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Properties of a Level Set Algorithm for the Visibility Problems

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Abstract

We study an implicit visibility formulation and show that the corresponding closed form formula satisfies a dynamic programming principle, and is the viscosity solution of a Hamilton-Jacobi type equation involving jump discontinuities in the Hamiltonian. We derive the corresponding discretization in multi-dimensions and prove convergence of the corresponding numerical approximations. Finally, we introduce a generalization of the original Hamilton-Jacobi equation and the corresponding discretization that can be solved efficiently using either the fast sweeping or the fast marching methods. Thus, the visibility of an observer in non-constant media can be computed. We also introduce a specialization of the algorithms for environments in which occluders are described by the graph of a function.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Chiu-Yen Kao

  2. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    Richard Tsai

Authors
  1. Chiu-Yen Kao
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  2. Richard Tsai
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Correspondence to Chiu-Yen Kao.

Additional information

This research is supported by NSF DMS-0513394 and The Sloan Foundation.

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Kao, CY., Tsai, R. Properties of a Level Set Algorithm for the Visibility Problems. J Sci Comput 35, 170–191 (2008). https://doi.org/10.1007/s10915-008-9197-5

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  • DOI: https://doi.org/10.1007/s10915-008-9197-5

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