fast sweeping method
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Computation of traveltimes and ray paths is important for anisotropic tomography inversions. The Eikonal-equation-based method outperforms traditional ray methods by producing more accurate results. However, most existing Eikonal solvers are formulated on structured regular meshes, which are no longer accurate for models with the presence of irregular topography and subsurface interfaces. To solve Eikonal equation in vertically transversely isotropic (VTI) or tilted transversely isotropic (TTI) models with irregular geometry, we formulate a new iterative fast sweeping method on unstructured triangular meshes. The fixed-point iteration is implemented to capture the high-order nonlinear terms therein and a fast sweeping method on unstructured triangular meshes is implemented to solve the resulting elliptically anisotropic Eikonal equation at every iteration. We test the new algorithm for direct arrivals and reflected arrivals, and then use the calculated traveltimes to track the ray path in VTI/TTI media. Numerical tests demonstrate the validity and accuracy of the new method for models with rough topography and subsurface interface.

Object shape reconstruction from images has been an active topic in computer vision. Shape-from-shading (SFS) is an important approach for inferring 3D surface from a single shading image. In this paper, we present a unified SFS approach for surfaces of various reflectance properties using fast eikonal solvers. The whole approach consists of three main components: a unified SFS model, a unified eikonal-type partial differential image irradiance (PDII) equation, and fast eikonal solvers for the PDII equation. The first component is designed to address different reflectance properties including diffuse, specular, and hybrid reflections in the imaging process of the camera. The second component is meant to derive the PDII equation under an orthographic camera projection and a single distant point light source whose direction is the same as the camera. Finally, the last component is targeted at solving the resultant PDII equation by using fast eikonal solvers. It comprises two Godunov-based schemes with fast sweeping method that can handle the eikonal-type PDII equation. Experiments on several synthetic and real images demonstrate that each type of the surfaces can be effectively reconstructed with more accurate results and less CPU running time.