# REVERSE SUBDIVISION FOR OPTIMIZING VISIBILITY TESTS

### Troy F. Alderson, Faramarz F. Samavati

#### Abstract

Certain applications require knowledge of whether two entities are visible to each other over a terrain, determined using a line-of-sight computation. Several fast algorithms exist for terrain line-of-sight computations. However, performing numerous line-of-sight computations, particularly over a large terrain data set, can be highly resource-intensive (in run time and/or memory). Methods from the field of terrain simplification can be used to reduce the resource impact of the visibility algorithms. To take advantage of the especially fast algorithms that exist for regular terrain models, we introduce regularity-preserving terrain simplification methods based on reverse subdivision, including a novel reverse subdivision algorithm designed to maximize visibility test accuracy, and compared the resulting visibility test output to several terrain simplification methods. Additionally, the positions of the entities after simplification can have a significant impact on the visibility test results. Hence, we have experimented with different functions that change the positions of the test points in an attempt to maximize visibility test accuracy after simplification.

#### References

- Andrade, M. V. A., Magalha˜es, S. V. G., Magalha˜es, M. A., Franklin, W. R., and Cutler, B. M. (2011). Efficient viewshed computation on terrain in external memory. Geoinformatica, 15(2):381-397.
- Bartels, R. H. and Samavati, F. F. (2000). Reversing subdivision rules: Local linear conditions and observations on inner products. Journal of Computational and Applied Mathematics, 119(1-2):29-67.
- Ben-Moshe, B., Mitchell, J. S. B., Katz, M. J., and Nir, Y. (2002). Visibility preserving terrain simplification: An experimental study. In Proceedings of SCG 2002.
- Bresenham, J. E. (1965). Algorithm for computer control of a digital plotter. IBM Systems Journal, 4(1):25-30.
- Chaikin, G. M. (1974). An algorithm for high-speed curve generation. Computer Graphics and Image Processing, 3(4):346-349.
- De Floriani, L. and Magillo, P. (1993). Algorithms for visibility computation on digital terrain models. In Proceedings of the 1993 ACM/SIGAPP Symposium on Applied Computing: States of the Art and Practice, SAC 7893. ACM, New York, NY, USA.
- Douglas, D. H. and Peucker, T. K. (1973). Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. The Canadian Cartographer, 10(2):112-122.
- Duvenhage, B. (2009). Using an implicit min/max kd-tree for doing efficient terrain line of sight calculations. In Proceedings of AFRIGRAPH 2009.
- Dyn, N., Levin, D., and Gregory, J. A. (1987). A 4- point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design, 4(4):257- 268.
- Garland, M. (1999). Quadric-Based Polygonal Surface Approximation. PhD thesis, Carnegie Mellon University.
- Garland, M. and Heckbert, P. S. (1995). Fast polygonal approximation of terrains and height fields. Technical Report CMU-CS-95-181, Carnegie Mellon University.
- Garland, M. and Heckbert, P. S. (1997). Surface simplification using quadric error metrics. In Proceedings of SIGGRAPH 1997.
- Heckbert, P. S. and Garland, M. (1997). Survey of polygonal surface simplification algorithms.
- Losasso, F. and Hoppe, H. (2004). Geometry clipmaps: Terrain rendering using nested regular grids. In ACM SIGGRAPH 2004 Papers.
- Prusinkiewicz, P., Samavati, F. F., Smith, C., and Karwowski, R. (2003). L-system description of subdivision curves. International Journal of Shape Modeling, 9(1):41-59.
- Samavati, F. F. and Bartels, R. H. (1999). Multiresolution curve and surface representation: Reversing subdivision rules by least-squares data fitting. Computer Graphics Forum, 18:97-120.
- Samavati, F. F., Bartels, R. H., and Olsen, L. (2007). Local b-spline multiresolution with examples in iris synthesis and volumetric rendering. In Image Pattern Recognition: Synthesis and Analysis in Biometrics, volume 67 of Series in Machine Perception and Artificial Intelligence, pages 65-102. World Scientific Publishing.
- Seixas, R. d. B., Mediano, M. R., and Gattass, M. (1999). Efficient line-of-sight algorithms for real terrain data. In Proceedings of SPOLM 7899.
- Silva, C. T. and Mitchell, J. S. B. (1998). Greedy cuts: An advancing front terrain triangulation algorithm. In Proceedings of GIS 1998.
- Silva, C. T., Mitchell, J. S. B., and Kaufman, A. E. (1995). Automatic generation of triangular irregular networks using greedy cuts. In Proceedings of the IEEE Conference on Visualization 1995, pages 201-208, 453.

#### Paper Citation

#### in Harvard Style

F. Alderson T. and F. Samavati F. (2012). **REVERSE SUBDIVISION FOR OPTIMIZING VISIBILITY TESTS** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)* ISBN 978-989-8565-02-0, pages 143-150. DOI: 10.5220/0003851501430150

#### in Bibtex Style

@conference{grapp12,

author={Troy F. Alderson and Faramarz F. Samavati},

title={REVERSE SUBDIVISION FOR OPTIMIZING VISIBILITY TESTS},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)},

year={2012},

pages={143-150},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003851501430150},

isbn={978-989-8565-02-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications and International Conference on Information Visualization Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2012)

TI - REVERSE SUBDIVISION FOR OPTIMIZING VISIBILITY TESTS

SN - 978-989-8565-02-0

AU - F. Alderson T.

AU - F. Samavati F.

PY - 2012

SP - 143

EP - 150

DO - 10.5220/0003851501430150