This Article 
 Bibliographic References 
 Add to: 
Supercubes: A High-Level Primitive for Diamond Hierarchies
November/December 2009 (vol. 15 no. 6)
pp. 1603-1610
Kenneth Weiss, University of Maryland, College Park
Leila De Floriani, University of Genova
Volumetric datasets are often modeled using a multiresolution approach based on a nested decomposition of the domain into a polyhedral mesh. Nested tetrahedral meshes generated through the longest edge bisection rule are commonly used to decompose regular volumetric datasets since they produce highly adaptive crack-free representations. Efficient representations for such models have been achieved by clustering the set of tetrahedra sharing a common longest edge into a structure called a diamond. The alignment and orientation of the longest edge can be used to implicitly determine the geometry of a diamond and its relations to the other diamonds within the hierarchy. We introduce the supercube as a high-level primitive within such meshes that encompasses all unique types of diamonds. A supercube is a coherent set of edges corresponding to three consecutive levels of subdivision. Diamonds are uniquely characterized by the longest edge of the tetrahedra forming them and are clustered in supercubes through the association of the longest edge of a diamond with a unique edge in a supercube. Supercubes are thus a compact and highly efficient means of associating information with a subset of the vertices, edges and tetrahedra of the meshes generated through longest edge bisection. We demonstrate the effectiveness of the supercube representation when encoding multiresolution diamond hierarchies built on a subset of the points of a regular grid. We also show how supercubes can be used to efficiently extract meshes from diamond hierarchies and to reduce the storage requirements of such variable-resolution meshes.

[1] P. Cignoni, F. Ganovelli, E. Gobbetti, F. Marton, F. Ponchio, and R. Scopigno, Adaptive tetrapuzzles: efficient out-of-core construction and visualization of gigantic multiresolution polygonal models. ACM Transactions on Graphics, 23 (3): 796–803, 2004.
[2] L. De Floriani and P. Magillo, Multi-resolution mesh representation: models and data structures. In M. Floater, A. Iske, and E. Quak editors, Principles of Multi-resolution Geometric Modeling, Lecture Notes in Mathematics, pages 364–418, Berlin, 2002. Springer Verlag.
[3] M. Duchaineau, M. Wolinsky, D. E. Sigeti, M. C. Miller, C. Aldrich, and M. B. Mineev-Weinstein, ROAMing terrain: real-time optimally adapting meshes. In R. Yagel, and H. Hagen editors, Proc. IEEE Visualization, pages 81–88, Phoenix, AZ, October 1997. IEEE Computer Society.
[4] S. F. Frisken, R. N. Perry, A. P. Rockwood, and T. R. Jones, Adaptively sampled distance fields: a general representation of shape for computer graphics. In Proceedings SIGGRAPH'00 Conference, pages 249–254, New Orleans, LA, July 2000. ACM Press.
[5] T. Gerstner, Multi-resolution visualization and compression of global topographic data. GeoInformatica, 7 (1): 7–32, 2003.
[6] T. Gerstner and R. Pajarola, Topology-preserving and controlled topology simplifying multi-resolution isosurface extraction. In Proceedings IEEE Visualization 2000, pages 259–266, 2000.
[7] T. Gerstner and M. Rumpf, Multiresolutional parallel isosurface extraction based on tetrahedral bisection. In Proceedings Symposium on Volume Visualization, pages 267–278. ACM Press, 1999.
[8] B. Gregorski, M. Duchaineau, P. Lindstrom, V. Pascucci, and K. Joy, Interactive view-dependent rendering of large isosurfaces. In Proceedings IEEE Visualization, pages 475–484. IEEE Computer Society Washington, DC, USA, 2002.
[9] B. Gregorski, J. Senecal, M. Duchaineau, and K. Joy, Adaptive extraction of time-varying isosurfaces. IEEE Transactions on Visualization and Computer Graphics, 10 (6): 683–694, 2004.
[10] D. J. Hebert, Symbolic local refinement of tetrahedral grids. Journal of Symbolic Computation, 17 (5): 457–472, May 1994.
[11] T. Ju, F. Losasso, S. Schaefer, and J. Warren, Dual contouring of hermite data. ACM Trans. Graph., 21 (3): 339–346, 2002.
[12] A. Kimura, Y. Takama, Y. Yamazoe, S. Tanaka, and H. Tanaka, Parallel volume segmentation with tetrahedral adaptive grid. International Conference on Pattern Recognition, 2: 281–286, 2004.
[13] M. Lee, L. De Floriani, and H. Samet, Constant-time neighbor finding in hierarchical tetrahedral meshes. In Proceedings International Conference on Shape Modeling, pages 286–295, Genova, Italy, May 2001. IEEE Computer Society.
[14] S. Lefebvre and H. Hoppe, Perfect spatial hashing. ACM Transactions on Graphics (TOG), 25 (3): 579–588, 2006.
[15] D. Luebke, M. Reddy, J. Cohen, A. Varshney, B. Watson, and R. Huebner, Level of Detail for 3D Graphics. Morgan-Kaufmann, San Francisco, 2002.
[16] S. Marchesin, J. Dischler, and C. Mongenet, 3D ROAM for scalable volume visualization. IEEE Symposium on Volume Visualization and Graphics, pages 79–86, 2004.
[17] J. M. Maubach, Local bisection refinement for n-simplicial grids generated by reflection. SIAM Journal on Scientific Computing, 16 (1): 210– 227, January 1995.
[18] V. Mello, L. Velho, and G. Taubin, Estimating the in/out function of a surface represented by points. Symposium on Solid Modeling and Applications 2003, pages 108–114, 2003.
[19] V. Pascucci, Slow Growing Subdivisions (SGS) in any dimension: towards removing the curse of dimensionality. Computer Graphics Forum, 21 (3): 451–460, 2002.
[20] V. Pascucci, Isosurface computation made simple: Hardware acceleration, adaptive refinement and tetrahedral stripping. Eurographics/IEEE TVCG Symposium on Visualization (VisSym), pages 293–300, 2004.
[21] R. N. Perry and S. F. Frisken, Kizamu: A system for sculpting digital characters. In ACM Computer Graphics, (Proc. SIGGRAPH 2001), pages 47–56, Los Angeles, CA, August 2001. ACM Press.
[22] M. Rivara and C. Levin, A 3D refinement algorithm for adaptive and multigrid techniques. Communications in Applied Numerical Methods, 8: 281–290, 1992.
[23] T. Roxborough and G. Nielson, Tetrahedron-based, least-squares, progressive volume models with application to freehand ultrasound data. In Proceedings IEEE Visualization 2000, pages 93–100. IEEE Computer Society, October 2000.
[24] H. Samet, Foundations of Multidimensional and Metric Data Structures. Elsevier, 2006.
[25] S. Takahashi, Y. Takeshima, G. Nielson, and I. Fujishiro, Topological volume skeletonization using adaptive tetrahedralization. Geometric Modeling and Processing, 2004. Proceedings, pages 227–236, 2004.
[26] K. Weiss and L. De, Floriani. Multiresolution Interval Volume Meshes. In H.-C. Hege, D. Laidlaw, R. Pajarola, and O. Staadt editors, , IEEE/ EG Symposium on Volume and Point-Based Graphics, pages 65–72, Los Angeles, California, USA, 2008. Eurographics Association.
[27] K. Weiss and L. De Floriani, Sparse Terrain Pyramids. In Proc. of the 16th ACM SIGSPATIAL Int. Conference on Advances in Geographic Information Systems, pages 115–124, New York, NY, USA, 2008. ACM.
[28] K. Weiss and L. De, Floriani. Diamond Hierarchies of Arbitrary Dimension. Computer Graphics Forum, 28 (5): 1289–1300, 2009.
[29] R. Westermann, L. Kobbelt, and T. Ertl, Real-time exploration of regular volume data by adaptive reconstruction of isosurfaces. The Visual Computer, 15 (2): 100–111, 1999.
[30] J. Wilhelms and A. Van Gelder, Topological considerations in isosur-face generation extended abstract. Proceedings of the 1990 workshop on Volume visualization, pages 79–86, 1990.
[31] Y. Zhou, B. Chen, and A. Kaufman, Multi-resolution tetrahedral framework for visualizing regular volume data. In R. Yagel, and H. Hagen editors, Proceedings IEEE Visualization '97, pages 135–142, Phoenix, AZ, October 1997. IEEE Computer Society.

Index Terms:
Longest edge bisection, diamonds, hierarchy of diamonds, multiresolution models, selective refinement.
Kenneth Weiss, Leila De Floriani, "Supercubes: A High-Level Primitive for Diamond Hierarchies," IEEE Transactions on Visualization and Computer Graphics, vol. 15, no. 6, pp. 1603-1610, Nov.-Dec. 2009, doi:10.1109/TVCG.2009.186
Usage of this product signifies your acceptance of the Terms of Use.