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Applied
Math Seminar
Quarter 2001
3:15 - 4:15 p.m.
Sloan Mathematics Corner
Building 380, Room 380-C
Friday, October 19, 2001
Stanley Osher
University of California, Los Angeles
Geometric Optics in a Phase Space Based
Level Set and Eulerian Framework
Abstract:
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(joint work with L.-T. Cheng, M. Kang, H. Shim and Y.-H.
Tsai)
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
projections to standard space gives the wavefronts, in a reduced phase
space under an Eulerian and partial differential equation based framework.
The bicharacteristic strips are represented using a level set approach
for handling higher codimensional objects and the PDE responsible for
the evolution is a reduced form of the Liouville equation. Results for
the 2 dimensional case for variable index of refraction are shown and
compared to other methods. Results are also shown involving reflection
and some 3 dimensional extensions
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