    Construction of voronoi diagram on the upper half-plane

    Kensuke Onishi and Nobuki Takayama. Construction of voronoi diagram on the upper half-plane. IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences, E79-A(4):533–539, 1996. 13 A PREPRINT - J ULY 2, 2019

    work page 1996

  4. [4]

    Delaunay triangulations and voronoi diagrams for riemannian manifolds

    Greg Leibon and David Letscher. Delaunay triangulations and voronoi diagrams for riemannian manifolds. In Proceedings of the Sixteenth Annual Symposium on Computational Geometry, SCG ’00, pages 341–349. ACM, 2000

    work page 2000

  5. [5]

    Sebastian Seung and Daniel D

    H. Sebastian Seung and Daniel D. Lee. The manifold ways of perception. Science, 290(5500):2268–2269, 2000

    work page 2000

  6. [6]

    An introduction to differentiable manifolds and Riemannian geometry

    William M Boothby. An introduction to differentiable manifolds and Riemannian geometry. Academic Press, 1975

    work page 1975

  7. [7]

    The discrete geodesic problem

    Joseph SB Mitchell, David M Mount, and Christos H Papadimitriou. The discrete geodesic problem. SIAM Journal on Computing, 16(4):647–668, 1987

    work page 1987

  8. [8]

    Centroidal voronoi tessellations: Applications and algorithms

    Qiang Du, Vance Faber, and Max Gunzburger. Centroidal voronoi tessellations: Applications and algorithms. Siam Review, 41(4):637–676, 1999

    work page 1999

  9. [9]

    Stuart P. Lloyd. Least squares quantization in pcm’s. IEEE Transactions on Information Theory, 28:129–136, 03 1982

    work page 1982

  10. [10]

    Gunzburger, and Lili Ju

    Qiang Du, Max D. Gunzburger, and Lili Ju. Constrained centroidal voronoi tessellations for surfaces. SIAM Journal on Scientific Computing, 24(5):1488–1506, 2003

    work page 2003

  11. [11]

    Construction of iso-contours, bisectors, and voronoi diagrams on triangulated surfaces

    Liu Yong-Jin, Chen Zhan-Qing, and Tang Kai. Construction of iso-contours, bisectors, and voronoi diagrams on triangulated surfaces. IEEE Transactions on Pattern Analysis & Machine Intelligence, 33(8):1502–1517, 2011

    work page 2011

  12. [12]

    Intrinsic manifold SLIC: A simple and efficient method for computing content-sensitive superpixels

    Yong-Jin Liu, Minjing Yu, Bing-Jun Li, and Ying He. Intrinsic manifold SLIC: A simple and efficient method for computing content-sensitive superpixels. IEEE Trans. Pattern Anal. Mach. Intell., 40(3):653–666, 2018

    work page 2018

  13. [13]

    Ming Yu, O

    L. Ming Yu, O. Tuzel, S. Ramalingam, and R. Chellappa. Entropy rate superpixel segmentation. In Computer Vision & Pattern Recognition, 2011

    work page 2011

  14. [14]

    Optimal contour closure by superpixel grouping

    Alex Levinshtein, Cristian Sminchisescu, and Sven Dickinson. Optimal contour closure by superpixel grouping. In European Conference on Computer Vision, 2010

    work page 2010

  15. [15]

    Class segmentation and object localization with superpixel neighborhoods

    Brian Fulkerson, Andrea Vedaldi, and Stefano Soatto. Class segmentation and object localization with superpixel neighborhoods. In IEEE International Conference on Computer Vision, 2009

    work page 2009

  16. [16]

    Superpixel tracking

    Wang Shu, Huchuan Lu, Yang Fan, and Ming Hsuan Yang. Superpixel tracking. In International Conference on Computer Vision, 2011

    work page 2011

  17. [17]

    Multi-view superpixel stereo in urban environments

    Branislav Miˇcušík and Jana Košecká. Multi-view superpixel stereo in urban environments. International Journal of Computer Vision, 89(1):106–119, 2010

    work page 2010

  18. [18]

    Superpixels: An evaluation of the state-of-the-art

    David Stutz, Alexander Hermans, and Bastian Leibe. Superpixels: An evaluation of the state-of-the-art. Computer Vision and Image Understanding, 166:1–27, 2018

    work page 2018

  19. [19]

    Manifold SLIC: a fast method to compute content-sensitive superpixels

    Yong-Jin Liu, Chengchi Yu, Minjing Yu, and Ying He. Manifold SLIC: a fast method to compute content-sensitive superpixels. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR ’16, pages 651–659, 2016

    work page 2016

  20. [20]

    Content-sensitive supervoxels via uniform tessellations on video manifolds

    Ran Yi, Yong-Jin Liu, and Yu-Kun Lai. Content-sensitive supervoxels via uniform tessellations on video manifolds. In IEEE Conference on Computer Vision and Pattern Recognition, CVPR ’18, pages 646–655, 2018

    work page 2018

  21. [21]

    Manifold differential evolution (MDE): A global optimization method for geodesic centroidal voronoi tessellations on meshes

    Yong-Jin Liu, Chun-Xu Xu, Ran Yi, Dian Fan, and Ying He. Manifold differential evolution (MDE): A global optimization method for geodesic centroidal voronoi tessellations on meshes. ACM Transactions on Graphics (SIGGRAPH ASIA 2016), 35(6):243:1–243:10, 2016

    work page 2016

  22. [22]

    Constructing intrinsic delaunay triangulations from the dual of geodesic voronoi diagrams

    Yong-Jin Liu, Dian Fan, Chun-Xu Xu, and Ying He. Constructing intrinsic delaunay triangulations from the dual of geodesic voronoi diagrams. ACM Trans. Graph., 36(2):15:1–15:15, 2017

    work page 2017

  23. [23]

    The complexity of geodesic voronoi diagrams on triangulated 2-manifold surfaces

    Yong-Jin Liu and Kai Tang. The complexity of geodesic voronoi diagrams on triangulated 2-manifold surfaces. Information Processing Letters, 113(4):132–136, 2013

    work page 2013

  24. [24]

    Semi-continuity of skeletons in 2-manifold and discrete voronoi approximation

    Yong-Jin Liu. Semi-continuity of skeletons in 2-manifold and discrete voronoi approximation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(9):1938–1944, 2015

    work page 1938

  25. [25]

    Herbert Edelsbrunner and Nimish R. Shah. Triangulating topological spaces. International Journal of Computa- tional Geometry and Applications, 7(4):365–378, 1997

    work page 1997

  26. [26]

    Global optimization of centroidal voronoi tessellation with monte carlo approach

    Lin Lu, Feng Sun, Hao Pan, and Wenping Wang. Global optimization of centroidal voronoi tessellation with monte carlo approach. IEEE Trans. Vis. Comput. Graph., 18(11):1880–1890, 2012

    work page 2012

  27. [27]

    Differential evolution: A survey of the state-of-the-art

    Swagatam Das and Ponnuthurai Nagaratnam Suganthan. Differential evolution: A survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation, 15(1):4–31, 2011

    work page 2011

  28. [28]

    B B. H. Korte and Jens Vygen. Combinatorial Optimization: Theory and Algorithms. Spinger, 2012. 14 A PREPRINT - J ULY 2, 2019

    work page 2012

  29. [29]

    Intrinsic computation of centroidal voronoi tessellation (CVT) on meshes

    Xiaoning Wang, Xiang Ying, Yong-Jin Liu, Shi-Qing Xin, Wenping Wang, Xianfeng Gu, Wolfgang Müller-Wittig, and Ying He. Intrinsic computation of centroidal voronoi tessellation (CVT) on meshes. Computer-Aided Design, 58:51–61, 2015

    work page 2015

  30. [30]

    Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements

    Xavier Pennec. Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements. Journal of Mathematical Imaging and Vision, 25(1):127–154, 2006

    work page 2006

  31. [31]

    Rustamov

    Raif M. Rustamov. Barycentric coordinates on surfaces. Comput. Graph. Forum, 29(5):1507–1516, 2010

    work page 2010

  32. [32]

    Tenenbaum

    Vin de Silva and Joshua B. Tenenbaum. Global versus local methods in nonlinear dimensionality reduction. In Advances in Neural Information Processing Systems (NIPS ’02), pages 705–712, 2002. 15

    work page 2002