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

4. Solid and Physical Modeling 2014
Motivation & Scope
Applications-
Reverse engineering
Cultural heritage
Rapid prototyping
Urban modeling etc…
Digitization-courtesy:
Solid and Physical Modeling 2014
City modeling-Poullis et.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. Solid and Physical Modeling 2014
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. Solid and Physical Modeling 2014
Divergent Concavity
Closed, planar and positively oriented curve
Solid and Physical Modeling 2014

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

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

11. Solid and Physical Modeling 2014
Divergent Concavity
BTP BT
BTP
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
Pseudo concavity IP IP
Solid and Physical Modeling 2014

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

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

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

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

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

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

19. Solid and Physical Modeling 2014
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)
Junction points
Connectedness Q
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)
SHG(S) is a connected triangulation, free of junction points and consists of a subset of Delaunay triangles such that the circumcenter of these triangles lie interior to its boundary.
Delaunay triangulation
Solid and Physical Modeling 2014
SHG

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

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

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

32. Solid and Physical Modeling 2014
Sculpting Algorithm
Solid and Physical Modeling 2014

33. Solid and Physical Modeling 2014
Results*-Bimba**
74K points, 250K tetrahedra
*implemented in CGAL (computational geometry algorithms library)
** Models from 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, 552K tetrahedra
Shark, 10K points, 20K Delaunay tetrahedra
Solid and Physical Modeling 2014

37. Solid and Physical Modeling 2014
Results-Down Sampling
Solid and Physical Modeling 2014

38. Solid and Physical Modeling 2014
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. Solid and Physical Modeling 2014
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. Solid and Physical Modeling 2014
References
AMENTA, N., CHOI, S., AND KOLLURI, R. K The power crust, unions of balls, and the medial axis transform. Computational Geometry: Theory and Applications 19, 127–153.
BOISSONNAT, J.-D Geometric structures for threedimensional shape representation. ACM Trans. Graph. 3, 4 (Oct.), 266–286.
DEY, T. K., AND GOSWAMI, S Provable surface reconstruction from noisy samples. Comput. Geom. Theory Appl. 35, 1 (Aug.), 124–141.
MANSON, J., PETROVA, G., AND SCHAEFER, S 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 Multi-level partition of unity implicits. In ACM SIGGRAPH 2003 Papers, ACM, New York, NY, USA, SIGGRAPH ’03, 463–470.
VELTKAMP, R. C Closed Object Boundaries from Scattered Points. Springer-Verlag New York, Inc., Secaucus, NJ, USA.
EDELSBRUNNER, H., AND M¨U CKE, E. P Threedimensional alpha shapes. ACM Trans. Graph. 13, 1 (Jan.), 43– 2.
KAZHDAN, M 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
Jiju Peethambaran
Solid and Physical Modeling 2014