A Survey on Implicit Surface Polygonization

Abstract

Implicit surfaces (IS) are commonly used in image creation, modeling environments, modeling objects, and scientific data visualization. In this article, we present a survey of different techniques for fast visualization of IS. The main classes of visualization algorithms are identified along with the advantages of each in the context of the different types of IS commonly used in computer graphics. We focus closely on polygonization methods, as they are the most suited to fast visualization. Classification and comparison of existing approaches are presented using criteria extracted from current research. This enables the identification of the best strategies according to the number of specific requirements, such as speed, accuracy, quality, or stylization.

References

1. CGAL. 2014. The Computational Geometry Algorithms Library Home Page. Retrieved April 1, 2015, from http://www.cgal.org.
2. ImplicitMesher. 2014. ImplicitMesher Home Page. Retrieved April 1, 2015, from http://www.dgp.toronto.edu/∼rms/software/ImplicitMesher/index.html.
3. VTK. 2014. VTK Home Page. Retrieved April 1, 2015, from http://www.vtk.org/.
4. Samir Akkouche and Eric Galin. 2001. Adaptive implicit surface polygonization using marching triangles. Computer Graphics Forum 20, 2, 67--80.
5. Lionel Alberti, Georges Comte, and Bernard Mourrain. 2005. Meshing implicit algebraic surfaces: The smooth case. In Mathematical Methods for Curves and Surfaces: Tromso ’04, L. L. Schumaker, M. Maehlen, and K. Morken (Eds.). Nashboro, 11--26.
6. Rémi Allègre, Eric Galin, Raphaëlle Chaine, and Samir Akkouche. 2006. The HybridTree: Mixing skeletal implicit surfaces, triangle meshes and point sets in a free-form modeling system. Graphical Models 1, 42--64. http://liris.cnrs.fr/publis/?id=1946.
7. Marco Attene, Bianca Falcidieno, Jarek Rossignac, and Michela Spagnuolo. 2003. Edge-sharpener: Recovering sharp features in triangulations of non-adaptively re-meshed surfaces. In Proceedings of the 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (SGP’03). 62--69.
8. Sergei Azernikov and Anath Fischer. 2005. Anisotropic meshing of implicit surfaces. In Proceedings of the International Conference on Shape Modeling and Applications (SMI’05). IEEE, Los Alamitos, CA, 94--103. DOI:http://dx.doi.org/10.1109/SMI.2005.5
9. Chandrajit L. Bajaj, Valerio Pascucci, and Daniel R. Schikore. 1999. Accelerated isocontouring of scalar fields. In Data Visualization Techniques. Trends in Software, Vol. 6. John Wiley and Sons, 31--47.
10. Loic Barthe, Benjamin Mora, Neil Dodgson, and Malcolm Sabin. 2002. Interactive implicit modelling based on C¹ continuous reconstruction of regular grids. International Journal of Shape Modeling 8, 2, 99--117.
11. Alexander G. Belyaev, Alexander A. Pasko, and Tosiyasu L. Kunii. 1998. Ridges and ravines on implicit surfaces. In Proceedings of the Computer Graphics International Conference (CGI’98). 530. DOI:http://dx.doi.org/10.1109/CGI.1998.694306
12. Alexander G. Belyaev and Elena V. Anoshkina. 2005. Detection of surface creases in range data. In Mathematics of Surfaces. Lecture Notes in Computer Science, vol. 3604. Springer, 50--61.
13. Matthew Berger, Andrea Tagliasacchi, Lee M. Seversky, Pierre Alliez, Joshua A. Levine, Andrei Sharf, and Claudio T. Silva. 2014. State of the art in surface reconstruction from point clouds. In Eurographics 2014—State of the Art Reports, S. Lefebvre and M. Spagnuolo (Eds.). Eurographics Association. DOI:http://dx.doi.org/10.2312/egst.20141040
14. James F. Blinn. 1982. A generalization of algebraic surface drawing. ACM Transactions on Graphics 1, 3, 235--256. DOI:http://dx.doi.org/10.1145/357306.357310
15. Jules Bloomenthal. 1988. Polygonization of implicit surfaces. Computer Aided Geometric Design 5, 4, 341--355. DOI:http://dx.doi.org/10.1016/0167-8396(88)90013-1
16. Jules Bloomenthal. 1994. An implicit surface polygonizer. In Graphics Gems IV, Paul Heckbert (Ed.). Academic Press, Boston, MA, 324--349.
17. Jules Bloomenthal. 1995. Modelling Natural Forms. Ph.D. Dissertation. University of Calgary.
18. Jules Bloomenthal and Chandrajit Bajaj. 1997. Introduction to Implicit Surfaces. Morgan Kaufmann, San Francisco, CA.
19. Jules Bloomenthal and Keith Ferguson. 1995. Polygonization of non-manifold implicit surfaces. In Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’95). ACM, New York, NY, 309--316. DOI:http://dx.doi.org/10.1145/218380.218462
20. Ilia Bogaevski, Veronique Lang, Alexander Belyaev, and Tosiyasu L. Kunii. 2003. Color ridges on implicit polynomial surfaces. In Proceedings of GraphiCon 2003. 161--164.
21. Jean-Daniel Boissonnat and Steve Oudot. 2005. Provably good sampling and meshing of surfaces. Graphical Models 67, 5, 405--451. DOI:http://dx.doi.org/10.1016/j.gmod.2005.01.004
22. Andrea Bottino, Wim Nuij, and Kees Van Overveld. 1996. How to shrinkwrap through a critical point: An algorithm for the adaptive triangulation of iso-surfaces with arbitrary topology. In Proceedings of Implicit Surfaces ’96. 53--72.
23. Antoine Bouthors and Matthieu Nesme. 2007. Twinned meshes for dynamic triangulation of implicit surfaces. In Proceedings of Graphics Interface 2007 (GI’07). ACM, New York, NY, 3--9. DOI:http://dx.doi.org/10.1145/1268517.1268521
24. David J. Bremer and John F. Hughes. 1998. Rapid approximate silhouette rendering of implicit surfaces. In Proceedings of Implicit Surfaces ’98. 155--164.
25. Fatih Calakli and Gabriel Taubin. 2011. SSD: Smooth signed distance surface reconstruction. Computer Graphics Forum 30, 7, 1993--2002. DOI:http://dx.doi.org/10.1111/j.1467-8659.2011.02058.x
26. Jonathan C. Carr, Rick K. Beatson, Jon B. Cherrie, Tim J. Mitchell, W. Richard Fright, Bruce 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’01). ACM, New York, NY, 67--76. DOI:http://dx.doi.org/10.1145/383259.383266
27. Martin Cermák and Václav Skala. 2002. Polygonization by the edge spinning. In Proceedings of the 16th Conference on Scientific Computing (Algoritmy’02).
28. Martin Cermák and Václav Skala. 2004. Adaptive edge spinning algorithm for polygonization of implicit surfaces. In Proceedings of the Computer Graphics International Conference (CGI’04). IEEE, Los Alamitos, CA, 36--43.
29. Martin Cermák and Vaclav Skala. 2007. Polygonisation of disjoint implicit surfaces by the adaptive edge spinning algorithm of implicit objects. International Journal of Computer Science Engineering 3, 1, 45--52. DOI:http://dx.doi.org/10.1504/IJCSE.2007.014464
30. Shek Ling Chan and Enrico O. Purisima. 1998. A new tetrahedral tesselation scheme for isosurface generation. Computers and Graphics 22, 1, 83--90.
31. Siu-Wing Cheng, Tamal K. Dey, Edgar A. Ramos, and Tathagata Ray. 2004. Sampling and meshing a surface with guaranteed topology and geometry. In Proceedings of the 20th Annual Symposium on Computational Geometry (SCG’04). ACM, New York, NY, 280--289. DOI:http://dx.doi.org/10.1145/997817.997861
32. Evgeni Chernyaev. 1995. Marching Cubes 33: Construction of Topologically Correct Isosurfaces. Technical Report Technical Report CN/95-17. CERN, Geneva, Switzerland. http://wwwinfo.cern.ch/asdoc/psdir/mc.ps.gz.
33. L. Paul Chew. 1993. Guaranteed-quality mesh generation for curved surfaces. In Proceedings of the 9th Annual Symposium on Computational Geometry (SCG’93). ACM, New York, NY, 274--280. DOI:http://dx.doi.org/10.1145/160985.161150
34. Benoit Crespin. 2002. Dynamic triangulation of variational implicit surfaces using incremental Delaunay tetrahedralization. In Proceedings of the IEEE Symposium on Volume Visualization and Graphics (VVS’02). IEEE, Los Alamitos, CA, 73--80.
35. Alvaro Cuno, Claudio Esperanca, Antonio Oliveira, and Paulo Roma Cavalcanti. 2004. Fast polygonization of variational implicit surfaces. In Proceedings of the XVII Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI’04). IEEE, Los Alamitos, CA, 258--265.
36. Bruno Rodrigues de Araújo and Joaquim Armando Pires Jorge. 2005a. Adaptive polygonization of implicit surfaces. Computers and Graphics 29, 5, 686--696. http://dblp.uni-trier.de/db/journals/cg/cg29.html#AraujoJ05.
37. Bruno Rodrigues de Araújo and Joaquim Armando Pires Jorge. 2005b. A calligraphic interface for interactive free-form modeling with large datasets. In Proceedings of the XVIII Brazilian Symposium on Computer Graphics and Image Processing (SIBGRAPI’05). IEEE, Los Alamitos, CA, 333. DOI:http://dx.doi.org/10.1109/SIBGRAPI.2005.2
38. Luiz Henrique de Figueiredo, Jonas de Miranda Gomes, Demetri Terzopoulos, and Luiz Velho. 1992. Physically-based methods for polygonization of implicit surfaces. In Proceedings of the Conference on Graphics Interface ’92. 250--257.
39. Mathieu Desbrun, Nicolas Tsingos, and Marie-Paule Gascuel. 1995. Adaptive sampling of implicit surfaces for interactive modeling and animation. In Proceedings of Implicit Surfaces ’95.
40. Tamal K. Dey and Joshua A. Levine. 2007. Delaunay meshing of isosurfaces. In Proceedings of the Shape Modeling International Conference. 241--250.
41. Carlos A. Dietrich, Carlos E. Scheidegger, João L. D. Comba, Luciana P. Nedel, and Cláudio T. Silva. 2009a. Marching cubes without skinny triangles. Computing in Science and Engineering 11, 2, 82--87. DOI:http://dx.doi.org/10.1109/MCSE.2009.34
42. Carlos A. Dietrich, Carlos E. Scheidegger, John Schreiner, João L. D. Comba, Luciana P. Nedel, and Cláudio T. Silva. 2009b. Edge transformations for improving mesh quality of marching cubes. IEEE Transactions on Visualization and Computer Graphics 15, 1, 150--159. DOI:http://dx.doi.org/10.1109/TVCG.2008.60
43. Kevin Foster, Pauline Jepp, Brian Wyvill, Mario Sousa, Callum Galbraith, and Joaquim Jorge. 2005. Pen-and-ink for BlobTree implicit models. Computer Graphics Forum 24, 3, 267--276.
44. Pascal Jean Frey and Paul-Louis George. 2010. Meshing Implicit Curves and Surfaces. ISTE, London, UK. DOI:http://dx.doi.org/10.1002/9780470611166.ch16
45. Eric Galin and Samir Akkouche. 2000. Incremental polygonization of implicit surfaces. Graphical Models 62, 1, 19--39.
46. Arnaud Gelas, Sébastien Valette, Rémy Prost, and Wieslaw L. Nowinski. 2009. Technical section: Variational implicit surface meshing. Compututers and Graphics 33, 3, 312--320. DOI:http://dx.doi.org/10.1016/j.cag.2009.03.016
47. Joao P. Gois, Valdecir Polizelli-Junior, Tiago Etiene, Eduardo Tejada, Antonio Castelo, Luis G. Nonato, and Thomas Ertl. 2008. Twofold adaptive partition of unity implicits. Visual Computer 24, 12, 1013--1023. DOI:http://dx.doi.org/10.1007/s00371-008-0297-x
48. Ron Goldman. 2005. Curvature formulas for implicit curves and surfaces. Computer Aided Geometric Design 22, 7, 632--658. DOI:http://dx.doi.org/10.1016/j.cagd.2005.06.005
49. Abel Gomes, Irina Voiculescu, Joaquim Jorge, Brian Wyvill, and Callum Galbraith. 2009. Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms. Springer.
50. Abel J. P. Gomes, Sérgio Dias, and José F. M. Morgado. 2010. Polygonization of non-homogeneous non-manifold implicit surfaces with tentative topological guarantees.. In Proceedings of the IEEE Congress on Evolutionary Computation. IEEE, Los Alamitos, CA, 1--8. http://dblp.uni-trier.de/db/conf/cec/cec2010.html#GomesDM10.
51. Alfred Gray. 1996. Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press, Boca Raton, FL.
52. Markus Hadwiger, Joe M. Kniss, Christof Rezk-salama, Daniel Weiskopf, and Klaus Engel. 2006. Real-Time Volume Graphics. A. K. Peters, Ltd., Natick, MA.
53. Mark Hall and Joe Warren. 1990. Adaptive polygonalization of implicitly defined surfaces. IEEE Computer Graphics and Applications 10, 6, 33--42. DOI:http://dx.doi.org/10.1109/38.62694
54. Charles D. Hansen and Paul Hinker. 1992. Massively parallel isosurface extraction. In Proceedings of the 3rd Conference on Visualization ’92 (VIS’92). 77--83. DOI:http://dx.doi.org/10.1109/VISUAL.1992.235223
55. John C. Hart. 1997. Morse theory for implicit surface modeling. In Mathematical Visualization. Springer-Verlag, 257--268.
56. Erich Hartmann. 1998. A marching method for the triangulation of surfaces. Visual Computer 14, 2, 95--108.
57. Paul S. Heckbert and Michael Garland. 1999. Optimal triangulation and quadric-based surface simplification. Computational Geometry: Theory and Applications 14, 1--3, 49--65. DOI:http://dx.doi.org/10.1016/S0925-7721(99)00030-9
58. Hans-Christian Hege, Martin Seebas, Detlev Stalling, and Malte Zockler. 1997. A Generalized Marching Cubes Algorithm Based on Non-Binary Classifications. Technical Report.
59. Adrian Hilton and John Illingworth. 1997. Marching Triangles: Delaunay Implicit Surface Triangulation. Technical Report 01. University of Surrey, Guidford, CVSSP.
60. Adrian Hilton, Andrew J. Stoddart, John Illingworth, and Terry Windeatt. 1996. Marching triangles: Range image fusion for complex object modelling. In Proceedings of the International Conference on Image Processing. 381--384.
61. Chien-Chang Ho, Fu-Che Wu, Bing-Yu Chen, Yung-Yu Chuang, and Ming Ouhyoung. 2005. Cubical marching squares: Adaptive feature preserving surface extraction from volume data. Computer Graphics Forum 24, 3, 537--545.
62. Kin Chuen Hui and Z. H. Jiang. 1999. Tetrahedra based adaptive polygonization of implicit surface patches. Computer Graphics Forum 18, 1, 57--68.
63. Pauline Jepp. 2007. Using MAS for Illustrative Rendering of Implicit Surfaces. Ph.D. Dissertation. University of Calgary, Calgary, Alberta.
64. Pauline Jepp, Jörg Denzinger, Brian Wyvill, and Mario Costa Sousa. 2008. Using multi-agent systems for sampling and rendering implicit surfaces. In Proceedings of the Brazilian Symposium on Computer Graphics and Image Processing. 255--262. DOI:http://dx.doi.org/10.1109/SIBGRAPI.2008.18
65. Gunnar Johansson and Hamish Carr. 2006a. Accelerating marching cubes with graphics hardware. In Proceedings of the Conference of the Center for Advanced Studies on Collaborative Research (CASCON’06). ACM, New York, NY, 39. DOI:http://dx.doi.org/10.1145/1188966.1189018
66. Gunnar Johansson and Hamish Carr. 2006b. Accelerating marching cubes with graphics hardware. In Proceedings of the Conference of the Center for Advanced Studies on Collaborative Research (CASCON’06). ACM, New York, NY, Article No. 39. DOI:http://dx.doi.org/10.1145/1188966.1189018
67. Tao Ju, Frank Losasso, Scott Schaefer, and Joe Warren. 2002. Dual contouring of hermite data. In Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’02). ACM, New York, NY, 339--346. DOI:http://dx.doi.org/10.1145/566570.566586
68. Devendra Kalra and Alan H. Barr. 1989. Guaranteed ray intersections with implicit surfaces. In Proceedings of the 16th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’89). ACM, New York, NY, 297--306. DOI:http://dx.doi.org/10.1145/74333.74364
69. Takashi Kanai, Yutaka Ohtake, Hiroaki Kawata, and Kiwamu Kase. 2006. GPU-based rendering of sparse low-degree implicit surfaces. In Proceedings of the 4thInternational Conference on Computer Graphics and Interactive Techniques in Australasia and Southeast Asia (GRAPHITE’06). ACM, New York, NY, 165--171. DOI:http://dx.doi.org/10.1145/1174429.1174455
70. Tasso Karkanis and A. James Stewart. 2001. Curvature-dependent triangulation of implicit surfaces. IEEE Computer Graphics and Applications 21, 2, 60--69. DOI:http://dx.doi.org/10.1109/38.909016
71. Michael Kazhdan, Allison Klein, Ketan Dalal, and Hugues Hoppe. 2007. Unconstrained isosurface extraction on arbitrary octrees. In Proceedings of the 5th Eurographics Symposium on Geometry Processing (SGP’07). 125--133. http://dl.acm.org/citation.cfm?id=1281991.1282009
72. Peter Kipfer and Rüdiger Westermann. 2005. GPU construction and transparent rendering of iso-surfaces. In Proceedings of Vision, Modeling, and Visualization 2005. 241--248.
73. Aaron Knoll. 2009. Ray Tracing Implicit Surfaces for Interactive Visualization. Ph.D. Dissertation. University of Utah.
74. Aaron Knoll, Ingo Wald, Steven G. Parker, and Charles D. Hansen. 2006. Interactive isosurface ray tracing of large octree volumes. In Proceedings of the IEEE Symposium on Interactive Ray Tracing. 115--124.
75. Leif Kobbelt and Mario Botsch. 2003. Feature sensitive mesh processing. In Proceedings of the 19th Spring Conference on Computer Graphics (SCCG’03). ACM, New York, NY, 17--22. DOI:http://dx.doi.org/10.1145/984952.984956
76. Leif Kobbelt and Mario Botsch. 2004. A survey of point-based techniques in computer graphics. Computers and Graphics 28, 6, 801--814. DOI:http://dx.doi.org/10.1016/j.cag.2004.08.009
77. Leif P. Kobbelt, Mario Botsch, Ulrich Schwanecke, and Hans-Peter Seidel. 2001. Feature sensitive surface extraction from volume data. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’01). ACM, New York, NY, 57--66. DOI:http://dx.doi.org/10.1145/383259.383265
78. Jan J. Koenderink. 1990. Solid Shape. MIT Press, Cambridge, MA.
79. Florian Levet, Julien Hadim, Patrick Reuter, and Christophe Schlick. 2005. Anisotropic sampling for differential point rendering of implicit surfaces. In Proceedings of the Winter School of Computer Graphics Conference (WSCG’05). 109--116.
80. Thomas Lewiner, Hélio Lopes, Antônio Wilson Vieira, and Geovan Tavares. 2003. Efficient implementation of marching cubes’ cases with topological guarantees. Journal of Graphics Tools 8, 1--15. http://cgal.inria.fr/Publications/2003/LLVT03.
81. Shengjun Liu, Xuehui Yin, Xiaogang Jin, and Jieqing Feng. 2005. High quality triangulation of implicit surfaces. In Proceedings of the 9th International Conference on Computer Aided Design and Computer Graphics (CAD-CG’05). IEEE, Los Alamitos, CA, 133--138. DOI:http://dx.doi.org/10.1109/CAD-CG.2005.46
82. Adriano Lopes and Ken Brodlie. 2003. Improving the robustness and accuracy of the marching cubes algorithm for isosurfacing. IEEE Transactions on Visualization and Computer Graphics 9, 1, 16--29. DOI:http://dx.doi.org/10.1109/TVCG.2003.1175094
83. Daniel S. Lopes, Mauro T. Silva, and Jorge A. Ambrósio. 2013. Tangent vectors to a 3-D surface normal: A geometric tool to find orthogonal vectors based on the Householder transformation. Computer-Aided Design 45, 3, 683--694. DOI:http://dx.doi.org/10.1016/j.cad.2012.11.003
84. William E. Lorensen and Harvey E. Cline. 1987. Marching cubes: A high resolution 3D surface construction algorithm. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’87). ACM, New York, NY, 163--169. DOI:http://dx.doi.org/10.1145/37401.37422
85. Sergey V. Matveyev. 1994. Approximation of isosurface in the marching cube: Ambiguity problem. In Proceedings of the Conference on Visualization ’94 (VIS’94). IEEE, Los Alamitos, CA, 288--292. http://portal.acm.org/citation.cfm?id=951087.951140
86. Neil H. McCormick and Robert B. Fisher. 2002. Edge-Constrained Marching Triangles. Technical Report EDI-INF-RR-0188. Division of Informatics, University of Edinburgh.
87. Miriah Meyer, Robert M. Kirby, and Ross Whitaker. 2007. Topology, accuracy, and quality of isosurface meshes using dynamic particles. IEEE Transactions on Visualization and Computer Graphics 13, 6, 1704--1711. DOI:http://dx.doi.org/10.1109/TVCG.2007.70604
88. Claudio Montani, Riccardo Scateni, and Robert Scopigno. 1994a. Discretized marching cubes. In Proceedings of the Conference on Visualization ’94 (VIS’94). IEEE, Los Alamitos, CA, 281--287.
89. Claudio Montani, Riccardo Scateni, and Roberto Scopigno. 1994b. A modified look-up table for implicit disambiguation of marching cubes. Visual Computer 10, 6, 353--355. http://www.crs4.it/vic/cgi-bin/bib-page.cgi?id=‘Montani:1994:MLT’
90. Doug Moore and Joe Warren. 1995. Mesh Displacement: An Improved Contouring Method for Trivariate Data. Technical Report.
91. Bryan S. Morse, Terry S. Yoo, David T. Chen, Penny Rheingans, and Kalpathi R. Subramanian. 2001. Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions. In Proceedings of the SMI International Conference on Shape Modeling and Applications (SMI’01). 89--98. DOI:http://dx.doi.org/10.1109/SMA.2001.923379
92. Shigeru Muraki. 1991. Volumetric shape description of range data using B&##x30E;lobby model. In Proceedings of the 18th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’91). ACM, New York, NY, 227--235. DOI:http://dx.doi.org/10.1145/122718.122743
93. Balas K. Natarajan. 1994. On generating topologically consistent isosurfaces from uniform samples. Visual Computer 11, 1, 52--62. DOI:http://dx.doi.org/10.1007/BF01900699
94. Peter Neugebauer and Konrad Klein. 1997. Adaptive triangulation of objects reconstructed from multiple range images. In Proceedings of the Conference on Visualization ’97 (VIS’97).
95. Gregory M. Nielson. 2003. On marching cubes. IEEE Transactions on Visualization and Computer Graphics 9, 3, 283--297. DOI:http://dx.doi.org/10.1109/TVCG.2003.1207437
96. Gregory M. Nielson. 2004. Dual marching cubes. In Proceedings of the Conference on Visualization ’04 (VIS’04). IEEELos Alamitos, CA, 489--496. DOI:http://dx.doi.org/10.1109/VIS.2004.28
97. Gregory M. Nielson and Bernd Hamann. 1991. The asymptotic decider: Resolving the ambiguity in marching cubes. In Proceedings of the 2nd Conference on Visualization ’91 (VIS’91). IEEE, Los Alamitos, CA, 83--91. http://dl.acm.org/citation.cfm?id=949607.949621.
98. Paul Ning and Jules Bloomenthal. 1993. An evaluation of implicit surface tilers. IEEE Computer Graphic and Applications 13, 6, 33--41. DOI:http://dx.doi.org/10.1109/38.252552
99. Hiromitsu Nishimura, Masashi Hirai, Tsuyoshi Kawai, Tory Kawata, Isao Shirakawa, and Kengo Omura. 1985. Object modelling by distribution function and a method of image generation. Transactions of the IEICE Japan J68-D, 4, 718--725.
100. Yutaka Ohtake, Alexander Belyaev, Marc Alexa, Greg Turk, and Hans-Peter Seidel. 2003. Multi-level partition of unity implicits. In ACM SIGGRAPH 2003 Papers. ACM, New York, NY, 463--470. DOI:http://dx.doi.org/10.1145/1201775.882293
101. Yutaka Ohtake, Alexander Belyaev, and Alexander Pasko. 2001. Dynamic meshes for accurate polygonization of implicit surfaces with sharp features. In Proceedings of the SMI International Conference on Shape Modeling and Applications. 74--81. DOI:http://dx.doi.org/10.1109/SMA.2001.923377
102. Yutaka Ohtake, Alexander Belyaev, and Hans-Peter Seidel. 2004. Ridge-valley lines on meshes via implicit surface fitting. ACM Transactions on Graphics 23, 3, 609--612. DOI:http://dx.doi.org/10.1145/1015706.1015768
103. Yutaka Ohtake, Alexander Belyaev, and Hans-Peter Seidel. 2005. 3D scattered data interpolation and approximation with multilevel compactly supported RBFs. Graphical Models 67, 3, 150--165. DOI:http://dx.doi.org/10.1016/j.gmod.2004.06.003
104. Yutaka Ohtake and Alexander G. Belyaev. 2002. Dual/primal mesh optimization for polygonized implicit surfaces. In Proceedings of the 7th ACM Symposium on Solid Modeling and Applications (SMA’02). ACM, New York, NY, 171--178. DOI:http://dx.doi.org/10.1145/566282.566308
105. Stanley Osher and Ronald P. Fedkiw. 2003. Level Set Methods and Dynamic Implicit Surfaces. Springer, New York, NY.
106. Stanley Osher and James A. Sethian. 1988. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79, 1, 12--49.
107. Afonso Paiva, Hélio Lopes, Thomas Lewiner, and Luiz Henrique de Figueiredo. 2006. Robust adaptive meshes for implicit surfaces. In Proceedings of the 19th Brazilian Symposium on Computer Graphics and Image Processing. 205--212. http://www.mat.puc-rio.br/∼tomlew/adaptive_implicit_sibgrapi.pdf.
108. Alexander Pasko, Valery Adzhiev, Alexei Sourin, and Vladimir Savchenko. 1995. Function representation in geometric modeling: Concepts, implementation and applications. Visual Computer 11, 8, 429--446.
109. Alexander A. Pasko, Victor V. Pilyugin, and V. N. Pokrovsky. 1988. Geometric modeling in the analysis of trivariate functions. Computers and Graphics 12, 3--4, 457--465.
110. Joaquim Peiró, Luca Formaggia, Mattia Gazzola, Alessandro Radaelli, and V. Rigamonti. 2007. Shape reconstruction from medical images and quality mesh generation via implicit surfaces. International Journal for Numerical Methods in Fluids 53, 8, 1339--1360. DOI:http://dx.doi.org/10.1002/fld.1362
111. Per-Olof Persson. 2005. Mesh Generation for Implicit Geometries. Ph.D. Dissertation. Massachusetts Institute of Technology, Cambridge, MA.
112. Per-Olof Persson. 2014. DistMesh—A Simple Mesh Generator in MATLAB. Retrieved April 1, 2015, from http://persson.berkeley.edu/distmesh/.
113. Simon Plantinga and Gert Vegter. 2004. Isotopic approximation of implicit curves and surfaces. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (SGP’04). ACM, New York, NY, 245--254. DOI:http://dx.doi.org/10.1145/1057432.1057465
114. Simon Plantinga and Gert Vegter. 2007. Isotopic meshing of implicit surfaces. Visual Computer 23, 1, 45--58. DOI:http://dx.doi.org/10.1007/s00371-006-0083-6
115. William H. Press, Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. 1986. Numerical Recipes—The Art of Scientific Computing. Cambridge University Press, Cambridge, MA.
116. Sundaresan Raman and Rephael Wenger. 2008. Quality isosurface mesh generation using an extended marching cubes lookup table. Computer Graphics Forum 27, 3, 791--798.
117. Xia Renbo, Liu Weijun, and Wang Yuechao. 2005. A robust and topological correct marching cube algorithm without look-up table. In Proceedings of the 5th International Conference on Computer and Information Technology (CIT’05). IEEE, Los Alamitos, CA, 565--569. DOI:http://dx.doi.org/10.1109/CIT.2005.44
118. Patrick Reuter. 2003. Reconstruction and Rendering of Implicit Surfaces from Large Unorganized Point Sets. Ph.D. Dissertation. Université Bordeaux 1, France.
119. Laurent Rineau and Mariette Yvinec. 2007. A generic software design for Delaunay refinement meshing. Computational Geometry 38, 1--2, 100--110. DOI:http://dx.doi.org/10.1016/j.comgeo.2006.11.008
120. Angela Rösch, Matthias Ruhl, and Dietmar Saupe. 1997. Interactive visualization of implicit surfaces with singularities. Eurographics Computer Graphics Forum 16, 5, 295--306.
121. Vladimir L. Rvachev. 1963. On the analytical description of some geometric objects. Reports of Ukrainian Academy of Sciences 153, 4, 765--767.
122. Scott Schaefer, Tao Ju, and Joe Warren. 2007. Manifold dual contouring. IEEE Transactions on Visualization and Computer Graphics 13, 3, 610--619. DOI:http://dx.doi.org/10.1109/TVCG.2007.1012
123. Scott Schaefer and Joe Warren. 2004. Dual marching cubes: Primal contouring of dual grids. In Proceedings of the 12th Pacific Conference on Computer Graphics and Applications (PG’04). IEEE, Los Alamitos, CA, 70--76.
124. Ryan Schmidt, Brian Wyvill, and Eric Galin. 2005. Interactive implicit modeling with hierarchical spatial caching. In Proceedings of the International Conference on Shape Modeling and Applications (SMI’05). IEEE, Los Alamitos, CA, 104--113. DOI:http://dx.doi.org/10.1109/SMI.2005.25
125. John Schreiner and Carlos Scheidegger. 2006. High-quality extraction of isosurfaces from regular and irregular grids. IEEE Transactions on Visualization and Computer Graphics 12, 5, 1205--1212. DOI:http://dx.doi.org/10.1109/TVCG.2006.149
126. John M. Schreiner, Carlos Eduardo Scheidegger, Shachar Fleishman, and Cláudio T. Silva. 2006. Direct (re)meshing for efficient surface processing. Computer Graphics Forum 25, 3, 527--536. http://dblp.uni-trier.de/db/journals/cgf/cgf25.html#SchreinerSFS06.
127. Vadim Shapiro. 2007. Semi-analytic geometry with R-functions. Acta Numerica 16, 239--303. DOI:http://dx.doi.org/10.1017/S096249290631001X
128. Pourya Shirazian, Brian Wyvill, and Jean-Luc Duprat. 2012. Polygonization of implicit surfaces on multi-core architectures with SIMD instructions. In Eurographics Symposium on Parallel Graphics and Visualization, H. Childs, T. Kuhlen, and F. Marton (Eds.). Eurographics Association, 89--98. http://dblp.uni-trier.de/db/conf/egpgv/egpgv2012.html#ShirazianWD12.
129. Christian Sigg. 2006. Representation and Rendering of Implicit Surfaces. Ph.D. Dissertation. ETH Zurich, Zurich, Switzerland.
130. Cláudio T. Silva, Joo L. D. Comba, Steven P. Callahan, and Fabio F. Bernardon. 2005. A survey of GPU-based volume rendering of unstructured grids. Brazilian Journal of Theoretic and Applied Computing 12, 9--29.
131. Marcelo F. Siqueira, S. R. Freitas, A. C. Filho, and G. Tavares. 1998. Speeding up adaptive polygonization. In Proceedings of the West Side Computer Graphics Conference (WSCG’98). 11--12.
132. Barton T. Stander and John C. Hart. 1995. Interactive re-polygonization of blobby implicit curves. In Proceedings of the Western Computer Graphics Symposium.
133. Barton T. Stander and John C. Hart. 1997. Guaranteeing the topology of an implicit surface polygonization for interactive modeling. In Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’97). ACM, New York, NY, 279--286. DOI:http://dx.doi.org/10.1145/258734.258868
134. Wen Y. Su and John C. Hart. 2005. A programmable particle system framework for shape modelling. In Proceedings of the International Conference on Shape Modeling and Applications. 114--123.
135. Gabriel Taubin. 2012. Smooth signed distance surface reconstruction and applications. In Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications. Lecture Notes in Computer Science, Vol. 7441. Springer, 38--45. DOI:http://dx.doi.org/10.1007/978-3-642-33275-3_4
136. Ireneusz Tobor, Patrick Reuter, and Christophe Schlick. 2006. Reconstructing multi-scale variational partition of unity implicit surfaces with attributes. Graphical Models 68, 1, 25--41. DOI:http://dx.doi.org/10.1016/j.gmod.2005.09.003
137. Graham M. Treece, Richard W. Prager, and Andrew H. Gee. 1999. Regularised marching tetrahedra: Improved iso-surface extraction. Computers and Graphics 23, 4, 583--598.
138. Frederic Triquet, Laurent Grisoni, Philippe Meseure, and Christophe Chaillou. 2003. Realtime visualization of implicit objects with contact control. In Proceedings of the International Conference on Computer Graphics and Interactive Techniques in Australia and South East Asia (GRAPHITE’03). http://www.anzgraph.org/graphite2003.
139. Frederic Triquet, Philippe Meseure, and Christophe Chaillou. 2001. Fast polygonization of implicit surfaces. In Proceedings of the 2001 WSCG International Conference (WSCG’01). 283--290. http://wscg.zcu.cz.
140. Greg Turk and James F. O’Brien. 1999. Shape transformation using variational implicit functions. In Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’99). ACM, New York, NY, 335--342. DOI:http://dx.doi.org/10.1145/311535.311580
141. Kees van Overveld and Brian Wyvill. 2004. Shrinkwrap: An efficient adaptive algorithm for triangulating an iso-surface. Visual Computer 20, 6, 362--379. DOI:http://dx.doi.org/10.1007/s00371-002-0197-4
142. Gokul Varadhan, Shankar Krishnan, Young J. Kim, and Dinesh Manocha. 2003. Feature-sensitive subdivision and isosurface reconstruction. In Proceedings of the 14th IEEE Visualization Conference (VIS’03). IEEE, Los Alamitos, CA, 14. DOI:http://dx.doi.org/10.1109/VISUAL.2003.1250360
143. Gokul Varadhan, Shankar Krishnan, Tvn Sriram, and Dinesh Manocha. 2004. Topology preserving surface extraction using adaptive subdivision. In Proceedings of the Eurographics/ACM SIGGRAPH Symposium on Geometry Processing (SGP’04). ACM, New York, NY, 235--244. DOI:http://dx.doi.org/10.1145/1057432.1057464
144. Gokul Varadhan, Shankar Krishnan, Liangjun Zhang, and Dinesh Manocha. 2006. Reliable implicit surface polygonization using visibility mapping. In Proceedings of the Eurographics Symposium on Geometry Processing. 211--221. DOI:http://dx.doi.org/10.2312/SGP/SGP06/211-221
145. Luiz Velho. 1996. Simple and efficient polygonization of implicit surfaces. Journal of Graphic Tools 1, 2, 5--24.
146. Luiz Velho, Luiz Henrique de Figueiredo, and Jonas Gomes. 1999. A unified approach for hierarchical adaptive tesselation of surfaces. ACM Transactions on Graphics 18, 4, 329--360. DOI:http://dx.doi.org/10.1145/337680.337717
147. Luiz Velho, Jonas Gomes, and Luiz Henrique de Figueiredo. 2002. Implicit Objects in Computer Graphics. Springer.
148. Jens Vorsatz, Christian Rössl, Leif Kobbelt, and Hans-Peter Seidel. 2001. Feature sensitive remeshing. Computer Graphics Forum 20, 3, 393--401.
149. Andrew P. Witkin and Paul S. Heckbert. 1994. Using particles to sample and control implicit surfaces. In Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH’94). ACM, New York, NY, 269--277. DOI:http://dx.doi.org/10.1145/192161.192227
150. Zoe Wood, Hugues Hoppe, Mathieu Desbrun, and Peter Schroder. 2002. Isosurface Topology Simplification. Technical Report MSR-TR-2002-28. Microsoft Research. http://www.labri.fr/publications/is/2004/RJTBS04.
151. Brian Wyvill, Eric Galin, and Andrew Guy. 1998. The blob tree, warping, blending and Boolean operations in an implicit surface modeling system. In Proceedings of Implicit Surfaces ’98.
152. Brian Wyvill and Kees van Overveld. 1996. Polygonization of implicit surfaces with constructive solid geometry. Journal of Shape Modelling 2, 4, 257--274.
153. Geoff Wyvill, Craig McPheeters, and Brian Wyvill. 1986. Data structure for soft objects. Visual Computer 2, 4, 227--234.
154. Yongjian Xi and Ye Duan. 2008. CAD and graphics: A novel region-growing based iso-surface extraction algorithm. Computers and Graphics 32, 6, 647--654. DOI:http://dx.doi.org/10.1016/j.cag.2008.09.007
155. Shuntaro Yamazaki, Kiwamu Kase, and Katsushi Ikeuchi. 2002. Non-manifold implicit surfaces based on discontinuous implicitization and polygonization. In Proceedings of the Geometric Modeling and Processing Conference. 138. DOI:http://dx.doi.org/10.1109/GMAP.2002.1027505
156. Yi Zhang, Xin Wang, and Xiao Jun Wu. 2006. Fast visualization algorithm for implicit surfaces. In Proceedings of the 16th International Conference on Artificial Reality and Telexistence (ICAT’06). 339--344. DOI:http://dx.doi.org/10.1109/ICAT.2006.62

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry