Geometric optics in a phase-space-based level set and Eulerian framework

  • Authors:

  • Stanley Osher;Li-Tien Cheng;Myungjoo Kang;Hyeseon Shim;Yen-Hsi Tsai

  • Affiliations:

  • Level Set Systems Inc., 1058 Embury Street, Pacific Palisades, California;Department of Mathematics, University of California San Diego, La Jolla, California;Level Set Systems Inc., 1058 Embury Street, Pacific Palisades, California;Level Set Systems Inc., 1058 Embury Street, Pacific Palisades, California;Department of Mathematics, University of California Los Angeles, Los Angeles, California

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2002

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Abstract

We introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics. This is important in a wide variety of applications in wave propagation. Our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts. This is achieved through solving for the bicharacteristic strips, whose projection to spatial space gives the wavefronts, in a reduced phase space under a Eulerian and partial differential-equation-based framework. The bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the Liouville equations. Results for the two-dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field. Results are also introduced to show the ability to handle reflection and to extend the method to the three-dimensional case.