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Fast Horizon Computation at All Points of a Terrain With Visibility and Shading Applications
January-March 1998 (vol. 4 no. 1)
pp. 82-93

Abstract—A terrain is most often represented with a digital elevation map consisting of a set of sample points from the terrain surface. This paper presents a fast and practical algorithm to compute the horizon, or skyline, at all sample points of a terrain. The horizons are useful in a number of applications, including the rendering of self-shadowing displacement maps, visibility culling for faster flight simulation, and rendering of cartographic data. Experimental and theoretical results are presented which show that the algorithm is more accurate that previous algorithms and is faster than previous algorithms in terrains of more than 100,000 sample points.

[1] P. Robertson, "Spatial Transformations for Rapid Scan-Line Surface Shadowing," IEEE Computer Graphics and Applications, pp. 30-38 Mar. 1989.
[2] G.S.P. Miller, "The Definition and Rendering of Terrain Maps," Computer Graphics (Proc. SIGGRAPH '86), vol. 20, pp. 39-48, Aug. 1986.
[3] N.L. Max, "Atmospheric Illumination and Shadows," Computer Graphics (Proc. SIGGRAPH '86), vol. 20, pp. 117-124, Aug. 1986.
[4] V. Ramachandran, "Perception of Shape From Shading," Nature, vol. 331, no. 6,152, pp. 163-165, 1988.
[5] M.S. Langer and H.H. Bülthoff, "Do Humans Perceive Judge Shape From Shading Better on Sunny Days or on Cloudy Days?" NECI Technical Report 97-130, NEC Research Inst., 1997.
[6] K. Kaneda, F. Kato, E. Nakamae, T. Nishita, H. Tanaka, and T. Noguchi, "Three Dimensional Terrain Modeling and Display for Environmental Assessment," Computer Graphics (Proc. SIGGRAPH '89), vol. 23, pp. 207-214, July 1989.
[7] S. Coquillart and M. Gangnet, "Shaded Display of Digital Maps," IEEE Computer Graphics and Applications, vol. 4, no. 7, pp. 35-42, July 1984.
[8] J. Kajiya, “The Rendering Equation,” Computer Graphics, pp. 143-150, 1986.
[9] G. Nagy, "Terrain Visibility," Computers and Graphics, vol. 18, no. 6, pp. 763-773, 1994.
[10] L. De Floriani and P. Magillo, "Visibility Algorithms on Triangulated Terrain Models," Int'l J. Geographic Information Systems, vol. 8, no. 1, pp. 13-41, 1994.
[11] A.J. Stewart, "Hierarchical Visibility in Terrains," Proc. Eurographics Rendering Workshop, pp. 217-228, June 1997.
[12] N.L. Max, "Shadows for Bump-Mapped Surfaces," Advanced Computer Graphics (Proc. Computer Graphics, Tokyo '86), T.L. Kunii, ed., pp. 145-156. Springer-Verlag, 1986.
[13] N.L. Max, "Horizon Mapping: Shadows for Bump-Mapped Surfaces," The Visual Computer, vol. 4, no. 2, pp. 109-117, July 1988.
[14] B. Cabral, N. Max, and R. Springmeyer, “Bi-Directional Reflection from Surface Bump Maps,” Computer Graphics, vol. 21, no. 4, pp. 273-282, 1987.
[15] D. Cohen-Or and A. Shaked, "Visibility and Dead-Zones in Digital Terrain Maps," Proc. Computer Graphics Forum (Eurographics '95), vol. 14, no. 3, 1995.
[16] C-H. Lee and Y.G. Shin, "An Efficient Ray Tracing Method for Terrain Rendering," Proc. Pacific Graphics '95, Aug. 1995.
[17] M.J. Atallah, "Dynamic Computational Geometry," Proc. 24th Ann. IEEE Symp. Foundations of Computer Science, pp. 92-99, 1983.
[18] J. Hershberger, "Finding the Upper Envelope of n Line Segments in O(n log n) Time," Information Processing Letters, vol. 33, pp. 169-174, 1989.
[19] L. De Floriani and E. Puppo, "Constrained Delaunay Triangulation for Multiresolution Surface Description," Proc. Ninth IEEE Int'l Conf. Pattern Recognition, pp. 566-569, 1988.
[20] L. Scarlatos and T. Pavlidis, "Hierarchical Triangulation Using Terrain Features," Proc. First 1990 IEEE Conf. Visualization (Visualization '90), pp. 168-175, 1990.
[21] M. de Berg and K.T.G. Dobrint, “On Levels of Detail in Terrains,” Proc. ACM Symp. Computational Geometry, pp. C26-C27, June 1995.
[22] L. De Floriani and P. Magillo, "Horizon Computation on a Hierarchical Triangulated Terrain Model," The Visual Computer, vol. 11, pp. 134-149, 1995.
[23] R. Cole and M. Sharir, "Visibility Problems for Polyhedral Terrains," J. Symbolic Computing, vol. 7, pp. 11-30, 1989.
[24] M. Bern, D. Dobkin, D. Eppstein, and R. Grossman, "Visibility With a Moving Point of View," Algorithmica, vol. 11, pp. 360-378, 1994.
[25] F.P. Preparata and M.I. Shamos, Computational Geometry. Springer-Verlag, 1985.
[26] M.H. Overmars and J. van Leeuwen, "Maintenance of Configurations in the Plane," J. Computer Systems Science, vol. 23, pp. 166-204, 1981.
[27] J. O'Rourke, Computational Geometry in C. Cambridge Univ. Press, 1993.
[28] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[29] A.J. Stewart and M.S. Langer, "Towards Accurate Recovery of Shape From Shading Under Diffuse Lighting," IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 19, no. 9, pp. 1,020-1,025, Sept. 1997.

Index Terms:
Terrain, digital elevation map, horizon, skyline, visibility, shadows, rendering, GIS.
A. James Stewart, "Fast Horizon Computation at All Points of a Terrain With Visibility and Shading Applications," IEEE Transactions on Visualization and Computer Graphics, vol. 4, no. 1, pp. 82-93, Jan.-March 1998, doi:10.1109/2945.675656
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