Boundary-sampled halfspaces: a new representation for constructive solid modeling

Abstract

We present a novel representation of solid models for shape design. Like Constructive Solid Geometry (CSG), the solid shape is constructed from a set of halfspaces without the need for an explicit boundary structure. Instead of using Boolean expressions as in CSG, the shape is defined by sparsely placed samples on the boundary of each halfspace. This representation, called Boundary-Sampled Halfspaces (BSH), affords greater agility and expressiveness than CSG while simplifying the reverse engineering process. We discuss theoretical properties of the representation and present practical algorithms for boundary extraction and conversion from other representations. Our algorithms are demonstrated on both 2D and 3D examples.

References

1. Pankaj K Agarwal and Micha Sharir. 2000. Arrangements and their applications. In Handbook of computational geometry. Elsevier, 49--119.
2. Lionel Alberti, Bernard Mourrain, and Julien Wintz. 2008. Topology and arrangement computation of semi-algebraic planar curves. Computer Aided Geometric Design 25, 8 (2008), 631--651.
3. Jean-Philippe Bauchet and Florent Lafarge. 2020. Kinetic shape reconstruction. ACM Transactions on Graphics (TOG) 39, 5 (2020), 1--14.
4. Eric Berberich, Pavel Emeliyanenko, Alexander Kobel, and Michael Sagraloff. 2012. Arrangement computation for planar algebraic curves. In Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation. 88--98.
5. Matthew Berger, Andrea Tagliasacchi, Lee M Seversky, Pierre Alliez, Gael Guennebaud, Joshua A Levine, Andrei Sharf, and Claudio T Silva. 2017. A survey of surface reconstruction from point clouds. In Computer Graphics Forum, Vol. 36. Wiley Online Library, 301--329.
6. Gilbert Bernstein and Don Fussell. 2009. Fast, exact, linear booleans. In Computer Graphics Forum, Vol. 28. Wiley Online Library, 1269--1278.
7. Alexandre Boulch, Martin de La Gorce, and Renaud Marlet. 2014. Piecewise-planar 3D reconstruction with edge and corner regularization. In Computer Graphics Forum, Vol. 33. Wiley Online Library, 55--64.
8. Suzanne F Buchele and Richard H Crawford. 2003. Three-dimensional halfspace constructive solid geometry tree construction from implicit boundary representations. In Proceedings of the eighth ACM symposium on Solid modeling and applications. 135--144.
9. J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell, W. R. Fright, B. C. McCallum, and T. R. Evans. 2001. Reconstruction and Representation of 3D Objects with Radial Basis Functions. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 2001). 67--76.
10. Malcolm S Casale and James E Bobrow. 1989. A set operation algorithm for sculptured solids modeled with trimmed patches. Computer Aided Geometric Design 6, 3 (1989), 235--247.
11. Anne-Laure Chauve, Patrick Labatut, and Jean-Philippe Pons. 2010. Robust piecewise-planar 3D reconstruction and completion from large-scale unstructured point data. In 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE, 1261--1268.
12. Zhiqin Chen, Andrea Tagliasacchi, and Hao Zhang. 2020. Bsp-net: Generating compact meshes via binary space partitioning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 45--54.
13. Gianmarco Cherchi, Marco Livesu, Riccardo Scateni, and Marco Attene. 2020. Fast and robust mesh arrangements using floating-point arithmetic. ACM Transactions on Graphics (TOG) 39, 6 (2020), 1--16.
14. Matthijs Douze, Jean-Sébastien Franco, and Bruno Raffin. 2017. QuickCSG: Fast Arbitrary Boolean Combinations of N Solids. arXiv preprint arXiv:1706.01558 (2017).
15. Tao Du, Jeevana Priya Inala, Yewen Pu, Andrew Spielberg, Adriana Schulz, Daniela Rus, Armando Solar-Lezama, and Wojciech Matusik. 2018. Inversecsg: Automatic conversion of 3d models to csg trees. ACM Transactions on Graphics (TOG) 37, 6 (2018), 1--16.
16. Laurent Dupont, Michael Hemmer, Sylvain Petitjean, and Elmar Schömer. 2007. Complete, exact and efficient implementation for computing the adjacency graph of an arrangement of quadrics. In European Symposium on Algorithms. Springer, 633--644.
17. Jack Edmonds and Richard M Karp. 1972. Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM (JACM) 19, 2 (1972), 248--264.
18. Pierre-Alain Fayolle and Alexander Pasko. 2016. An evolutionary approach to the extraction of object construction trees from 3D point clouds. Computer-Aided Design 74 (2016), 1--17.
19. Pierre-Alain Fayolle, Alexander Pasko, Elena Kartasheva, Christophe Rosenberger, and Christian Toinard. 2008. Automation of the volumetric models construction. In Heterogeneous objects modelling and applications. Springer, 214--238.
20. Francisco R Feito, Carlos J Ogáyar, Rafael Jesús Segura, and ML Rivero. 2013. Fast and accurate evaluation of regularized Boolean operations on triangulated solids. Computer-Aided Design 45, 3 (2013), 705--716.
21. Karim Hamza and Kazuhiro Saitou. 2004. Optimization of constructive solid geometry via a tree-based multi-objective genetic algorithm. In Genetic and Evolutionary Computation Conference. Springer, 981--992.
22. Christoph Martin Hoffmann. 1989. Geometric and solid modeling. (1989).
23. Zhiyang Huang, Nathan Carr, and Tao Ju. 2019. Variational implicit point set surfaces. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1--13.
24. Adrien Kaiser, Jose Alonso Ybanez Zepeda, and Tamy Boubekeur. 2019. A survey of simple geometric primitives detection methods for captured 3D data. In Computer Graphics Forum, Vol. 38. Wiley Online Library, 167--196.
25. John Keyser, Tim Culver, Mark Foskey, Shankar Krishnan, and Dinesh Manocha. 2004. ESOLID - a system for exact boundary evaluation. Computer-Aided Design 36, 2 (2004), 175--193.
26. Pierre-Alain Langlois, Alexandre Boulch, and Renaud Marlet. 2019. Surface reconstruction from 3d line segments. In 2019 International Conference on 3D Vision (3DV). IEEE, 553--563.
27. Jyh-Ming Lien, Vikram Sharma, Gert Vegter, and Chee Yap. 2014. Isotopic arrangement of simple curves: An exact numerical approach based on subdivision. In International Congress on Mathematical Software. Springer, 277--282.
28. Gabor Lukács, Ralph Martin, and Dave Marshall. 1998. Faithful least-squares fitting of spheres, cylinders, cones and tori for reliable segmentation. In European conference on computer vision. Springer, 671--686.
29. Bernard Mourrain, Jean-Pierre Técourt, and Monique Teillaud. 2005. On the computation of an arrangement of quadrics in 3d. Computational Geometry 30, 2 (2005), 145--164.
30. Gregory M Nielson. 2004. Radial hermite operators for scattered point cloud data with normal vectors and applications to implicitizing polygon mesh surfaces for generalized CSG operations and smoothing. In IEEE Visualization 2004. IEEE, 203--210.
31. Sven Oesau, Florent Lafarge, and Pierre Alliez. 2014. Indoor scene reconstruction using feature sensitive primitive extraction and graph-cut. ISPRS journal of photogrammetry and remote sensing 90 (2014), 68--82.
32. Alexander Pasko, Valery Adzhiev, Alexei Sourin, and Vladimir Savchenko. 1995. Function representation in geometric modeling: concepts, implementation and applications. The visual computer 11, 8 (1995), 429--446.
33. Darko Pavić, Marcel Campen, and Leif Kobbelt. 2010. Hybrid booleans. In Computer Graphics Forum, Vol. 29. Wiley Online Library, 75--87.
34. Aristides AG Requicha and Herbert B Voelcker. 1977. Constructive solid geometry. (1977).
35. Aristides G Requicha. 1980. Representations for rigid solids: Theory, methods, and systems. ACM Computing Surveys (CSUR) 12, 4 (1980), 437--464.
36. Jaroslaw Rossignac and Aristides Requicha. 1984. Constant-radius blending in solid modelling. (1984).
37. Jaroslaw R Rossignac and Aristides AG Requicha. 1986. Offsetting operations in solid modelling. Computer Aided Geometric Design 3, 2 (1986), 129--148.
38. Jaroslaw R Rossignac and Aristides AG Requicha. 1999. Solid modeling. Technical Report. Georgia Institute of Technology.
39. Ruwen Schnabel, Roland Wahl, and Reinhard Klein. 2007. Efficient RANSAC for point-cloud shape detection. In Computer graphics forum, Vol. 26. Wiley Online Library, 214--226.
40. Elmar Schömer and Nicola Wolpert. 2006. An exact and efficient approach for computing a cell in an arrangement of quadrics. Computational Geometry 33, 1--2 (2006), 65--97.
41. Ariel Shamir. 2008. A survey on mesh segmentation techniques. In Computer graphics forum, Vol. 27. Wiley Online Library, 1539--1556.
42. Vadim Shapiro. 2002. Solid Modeling. Handbook of computer aided geometric design 20 (2002), 473--518.
43. Vadim Shapiro and Donald L Vossler. 1991. Construction and optimization of CSG representations. Computer-Aided Design 23, 11 (1991), 4--20.
44. Vadim Shapiro and Donald L Vossler. 1993. Separation for boundary to CSG conversion. ACM Transactions on Graphics (TOG) 12, 1 (1993), 35--55.
45. Sara Silva, Pierre-Alain Fayolle, Johann Vincent, Guillaume Pauron, Christophe Rosenberger, and Christian Toinard. 2005. Evolutionary computation approaches for shape modelling and fitting. In Portuguese Conference on Artificial Intelligence. Springer, 144--155.
46. JM Smith and Neil A Dodgson. 2007. A topologically robust algorithm for Boolean operations on polyhedral shapes using approximate arithmetic. Computer-Aided Design 39, 2 (2007), 149--163.
47. Yannick Verdie, Florent Lafarge, and Pierre Alliez. 2015. LOD generation for urban scenes. ACM Transactions on Graphics 34, ARTICLE (2015), 30.
48. Charlie CL Wang. 2010. Approximate boolean operations on large polyhedral solids with partial mesh reconstruction. IEEE transactions on visualization and computer graphics 17, 6 (2010), 836--849.
49. Qiaoyun Wu, Kai Xu, and Jun Wang. 2018. Constructing 3D CSG models from 3D raw point clouds. In Computer Graphics Forum, Vol. 37. Wiley Online Library, 221--232.
50. Brian Wyvill, Andrew Guy, and Eric Galin. 1999. Extending the csg tree. warping, blending and boolean operations in an implicit surface modeling system. In Computer Graphics Forum, Vol. 18. Wiley Online Library, 149--158.
51. Songgang Xu and John Keyser. 2013. Fast and robust Booleans on polyhedra. Computer-Aided Design 45, 2 (2013), 529--534.
52. Qingnan Zhou, Eitan Grinspun, Denis Zorin, and Alec Jacobson. 2016. Mesh arrangements for solid geometry. ACM Transactions on Graphics (TOG) 35, 4 (2016), 1--15.
53. Yixin Zhuang, Hang Dou, Nathan Carr, and Tao Ju. 2017. Feature-aligned segmentation using correlation clustering. In Computational Visual Media, Vol. 2, 147--160.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

      • Shape analysis

      • Volumetric models