Scene Simpli ® cation TOPOLOGY PRESERVING DATA SIMPLIFICATION WITH ERROR BOUNDS

ÐMany approaches to simpli®cation of triangulated terrains and surfaces have been proposed which permit bounds on the error introduced. A few algorithms additionally bound errors in auxiliary functions de®ned over the triangulation. We present an approach to simpli®cation of scalar ®elds over unstructured grids which preserves the topology of functions de®ned over the triangulation, in addition to bounding of the errors. The topology of a 2D scalar ®eld is de®ned by critical points (local maxima, local minima, saddle points), in addition to integral curves between them, which together segment the ®eld into regions which vary monotonically. By preserving this shape description, we guarantee that isocontours of the scalar function maintain the correct topology in the simpli®ed model. Methods for topology preserving simpli®cation by both point-insertion (re®nement) and point-deletion (coarsening) are presented and compared. # 1998 Elsevier Science Ltd. All rights reserved

[1]  Bernd Hamann,et al.  A data reduction scheme for triangulated surfaces , 1994, Comput. Aided Geom. Des..

[2]  Christopher B. Jones,et al.  A Multiresolution topographic surface database , 1992, Int. J. Geogr. Inf. Sci..

[3]  Arie E. Kaufman,et al.  Voxel based object simplification , 1995, Proceedings Visualization '95.

[4]  Mark Gahegan,et al.  Feature-Based Derivation of Drainage Networks , 1993, Int. J. Geogr. Inf. Sci..

[5]  Russell H. Taylor,et al.  Superfaces: polygonal mesh simplification with bounded error , 1996, IEEE Computer Graphics and Applications.

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

[7]  A. Guéziec.  Surface simplification inside a tolerance volume , 1996 .

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

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

[10]  Leila De Floriani,et al.  A hierarchical structure for surface approximation , 1984, Comput. Graph..

[11]  E. J. Stollnitz,et al.  Wavelets for Computer Graphics : A Primer , 1994 .

[12]  Victor J. D. Tsai,et al.  Delaunay Triangulations in TIN Creation: An Overview and a Linear-Time Algorithm , 1993, Int. J. Geogr. Inf. Sci..

[13]  Larry S. Davis,et al.  Parallel Terrain Triangulation , 1994, Int. J. Geogr. Inf. Sci..

[14]  Lambertus Hesselink,et al.  Automated analysis of fluid flow topology , 1989, Photonics West - Lasers and Applications in Science and Engineering.

[15]  Al Globus,et al.  A tool for visualizing the topology of three-dimensional vector fields , 1991, Proceeding Visualization '91.

[16]  Victoria Interrante,et al.  Enhancing transparent skin surfaces with ridge and valley lines , 1995, Proceedings Visualization '95.

[17]  Koji Koyamada,et al.  Automatic Isosurface Propagation Using an Extrema Graph and Sorted Boundary Cell Lists , 1995, IEEE Trans. Vis. Comput. Graph..

[18]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[19]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  H. B. Griffiths.  CRITICAL POINT THEORY IN GLOBAL ANALYSIS AND DIFFERENTIAL TOPOLOGY , 1971 .

[21]  William H. Press,et al.  Numerical Recipes in Fortran 77: The Art of Scientific Computing 2nd Editionn - Volume 1 of Fortran Numerical Recipes , 1992 .

[22]  William E. Lorensen,et al.  Decimation of triangle meshes , 1992, SIGGRAPH.

[23]  James J. Little,et al.  Automatic extraction of Irregular Network digital terrain models , 1979, SIGGRAPH.

[24]  Hugues Hoppe,et al.  Progressive meshes , 1996, SIGGRAPH.

[25]  Chandrajit L. Bajaj,et al.  Decimation of 2D Scalar Data with Error Control , 1995 .

[26]  Valerio Pascucci,et al.  Visualization of scalar topology for structural enhancement , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[27]  Charles D. Hansen,et al.  Geometric optimization , 1993, Proceedings Visualization '93.

[28]  Ronald A. DeVore,et al.  Surface compression , 1992, Comput. Aided Geom. Des..

[29]  B. Lucier.  Wavelets and image compression , 1992 .

[30]  Shigeru Muraki,et al.  Approximation and rendering of volume data using wavelet transforms , 1992, Proceedings Visualization '92.

[31]  Joseph S. B. Mitchell,et al.  Automatic generation of triangular irregular networks using greedy cuts , 1995, Proceedings Visualization '95.

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

[33]  Chandrajit L. Bajaj,et al.  Error-bounded reduction of triangle meshes with multivariate data , 1996, Electronic Imaging.

[34]  R. Bader.  Atoms in molecules , 1990 .

[35]  Ronald A. DeVore,et al.  Image compression through wavelet transform coding , 1992, IEEE Trans. Inf. Theory.

[36]  Leila De Floriani,et al.  Delaunay-based representation of surfaces defined over arbitrarily shaped domains , 1985, Comput. Vis. Graph. Image Process..