Reconstruction of Water-tight Surfaces through Delaunay Sculpting Jiju Peethambaran and Ramanathan...

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

Surface Reconstruction Problem

Generate surface mesh from surface samples

Reconstruction Algorithm


Motivation & Scope

Require a surface mesh for Effective rendering of the model Computational analysis Parameterization- Morphing, blending etc..

Morphing-Kreavoy et. al 2004 Blending-Kreavoy et. al 2004Solid and Physical Modeling 2014

Motivation & Scope

Applications- Reverse engineering Cultural heritage Rapid prototyping Urban modeling etc…

Digitization-courtesy: http://graphics.stanford.edu/

City modeling-Poullis et.al 2011Solid and Physical Modeling 2014

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…


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

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


Divergent Concavity

Closed, planar and positively oriented curve


Divergent Concavity

Closed, planar and counter clock wise oriented curve

Inflection points and curvature

Concave portion (green colored)

IP IPConcavity


Divergent Concavity




BT-bi-tangent, BTP-bi-tangent points

IP IP

BT BTPBTP

Concavity


Divergent Concavity




BT-bi-tangent, BTP-bi-tangent points

Pseudo-concavity

IP IP

BT BTPBTP

Pseudo concavity


Divergent Concavity

Extremal and non-extremal BTs


Divergent Concavity

Divergent pseudo-concavity

Medial balls


Divergent Concavity

Medial balls

If all the pseudo-concavities are divergent, then it is divergent concave

Divergent Non-divergent


Divergent Concavity

Implications:

Point set, S sampled from a divergent concave curve


Divergent Concavity

Implications:

Delaunay triangulation of S


Divergent Concavity

Implications:

Divergent concave portion


Divergent Concavity

Implications:

Triangles in divergent concave region are


Divergent Concavity

Implications:

Triangles in divergent concave region are ObtuseTheir longest edge faces towards the extremal BTSolid and Physical Modeling 2014

Shape-hull Graph (SHG)

Junction points



Junction points

Connectedness

P

Q



Point set



Point set Del(S)



Point set Del(S)

SHG(S)

Del(S)-Delaunay trianglesin divergent concave regions=SHG(S)



Point set Del(S)

SHG(S)

Triangulation - sub graph of Del(S)ConnectedNo junction pointsConsists of Delaunay triangles whose circumcenter lies inside the boundary of SHG



Point set Del(S)

SHG(S) SH(S)Solid and Physical Modeling 2014


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 SHGSolid and Physical Modeling 2014

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

Point setShape-hull

Divergent concave curve


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


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)

Sculpting Algorithm


Sculpting Algorithm

Selection criterion- circumcenter of tetrahedra

Circumradius/shortest edge length Random removal

CircumradiusVolumeSolid and Physical Modeling 2014

Sculpting Algorithm


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

250K points, 500K Delaunay tetrahedra

Results-Budha


Caesar, 25K points, 84K tetrahedra

Foot, 10K points, 20K Delaunay tetrahedra

Results


Sheep, 159K points, 552K tetrahedra

Shark, 10K points, 20K Delaunay tetrahedra

Results


Results-Down Sampling


Results- Sharp Features

Powercrust R cocone Screened poisson Our methodSolid and Physical Modeling 2014

Conclusions

Divergent concavity for 2D curves Shape-hull graph Sculpting Algorithm for closed surface

reconstruction Future work-

1. Genus construction

2. Extension to non-divergent concave curves/surfaces


References1. 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.

2. BOISSONNAT, J.-D. 1984. Geometric structures for threedimensional shape representation. ACM Trans. Graph. 3, 4 (Oct.), 266–286.

3. DEY, T. K., AND GOSWAMI, S. 2006. Provable surface reconstruction from noisy samples. Comput. Geom. Theory Appl. 35, 1 (Aug.), 124–141.

4. 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.

5. 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.

6. VELTKAMP, R. C. 1994. Closed Object Boundaries from Scattered Points. Springer-Verlag New York, Inc., Secaucus, NJ, USA.

7. EDELSBRUNNER, H., AND M¨U CKE, E. P. 1994. Threedimensional alpha shapes. ACM Trans. Graph. 13, 1 (Jan.), 43– 2.

8. 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.


Thank YouQuestions?

Contact Information:

Ramanathan Muthuganapathy ([email protected], http://ed.iitm.ac.in/~raman)

Jiju Peethambaran ([email protected])


Reconstruction of Water-tight Surfaces through Delaunay Sculpting Jiju Peethambaran and Ramanathan...

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