Applications of the level set method to geometrical optics, transmission tomography, image processing and crystal growth modeling

The level set method is a successful numerical method. This dissertation focuses on its applications to geometrical optics, transmission tomography, image processing, and crystal growth modeling. In geometrical optics, we have proposed a level set framework to compute multivalued solutions of the paraxial eikonal equation. To further improve the computational efficiency and the memory requirement of this algorithm, we combine it with a local level set method and a semi-Lagrangian method. Using these tools, we then solve an inverse problem of geometrical optics, the so-called transmission traveltime tomography. Namely, we are inverting the velocity in a medium using possibly multivalued traveltime measurements. We have also studied two other applications of the level set method. In image processing, we have developed a variational method for image segmentation which determines the global minimizer of the active contour model. In crystal growth modeling, we have proposed an adaptive level set method. All these examples show that the level set method is an effective numerical method and can be easily applied to various fields in applied mathematics.