Shape complexity

The complexity of 3D shapes that are represented in digital form and processed in CAD/CAM/CAE, entertainment, biomedical, and other applications has increased considerably. Much research was focused on coping with or on reducing shape complexity. However, what exactly is shape complexity? We discuss several complexity measures and the corresponding complexity reduction techniques. Algebraic complexity measures the degree of polynomials needed to represent the shape exactly in its implicit or parametric form. Topological complexity measures the number of handles and components or the existence of non-manifold singularities, non-regularized components, holes or self-intersections. Morphological complexity measures smoothness and feature size. Combinatorial complexity measures the vertex count in polygonal meshes. Representational complexity measures the footprint and ease-of-use of a data structure, or the storage size of a compressed model. The latter three vary as a function of accuracy.

[1]  David Salomon,et al.  Data Compression: The Complete Reference , 2006 .

[2]  Jarek Rossignac.  Surface simplification and 3D geometry compression , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[3]  Martti Mäntylä,et al.  Introduction to Solid Modeling , 1988 .

[4]  Tosiyasu L. Kunii,et al.  Rendering trimmed implicit surfaces and curves , 2004, AFRIGRAPH '04.

[5]  Gabriel Taubin,et al.  Special Issue on Processing of Large Polygonal Meshes , 2002, Graph. Model..

[6]  M. F.,et al.  Bibliography , 1985, Experimental Gerontology.

[7]  Jarek Rossignac,et al.  Wrap&Zip decompression of the connectivity of triangle meshes compressed with Edgebreaker , 1999, Comput. Geom..

[8]  Marco Attene,et al.  SwingWrapper: Retiling triangle meshes for better edgebreaker compression , 2003, TOGS.

[9]  Jarek Rossignac.  Structured topological complexes: a feature-based API for non-manifold topologies , 1997, SMA '97.

[10]  N. Dyn,et al.  A butterfly subdivision scheme for surface interpolation with tension control , 1990, TOGS.

[11]  Peter Schröder,et al.  Normal meshes , 2000, SIGGRAPH.

[12]  Sylvain Lazard,et al.  Intersecting quadrics: an efficient and exact implementation , 2004, SCG '04.

[13]  Gabriel Taubin,et al.  Geometric compression through topological surgery , 1998, TOGS.

[14]  Aristides A. G. Requicha,et al.  Constructive non-regularized geometry , 1991, Comput. Aided Des..

[15]  Nicholas M. Patrikalakis,et al.  Topological and differential-equation methods for surface intersections , 1992, Comput. Aided Des..

[16]  Rémi Ronfard,et al.  Full‐range approximation of triangulated polyhedra. , 1996, Comput. Graph. Forum.

[17]  Charles T. Loop,et al.  Smooth spline surfaces over irregular meshes , 1994, SIGGRAPH.

[18]  Michael Garland,et al.  Surface simplification using quadric error metrics , 1997, SIGGRAPH.

[19]  Robert B. Tilove,et al.  Set Membership Classification: A Unified Approach to Geometric Intersection Problems , 1980, IEEE Transactions on Computers.

[20]  Jarek Rossignac,et al.  Active zones in CSG for accelerating boundary evaluation, redundancy elimination, interference detection, and shading algorithms , 1988, TOGS.

[21]  Jarek Rossignac,et al.  Mason: morphological simplification , 2005, Graph. Model..

[22]  P. S. Aleksandrov,et al.  Elementary concepts of topology , 1961 .

[23]  Robert B. Tilove.  A null-object detection algorithm for constructive solid geometry , 1984, CACM.

[24]  Jarek Rossignac,et al.  Piecewise Regular Meshes: Construction and Compression , 2002, Graph. Model..

[25]  Paolo Cignoni,et al.  Metro: Measuring Error on Simplified Surfaces , 1998, Comput. Graph. Forum.

[26]  Jarek Rossignac,et al.  Matchmaker: manifold BReps for non-manifold r-sets , 1999, SMA '99.

[27]  Jarek Rossignac,et al.  Edgebreaker on a Corner Table: A Simple Technique for Representing and Compressing Triangulated Surfaces , 2003 .

[28]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods , 1999 .

[29]  Ron Goldman,et al.  Geometric Algorithms for Detecting and Calculating All Conic Sections in the Intersection of Any 2 Natural Quadric Surfaces , 1995, CVGIP Graph. Model. Image Process..

[30]  Jarek Rossignac,et al.  Through the Cracks of the Solid Modeling Milestone , 1994 .

[31]  Henning Biermann,et al.  Approximate Boolean operations on free-form solids , 2001, SIGGRAPH.

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

[33]  Peter Schröder,et al.  Interpolating Subdivision for meshes with arbitrary topology , 1996, SIGGRAPH.

[34]  A. Safonova,et al.  3D Compression Made Simple: Edgebreaker on a Corner-Table , 2001 .

[35]  Jung Hong Chuang.  Level of Detail for 3D Graphics , 2002 .

[36]  David P. Dobkin,et al.  MAPS: multiresolution adaptive parameterization of surfaces , 1998, SIGGRAPH.

[37]  Aristides A. G. Requicha,et al.  Geometric Modelling Systems for mechanical design and manufacturing , 1978, ACM '78.

[38]  Jarek Rossignac,et al.  Edgebreaker: Connectivity Compression for Triangle Meshes , 1999, IEEE Trans. Vis. Comput. Graph..

[39]  Joe Warren,et al.  Subdivision Methods for Geometric Design: A Constructive Approach , 2001 .

[40]  Jarek Rossignac,et al.  Optimal Iso-Surfaces , 2004 .

[41]  Andrei Khodakovsky,et al.  Progressive geometry compression , 2000, SIGGRAPH.

[42]  Gabriel Taubin,et al.  Estimating the tensor of curvature of a surface from a polyhedral approximation , 1995, Proceedings of IEEE International Conference on Computer Vision.

[43]  Jarek Rossignac,et al.  Relationship between s-bounds and active zones in constructive solid geometry , 1989 .

[44]  Martti Mäntylä,et al.  Boolean operations of 2-manifolds through vertex neighborhood classification , 1986, TOGS.

[45]  Jarek Rossignac,et al.  Topologically exact evaluation of polyhedra defined in CSG with loose primitives , 1996, Comput. Graph. Forum.

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

[47]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[48]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[49]  Peter Liepa,et al.  Filling Holes in Meshes , 2003, Symposium on Geometry Processing.

[50]  James Vanderhyde,et al.  Extraction of topologically simple isosurfaces from volume datasets , 2003, IEEE Visualization, 2003. VIS 2003..

[51]  Jarek Rossignac,et al.  Simplifying interactive design of solid models: A hypertext approach , 2005, The Visual Computer.

[52]  Jarek Rossignac,et al.  An Edgebreaker-Based Efficient Compression Scheme for Connectivity of Regular Meshes , 2000, CCCG.

[53]  Ignacio Llamas,et al.  Twister: a space-warp operator for the two-handed editing of 3D shapes , 2003, ACM Trans. Graph..

[54]  Tamal K. Dey,et al.  Provable surface reconstruction from noisy samples , 2006, Comput. Geom..

[55]  Josef Hoschek,et al.  Handbook of Computer Aided Geometric Design , 2002 .

[56]  Volker Coors,et al.  Delphi Encoding: Improving Edgebreaker 3D Compression by Geometry-based Connectivity Prediction , 2003 .

[57]  Jarek Rossignac,et al.  Education-driven research in CAD , 2004, Comput. Aided Des..

[58]  Jarek Rossignac,et al.  Tightening: curvature-limiting morphological simplification , 2005, SPM '05.

[59]  Jarek Rossignac,et al.  Guaranteed 3.67v bit encoding of planar triangle graphs , 1999, CCCG.

[60]  T. M. Murali,et al.  Consistent solid and boundary representations from arbitrary polygonal data , 1997, SI3D.

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

[62]  A.A.G. Requicha,et al.  Boolean operations in solid modeling: Boundary evaluation and merging algorithms , 1985, Proceedings of the IEEE.

[63]  Alexei Sourin,et al.  Function representation in geometric modeling: concepts, implementation and applications , 1995, The Visual Computer.

[64]  Craig Gotsman,et al.  Triangle Mesh Compression , 1998, Graphics Interface.

[65]  Jarek Rossignac,et al.  Multi-resolution 3D approximations for rendering complex scenes , 1993, Modeling in Computer Graphics.