Visibility and its dynamics in a PDE based implicit framework

  • Authors:

  • Y.-H. R. Tsai;L.-T. Cheng;S. Osher;P. Burchard;G. Sapiro

  • Affiliations:

  • Institute for Advanced Study, and Department of Mathematics, Princeton University, Princeton, NJ;Department of Mathematics, UCSD, La Jolla, CA;Department of Mathematics, UCLA, Los Angeles, CA; ;Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2004

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Abstract

We investigate the problem of determining visible regions given a set of (moving) obstacles and a (moving) vantage point. Our approach to this problem is through an implicit framework, where the obstacles are represented by a level set function. The visibility problem is formally formulated as a boundary value problem (BVP) of a first order partial differential equation. It is based on the continuation of values along the given ray field. We propose a one-pass, multi-level algorithm for the construction of the solution on a grid. Furthermore, we study the dynamics of shadow boundaries on the surfaces of the obstacles when the vantage point moves along a given trajectory. In all of these situations, topological changes such as merging and breaking occur in the regions of interest. These are automatically handled by the level set framework proposed here. Finally, we obtain additional useful information through simple operations in the level set framework.