Reflection in a Level Set Framework for Geometric Optics

Geometric optics makes its impact both in mathematics and real world applications related to ray tracing, migration, and tomography. Of special importance in these problems are the wavefronts, or points of constant traveltime away from sources, in the medium. Previously in [Osher, Cheng, Kang, Shim, and Tsai (2002)], we initiated a level set approach for the construction of wavefronts in isotropic media that handled the two major algorithmic issues involved with this problem: resolution and multivalued solutions. This approach was quite general and we were able to construct wavefronts in the presence of refraction, reflection, higher dimensions, and, in [Qian, Cheng, and Osher (2003)], anisotropy as well. However, the technique proposed for handling reflections of waves off objects, an important phenomenon involved in all applications of geometric optics, was inefficient and unwieldy to the point of being unusable, especially in the presence of multiple reflections. We introduce here an alternative approach based on the foundation presented in [Osher, Cheng, Kang, Shim, and Tsai (2002)]. This reworking allows the level set method to be considered for realistic applications involving reflecting surfaces in geometric optics.

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