Computational geometry and visualization: problems at the interface

In this paper we survey certain geometric problems that arise in volume visualization and discuss how techniques from computational geometry can be applied to them. Such problems include depth-sorting of polyhedral complexes, point-location, ray shooting and tracing, and others. We give a few worked-out illustrative examples, as well as references to the extant literature.

[1]  Bernard Chazelle,et al.  How to Search in History , 1983, Inf. Control..

[2]  J. Stolfi Primitives for computational geometry , 1988 .

[3]  Michael P. Garrity Raytracing irregular volume data , 1990, SIGGRAPH 1990.

[4]  Henry Neeman A decomposition algorithm for visualizing irregular grids , 1990, SIGGRAPH 1990.

[5]  Wolfgang Leister,et al.  Ray Tracing Algorithms — Theory and Practice , 1988 .

[6]  P. Hanrahan,et al.  Area and volume coherence for efficient visualization of 3D scalar functions , 1990, SIGGRAPH 1990.

[7]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[8]  A. R. Forrest II. Current developments in the design and production of three- dimensional curved objects - Computational geometry , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  Robert E. Tarjan,et al.  Making data structures persistent , 1986, STOC '86.

[10]  Arie E. Kaufman,et al.  Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes , 1987, SIGGRAPH.

[11]  Roberto Tamassia,et al.  Efficient Spatial Point Location (Extended Abstract) , 1989, WADS.

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

[13]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[14]  E. Brisson,et al.  Representation ofd-dimensional geometric objects , 1990 .

[15]  Micha Sharir,et al.  Output-sensitive hidden surface removal , 1989, 30th Annual Symposium on Foundations of Computer Science.

[16]  David P. Dobkin,et al.  Contour tracing by piecewise linear approximations , 1990, TOGS.

[17]  Michael T. Goodrich,et al.  Dynamic trees and dynamic point location , 1991, STOC '91.

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

[19]  Pankaj K. Agarwal,et al.  A deterministic algorithm for partitioning arrangements of lines and its application , 1989, SCG '89.

[20]  Herbert Edelsbrunner,et al.  Halfplanar Range Search in Linear Space and O(n^(0.695)) Query Time , 1986, Inf. Process. Lett..

[21]  Roberto Tamassia,et al.  Fully Dynamic Point Location in a Monotone Subdivision , 1989, SIAM J. Comput..

[22]  Andrew S. Glassner,et al.  An introduction to ray tracing , 1989 .

[23]  Herbert Edelsbrunner,et al.  An acyclicity theorem for cell complexes in d dimensions , 1989, SCG '89.

[24]  David P. Dobkin,et al.  Primitives for the manipulation of three-dimensional subdivisions , 1987, SCG '87.

[25]  Emo Welzl,et al.  Partition trees for triangle counting and other range searching problems , 1988, SCG '88.

[26]  Lance Williams,et al.  Pyramidal parametrics , 1983, SIGGRAPH.

[27]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[28]  Marc Levoy,et al.  Display of surfaces from volume data , 1988, IEEE Computer Graphics and Applications.

[29]  Azriel Rosenfeld,et al.  Three-Dimensional Digital Topology , 1981, Inf. Control..

[30]  Lambertus Hesselink,et al.  Surface representations of two- and three-dimensional fluid flow topology , 1990, Proceedings of the First IEEE Conference on Visualization: Visualization `90.

[31]  R. Schmacher,et al.  Study for Applying Computer-Generated Images to Visual Simulation: (510842009-001) , 1969 .

[32]  Tom Duff,et al.  Compositing digital images , 1984, SIGGRAPH.

[33]  G. Herman,et al.  Three-dimensional display of human organs from computed tomograms , 1979 .

[34]  Pankaj K. Agarwal,et al.  Ray shooting and other applications of spanning trees with low stabbing number , 1992, SCG '89.

[35]  J. Wilhelms,et al.  Topological considerations in isosurface generation extended abstract , 1990, SIGGRAPH 1990.

[36]  Don Speray,et al.  Volume probes: interactive data exploration on arbitrary grids , 1990, SIGGRAPH 1990.

[37]  Leonidas J. Guibas,et al.  Counting and cutting cycles of lines and rods in space , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[38]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[39]  Henry Fuchs,et al.  On visible surface generation by a priori tree structures , 1980, SIGGRAPH '80.