Efficient and small representation of line arrangements with applications

This paper addresses the problem of lossy compression of arrangements. Given an arrangement of $n$ lines in the plane, we show how to construct another arrangement consisting of many fewer lines. We give theoretical and empirical bounds to demonstrate the tradeoffs between the size of the new arrangement and the error from lossiness. For the specific application of computing discrepancies of point sets, we demonstrate that speedups by factors of several hundred are possible while introducing small errors. This research has been enabled by various visualization techniques and is accompanied by a video.

[1]  J. Latombe,et al.  Probabilistic Roadm Aps for Path Planning in High-dimensional Connguration Spaces , 1997 .

[2]  Micha Sharir,et al.  On the number of views of polyhedral terrains , 1993, CCCG.

[3]  Valerio Pascucci,et al.  Single Resolution Compression of Arbitrary Triangular Meshes with Properties , 1999, Data Compression Conference.

[4]  Joan L. Mitchell,et al.  JPEG: Still Image Data Compression Standard , 1992 .

[5]  Steven Skiena,et al.  Ranger: a tool for nearest neighbor search in high dimensions , 1993, SCG '93.

[6]  Tony DeRose,et al.  Multiresolution analysis of arbitrary meshes , 1995, SIGGRAPH.

[7]  Mark Nelson,et al.  The Data Compression Book , 2009 .

[8]  David P. Dobkin,et al.  Visualization of Geometric Algorithms , 1995, IEEE Trans. Vis. Comput. Graph..

[9]  Daniel P. Huttenlocher,et al.  Computing the minimum Hausdorff distance for point sets under translation , 1990, SCG '90.

[10]  Dan Halperin.  Robot motion planning and the single cell problem in arrangements , 1994, J. Intell. Robotic Syst..

[11]  Y. Meyer,et al.  Wavelets and Filter Banks , 1991 .

[12]  Bernard Chazelle.  Geometric discrepancy revisited , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[13]  David P. Dobkin,et al.  Searching for empty convex polygons , 1988, SCG '88.

[14]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[15]  Raimund Seidel,et al.  Efficiently Computing and Representing Aspect Graphs of Polyhedral Objects , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Michael Deering,et al.  Geometry compression , 1995, SIGGRAPH.

[17]  Peter Shirley,et al.  Discrepancy as a Quality Measure for Sample Distributions , 1991, Eurographics.

[18]  H. Coxeter,et al.  Geometry Revisited (New Mathematical Library) , 1967 .

[19]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[20]  Martin Isenburg.  Triangle Fixer: Edge-based Connectivity Compression , 2000, EuroCG.

[21]  Paul S. Heckbert,et al.  Survey of Polygonal Surface Simplification Algorithms , 1997 .

[22]  Craig Gotsman,et al.  Spectral compression of mesh geometry , 2000, EuroCG.

[23]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[24]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[25]  David P. Dobkin,et al.  Small representation of line arrangements , 2001, SCG '01.

[26]  GigusZiv,et al.  Efficiently Computing and Representing Aspect Graphs of Polyhedral Objects , 1991 .

[27]  Micha Sharir,et al.  Arrangements and Their Applications , 2000, Handbook of Computational Geometry.

[28]  H. Alt.  Nearest Neighbor Search in High Dimensions , .

[29]  Mark H. Overmars,et al.  Motion planning amidst fat obstacles (extended abstract) , 1994, SCG '94.

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

[31]  David Eppstein,et al.  Computing the discrepancy , 1993, SCG '93.

[32]  Lydia E. Kavraki,et al.  A Random Sampling Scheme for Path Planning , 1997, Int. J. Robotics Res..

[33]  Leonidas J. Guibas,et al.  Polyhedral Assembly Partitioning Using Maximally Covered Cells in Arrangements of Convex Polytopes , 1998, Int. J. Comput. Geom. Appl..

[34]  Jean-Claude Latombe,et al.  A General Framework for Assembly Planning: The Motion Space Approach , 1998, SCG '98.

[35]  B. Faverjon,et al.  Probabilistic Roadmaps for Path Planning in High-Dimensional Con(cid:12)guration Spaces , 1996 .

[36]  Mark H. Overmars,et al.  Spheres, molecules, and hidden surface removal , 1994, SCG '94.

[37]  Jean-Claude Latombe,et al.  Motion Planning with Uncertainty: A Landmark Approach , 1995, Artif. Intell..

[38]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

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

[40]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[41]  Jelena Kovacevic,et al.  Wavelets and Subband Coding , 2013, Prentice Hall Signal Processing Series.

[42]  Tony DeRose,et al.  Multiresolution analysis for surfaces of arbitrary topological type , 1997, TOGS.

[43]  Peter E. Hart,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[44]  Lydia E. Kavraki,et al.  Probabilistic roadmaps for path planning in high-dimensional configuration spaces , 1996, IEEE Trans. Robotics Autom..

[45]  A. Frank van der Stappen,et al.  Motion planning amidst fat obstacles , 1993 .

[46]  Michael McKenna.  Worst-case optimal hidden-surface removal , 1987, TOGS.

[47]  David P. Dobkin,et al.  Random-edge discrepancy of supersampling patterns , 1993 .

[48]  J. O'Rourke.  Art gallery theorems and algorithms , 1987 .