Polycube splines

ABSTRACT

This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.

REFERENCES

• 1: Chow, B., and Luo, F. 2003. Combinatorial ricci flows on surfaces. J. Differential Geometry, 97--129.
• 2: Floater, M. S., and Hormann, K. 2005. Surface parameterization: a tutorial and survey. In Advances in multiresolution for geometric modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin, Eds. Springer Verlag, 157--186.
• 3: Craig Gotsman , Xianfeng Gu , Alla Sheffer, Fundamentals of spherical parameterization for 3D meshes, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 4: Xianfeng Gu , Shing-Tung Yau, Global conformal surface parameterization, Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, June 23-25, 2003, Aachen, Germany
• 5: Gu, X., Wang, Y., Chan, T. F., Thompson, P. M., and Yau, S.-T. 2004. Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Transactions on Medical Imaging 23, 8 (Aug.), 945--958.
• 6: Xianfeng Gu , Ying He , Hong Qin, Manifold splines, Proceedings of the 2005 ACM symposium on Solid and physical modeling, p.27-38, June 13-15, 2005, Cambridge, Massachusetts [doi>10.1145/1060244.1060249]
• 7: He, Y., Jin, M., Gu, X., and Qin, H. 2005. A C¹ globally interpolatory spline of arbitrary topology. In LNCS, vol. 3752, 295--306.
• 8: He, Y., Gu, X.-F., and Qin, H. 2006. Automatic shape control of triangular B-splines of arbitrary topology. J. Comput. Sci. Technol. 21, 2, 232--237.
• 9: He, Y., Wang, K., Wang, H., Gu, X., and Qin, H. 2006. Manifold T-spline. In Proceedings of Geometric Modeling and Processing, 409--422.
• 10: Miao Jin , Yalin Wang , Shing-Tung Yau , Xianfeng Gu, Optimal Global Conformal Surface Parameterization, Proceedings of the conference on Visualization '04, p.267-274, October 10-15, 2004 [doi>10.1109/VIS.2004.75]
• 11: Miao Jin , Feng Luo , Xianfeng Gu, Computing surface hyperbolic structure and real projective structure, Proceedings of the 2006 ACM symposium on Solid and physical modeling, June 06-08, 2006, Cardiff, Wales, United Kingdom [doi>10.1145/1128888.1128904]
• 12: Jost, J., and Simha, R. R. 1997. Compact Riemann Surfaces: An Introduction to Contemporary Mathematics. Springer-Verlag Telos.
• 13: Liliya Kharevych , Boris Springborn , Peter Schröder, Discrete conformal mappings via circle patterns, ACM Transactions on Graphics (TOG), v.25 n.2, p.412-438, April 2006 [doi>10.1145/1138450.1138461]
• 14: Bruno Lévy , Sylvain Petitjean , Nicolas Ray , Jérome Maillot, Least squares conformal maps for automatic texture atlas generation, Proceedings of the 29th annual conference on Computer graphics and interactive techniques, July 23-26, 2002, San Antonio, Texas
• 15: Li, W.-C., Ray, N., and Lévy, B. 2006. Automatic and interactive mesh to T-spline conversion. In EG/ACM Symposium on Geometry Processing.
• 16: Jérôme Maillot , Hussein Yahia , Anne Verroust, Interactive texture mapping, Proceedings of the 20th annual conference on Computer graphics and interactive techniques, p.27-34, September 1993 [doi>10.1145/166117.166120]
• 17: Nicolas Ray , Wan Chiu Li , Bruno Lévy , Alla Sheffer , Pierre Alliez, Periodic global parameterization, ACM Transactions on Graphics (TOG), v.25 n.4, p.1460-1485, October 2006 [doi>10.1145/1183287.1183297]
• 18: Thomas W. Sederberg , Jianmin Zheng , Almaz Bakenov , Ahmad Nasri, T-splines and T-NURCCs, ACM Transactions on Graphics (TOG), v.22 n.3, July 2003
• 19: Thomas W. Sederberg , David L. Cardon , G. Thomas Finnigan , Nicholas S. North , Jianmin Zheng , Tom Lyche, T-spline simplification and local refinement, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 20: Alla Sheffer , Bruno Lévy , Maxim Mogilnitsky , Alexander Bogomyakov, ABF++: fast and robust angle based flattening, ACM Transactions on Graphics (TOG), v.24 n.2, p.311-330, April 2005 [doi>10.1145/1061347.1061354]
• 21: Vitaly Surazhsky , Craig Gotsman, Explicit surface remeshing, Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing, June 23-25, 2003, Aachen, Germany
• 22: Marco Tarini , Kai Hormann , Paolo Cignoni , Claudio Montani, PolyCube-Maps, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004
• 23: Wang, Y., and Zheng, J. 2006. Control point removal algorithm for T-spline surfaces. In GMP, 385--396.
• 24: Jianmin Zheng , Yimin Wang , Hock Soon Seah, Adaptive T-spline surface fitting to z-map models, Proceedings of the 3rd international conference on Computer graphics and interactive techniques in Australasia and South East Asia, November 29-December 02, 2005, Dunedin, New Zealand [doi>10.1145/1101389.1101468]

INDEX TERMS

Keywords:
affine atlas, manifold splines, polycube map, polycube splines, shape computing, solid modeling