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Reconstruction of Water-tight Surfaces through Delaunay Sculpting

This paper presents a reconstruction algorithm using Delaunay Sculpting to generate a water-tight surface mesh from surface samples, addressing the surface reconstruction problem. The algorithm is applicable in various domains such as reverse engineering, cultural heritage, rapid prototyping, and urban modeling. The technique involves characterizing divergent concavity, constructing a shape-hull graph, and performing sculpting using Delaunay triangulation. The approach is fully automatic, simple, and single-stage, providing an effective solution for generating accurate and detailed surface models.

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Reconstruction of Water-tight Surfaces through Delaunay Sculpting

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  1. Reconstruction of Water-tight Surfaces through Delaunay Sculpting Jiju Peethambaran and Ramanathan Muthuganapathy Advanced Geometric Computing Lab, Department of Engineering Design, Indian Institute of Technology, Madras, India Solid and Physical Modeling 2014

  2. Surface Reconstruction Problem Reconstruction Algorithm • Generate surface mesh from surface samples Solid and Physical Modeling 2014

  3. Motivation & Scope • Require a surface mesh for • Effective rendering of the model • Computational analysis • Parameterization- Morphing, blending etc.. Solid and Physical Modeling 2014 Blending-Kreavoy et. al 2004 Morphing-Kreavoy et. al 2004

  4. Motivation & Scope Digitization-courtesy: http://graphics.stanford.edu/ • Applications- • Reverse engineering • Cultural heritage • Rapid prototyping • Urban modeling etc… Solid and Physical Modeling 2014 City modeling-Poulliset.al 2011

  5. Related Work-Implicit Surfaces • Represent the surface by a function defined over the space • Extract the zero-set • Examples • Poisson [Kazhdan. 2005] • RBF [Carr et al. 2001] • MPU [Ohtake et al. 2003] • Wavelet [Manson et al. 2008] etc… Solid and Physical Modeling 2014

  6. Related Work-Delaunay/Voronoi • Under dense sampling, neighboring points on the surface is also neighbors in the space • Examples • Alpha shape [Edelsbrunner and Mucke 1994] • Sculpture by Boissonat [Boissonnat 1984] • Powercrust [Amenta et al. 2000] • Cocone [Dey et.al, 2006] • Constriction by Veltkamp [Veltkampl, 1994] etc… • Each has its own strengths and weaknesses!!! Solid and Physical Modeling 2014

  7. Our Contributions • Characterization of Divergent concavity for closed, planar curves • Shape-hull graph (SHG)-a proximity graph that captures the geometric shape • Surface reconstruction technique • Un-oriented point cloud • Fully automatic, simple and single stage • Delaunay Sculpting • Triangulated water-tight surface mesh Solid and Physical Modeling 2014

  8. Divergent Concavity Closed, planar and positively oriented curve Solid and Physical Modeling 2014

  9. Divergent Concavity IP IP Concavity Closed, planar and counter clock wise oriented curve Inflection points and curvature Concave portion (green colored) Solid and Physical Modeling 2014

  10. Divergent Concavity BTP BT BTP IP IP Concavity Closed, planar and counter clock wise oriented curve Inflection points and curvature Concave portion (green colored) BT-bi-tangent, BTP-bi-tangent points Solid and Physical Modeling 2014

  11. Divergent Concavity BTP BT BTP Pseudo concavity IP IP Closed, planar and counter clock wise oriented curve Inflection points and curvature Concave portion (green colored) BT-bi-tangent, BTP-bi-tangent points Pseudo-concavity Solid and Physical Modeling 2014

  12. Divergent Concavity Extremal and non-extremal BTs Solid and Physical Modeling 2014

  13. Divergent Concavity Medial balls Divergent pseudo-concavity Solid and Physical Modeling 2014

  14. Divergent Concavity Medial balls Non-divergent Divergent • If all the pseudo-concavities are divergent, then it is divergent concave Solid and Physical Modeling 2014

  15. Divergent Concavity Implications: Point set, S sampled from a divergent concave curve Solid and Physical Modeling 2014

  16. Divergent Concavity Implications: Delaunay triangulation of S Solid and Physical Modeling 2014

  17. Divergent Concavity Implications: Divergent concave portion Solid and Physical Modeling 2014

  18. Divergent Concavity Implications: Triangles in divergent concave region are Solid and Physical Modeling 2014

  19. Divergent Concavity Implications: • Triangles in divergent concave region are • Obtuse • Their longest edge faces towards the extremal BT Solid and Physical Modeling 2014

  20. Shape-hull Graph (SHG) Junction points Solid and Physical Modeling 2014

  21. Shape-hull Graph (SHG) Q Junction points Connectedness Solid and Physical Modeling 2014 P

  22. Shape-hull Graph (SHG) Point set Solid and Physical Modeling 2014

  23. Shape-hull Graph (SHG) Del(S) Point set Solid and Physical Modeling 2014

  24. Shape-hull Graph (SHG) Del(S) Point set Del(S)-Delaunay triangles in divergent concave regions =SHG(S) SHG(S) Solid and Physical Modeling 2014

  25. Shape-hull Graph (SHG) Del(S) Point set • Triangulation - sub graph of Del(S) • Connected • No junction points • Consists of Delaunay triangles whose circumcenter lies inside the boundary of SHG SHG(S) Solid and Physical Modeling 2014

  26. Shape-hull Graph (SHG) Del(S) Point set SHG(S) SH(S) Solid and Physical Modeling 2014

  27. Shape-hull Graph (SHG) Delaunay triangulation SHG • SHG(S) is a connectedtriangulation, free of junction points and consists of a subset of Delaunay triangles such that the circumcenter of these triangles lie interior to its boundary. Solid and Physical Modeling 2014

  28. Shape-hull Graph (SHG) Divergent concave curve Shape-hull Point set Lemma---SH(S), where S is densely sampled from a divergent concave curve Ω, represents piece-wise linear approximation of Ω Solid and Physical Modeling 2014

  29. Sculpting Algorithm • Construct Delaunay tetrahedral mesh • Repeatedly eliminate (or sculpt) boundary tetrahedra, T subjected to the following: • circumcenter of T lies outside the intermediate surface • T satisfies tetrahedral removal rules Solid and Physical Modeling 2014

  30. Sculpting Algorithm 1-boundary facet (abc) 2-boundary facets (abc) & (abd) • Tetrahedral removal rules- Remove the tetrahedra with one/ two boundary facets if it satisfy the constraints [Boissonnat,1984 ] Solid and Physical Modeling 2014

  31. Sculpting Algorithm Random removal Circumradius/shortest edge length • Selection criterion- circumcenter of tetrahedra Solid and Physical Modeling 2014 Volume Circumradius

  32. Sculpting Algorithm Solid and Physical Modeling 2014

  33. Results*-Bimba** 74K points, 250K tetrahedra *implemented in CGAL (computational geometry algorithms library) ** Models from Aim@shape or Stanford 3D scanning repository Solid and Physical Modeling 2014

  34. Results-Budha 250K points, 500K Delaunay tetrahedra Solid and Physical Modeling 2014

  35. Results Caesar, 25K points, 84K tetrahedra Foot, 10K points, 20K Delaunay tetrahedra Solid and Physical Modeling 2014

  36. Results Sheep, 159K points, 552Ktetrahedra Shark, 10K points, 20K Delaunay tetrahedra Solid and Physical Modeling 2014

  37. Results-Down Sampling Solid and Physical Modeling 2014

  38. Results-Down Sampling Solid and Physical Modeling 2014

  39. Results- Sharp Features Solid and Physical Modeling 2014 Powercrust R cocone Screened poisson Our method

  40. Conclusions • Divergent concavity for 2D curves • Shape-hull graph • Sculpting Algorithm for closed surface reconstruction • Future work- • Genus construction • Extension to non-divergent concave curves/surfaces Solid and Physical Modeling 2014

  41. References AMENTA, N., CHOI, S., AND KOLLURI, R. K. 2000. The power crust, unions of balls, and the medial axis transform. Computational Geometry: Theory and Applications 19, 127–153. BOISSONNAT, J.-D. 1984. Geometric structures for threedimensional shape representation. ACM Trans. Graph. 3, 4 (Oct.), 266–286. DEY, T. K., AND GOSWAMI, S. 2006. Provable surface reconstruction from noisy samples. Comput. Geom. Theory Appl. 35, 1 (Aug.), 124–141. MANSON, J., PETROVA, G., AND SCHAEFER, S. 2008. Streaming surface reconstruction using wavelets. Computer Graphics Forum (Proceedings of the Symposium on Geometry Processing) 27, 5, 1411–1420. OHTAKE, Y., BELYAEV, A., ALEXA, M., TURK, G., AND SEIDEL, H.-P. 2003. Multi-level partition of unity implicits. In ACM SIGGRAPH 2003 Papers, ACM, New York, NY, USA, SIGGRAPH ’03, 463–470. VELTKAMP, R. C. 1994. Closed Object Boundaries from Scattered Points. Springer-Verlag New York, Inc., Secaucus, NJ, USA. EDELSBRUNNER, H., AND M¨U CKE, E. P. 1994. Threedimensional alpha shapes. ACM Trans. Graph. 13, 1 (Jan.), 43– 2. KAZHDAN, M. 2005. Reconstruction of solid models from oriented point sets. In Proceedings of the Third Eurographics Symposium on Geometry Processing, Eurographics Association, Airela-Ville, Switzerland, Switzerland, SGP ’05. Solid and Physical Modeling 2014

  42. Thank You Questions? Contact Information: Ramanathan Muthuganapathy (emry01@gmail.com, http://ed.iitm.ac.in/~raman) Jiju Peethambaran (jijupnair2000@gmail.com) Solid and Physical Modeling 2014

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