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 Solid and Physical Modeling 2014

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 2004 Solid and Physical Modeling 2014

Motivation & Scope Applications-  Reverse engineering  Cultural heritage  Rapid prototyping  Urban modeling etc… Digitization-courtesy: / City modeling-Poullis et.al 2011 Solid 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… Solid and Physical Modeling 2014

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 Solid and Physical Modeling 2014

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

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

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

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

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

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

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

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

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

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

Divergent Concavity Implications: Triangles in divergent concave region are 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

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

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

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

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

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

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

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

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 SHG Solid 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 set Shape-hull Divergent concave curve 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

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 Solid and Physical Modeling 2014

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

Sculpting Algorithm 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

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

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

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

Results-Down Sampling Solid and Physical Modeling 2014

Results-Down Sampling Solid and Physical Modeling 2014

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

References 1. 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– BOISSONNAT, J.-D Geometric structures for threedimensional shape representation. ACM Trans. Graph. 3, 4 (Oct.), 266– DEY, T. K., AND GOSWAMI, S Provable surface reconstruction from noisy samples. Comput. Geom. Theory Appl. 35, 1 (Aug.), 124– 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– 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– VELTKAMP, R. C Closed Object Boundaries from Scattered Points. Springer-Verlag New York, Inc., Secaucus, NJ, USA. 7. EDELSBRUNNER, H., AND M¨U CKE, E. P Threedimensional alpha shapes. ACM Trans. Graph. 13, 1 (Jan.), 43– 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

Thank You Questions? Contact Information: Ramanathan Muthuganapathy Jiju Peethambaran Solid and Physical Modeling 2014