Computing and Rendering Point Set Surfaces

We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an out-of-core technique that can handle any point set. We show that the approximation error is bounded and present tools to increase or decrease the density of the points, thus allowing an adjustment of the spacing among the points to control the error. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.

[1]  Gabriel Taubin,et al.  Converting sets of polygons to manifold surfaces by cutting and stitching , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[2]  Marc Levoy,et al.  The Use of Points as a Display Primitive , 2000 .

[3]  Marc Levoy,et al.  Streaming QSplat: a viewer for networked visualization of large, dense models , 2001, I3D '01.

[4]  Tony DeRose,et al.  Piecewise smooth surface reconstruction , 1994, SIGGRAPH.

[5]  Paolo Cignoni,et al.  A comparison of mesh simplification algorithms , 1998, Comput. Graph..

[6]  Martin Rutishauser,et al.  Merging range images of arbitrarily shaped objects , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[7]  Arie E. Kaufman,et al.  Volume sculpting , 1995, I3D '95.

[8]  Mark Meyer,et al.  Implicit fairing of irregular meshes using diffusion and curvature flow , 1999, SIGGRAPH.

[9]  Marc Levoy,et al.  A volumetric method for building complex models from range images , 1996, SIGGRAPH.

[10]  James Arvo,et al.  Creating generative models from range images , 1999, SIGGRAPH.

[11]  Tony DeRose,et al.  Mesh optimization , 1993, SIGGRAPH.

[12]  Baoquan Chen,et al.  POP: a hybrid point and polygon rendering system for large data , 2001, Proceedings Visualization, 2001. VIS '01..

[13]  TaubinGabriel,et al.  The Ball-Pivoting Algorithm for Surface Reconstruction , 1999 .

[14]  Marc Alexa,et al.  Point set surfaces , 2001, Proceedings Visualization, 2001. VIS '01..

[15]  Leila De Floriani,et al.  Efficient implementation of multi-triangulations , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[16]  Greg Turk,et al.  Re-tiling polygonal surfaces , 1992, SIGGRAPH.

[17]  Hans-Peter Seidel,et al.  Multiresolution Shape Deformations for Meshes with Dynamic Vertex Connectivity , 2000, Comput. Graph. Forum.

[18]  Tony DeRose,et al.  Surface reconstruction from unorganized points , 1992, SIGGRAPH.

[19]  A. Ardeshir Goshtasby,et al.  Surface fitting to scattered data by a sum of Gaussians , 1993, Comput. Aided Geom. Des..

[20]  Enrico Puppo,et al.  Building and traversing a surface at variable resolution , 1997 .

[21]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[22]  Henrik Wann Jensen,et al.  Ray Tracing Point Sampled Geometry , 2000, Rendering Techniques.

[23]  Patricia Crossno,et al.  Isosurface extraction using particle systems , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[24]  Denis Laurendeau,et al.  Surface modeling from dynamic integration of multiple range views , 1992, [1992] Proceedings. 11th IAPR International Conference on Pattern Recognition.

[25]  Luiz Velho,et al.  Physically-based methods for polygonization of implicit surfaces , 1992 .

[26]  W. Press,et al.  Numerical Recipes in C++: The Art of Scientific Computing (2nd edn)1 Numerical Recipes Example Book (C++) (2nd edn)2 Numerical Recipes Multi-Language Code CD ROM with LINUX or UNIX Single-Screen License Revised Version3 , 2003 .

[27]  Marc Levoy,et al.  The digital Michelangelo project , 1999, Second International Conference on 3-D Digital Imaging and Modeling (Cat. No.PR00062).

[28]  Leila De Floriani,et al.  Building and traversing a surface at variable resolution , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[29]  D. Levin,et al.  Mesh-Independent Surface Interpolation , 2004 .

[30]  Denis Laurendeau,et al.  Scene reconstruction and description: geometric primitive extraction from multiple viewed scattered data , 1993, Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[31]  Marc Levoy,et al.  Zippered polygon meshes from range images , 1994, SIGGRAPH.

[32]  Marshall W. Bern,et al.  A new Voronoi-based surface reconstruction algorithm , 1998, SIGGRAPH.

[33]  Paul S. Heckbert,et al.  Using particles to sample and control implicit surfaces , 1994, SIGGRAPH.

[34]  Daniel G. Aliaga,et al.  Hybrid simplification: combining multi-resolution polygon and point rendering , 2001, Proceedings Visualization, 2001. VIS '01..

[35]  Matthias Zwicker,et al.  Surfels: surface elements as rendering primitives , 2000, SIGGRAPH.

[36]  Lars Linsen,et al.  Point cloud representation , 2001 .

[37]  Amitabh Varshney,et al.  Differential Point Rendering , 2001, Rendering Techniques.

[38]  Meenakshisundaram Gopi,et al.  Surface Reconstruction based on Lower Dimensional Localized Delaunay Triangulation , 2000, Comput. Graph. Forum.

[39]  David Levin,et al.  The approximation power of moving least-squares , 1998, Math. Comput..

[40]  H. Barlow.  Vision: A computational investigation into the human representation and processing of visual information: David Marr. San Francisco: W. H. Freeman, 1982. pp. xvi + 397 , 1983 .

[41]  Gabriel Taubin,et al.  The ball-pivoting algorithm for surface reconstruction , 1999, IEEE Transactions on Visualization and Computer Graphics.

[42]  Voicu Popescu,et al.  Real-Time Rendering of Real World Environments , 1999, Rendering Techniques.

[43]  Dinesh Manocha,et al.  Hierarchical back-face computation , 1996, Comput. Graph..

[44]  Marc Levoy,et al.  QSplat: a multiresolution point rendering system for large meshes , 2000, SIGGRAPH.

[45]  William J. Dally,et al.  Point Sample Rendering , 1998, Rendering Techniques.

[46]  Jean-Daniel Boissonnat,et al.  Geometric structures for three-dimensional shape representation , 1984, TOGS.

[47]  Marc Levoy,et al.  The digital Michelangelo project: 3D scanning of large statues , 2000, SIGGRAPH.

[48]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[49]  Katsushi Ikeuchi,et al.  Consensus surfaces for modeling 3D objects from multiple range images , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[50]  Vaughan R. Pratt,et al.  Direct least-squares fitting of algebraic surfaces , 1987, SIGGRAPH.

[51]  David B. Cooper,et al.  3L fitting of higher degree implicit polynomials , 1996, Proceedings Third IEEE Workshop on Applications of Computer Vision. WACV'96.

[52]  Chandrajit L. Bajaj,et al.  Automatic reconstruction of surfaces and scalar fields from 3D scans , 1995, SIGGRAPH.

[53]  Marc Levoy,et al.  Fitting smooth surfaces to dense polygon meshes , 1996, SIGGRAPH.

[54]  Gabriel Taubin,et al.  A signal processing approach to fair surface design , 1995, SIGGRAPH.

[55]  Daniel Cohen-Or,et al.  Volume graphics , 1993, Computer.

[56]  Paul J. Besl,et al.  A Method for Registration of 3-D Shapes , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[57]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[58]  In-Kwon Lee,et al.  Curve reconstruction from unorganized points , 2000, Comput. Aided Geom. Des..

[59]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[60]  Arie E. Kaufman,et al.  Alias-Free Voxelization of Geometric Objects , 1999, IEEE Trans. Vis. Comput. Graph..