A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TRIANGULATED SURFACES

Abstract

This paper presents an efficient, fine-grained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton-Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on state-of-the-art, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512-2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of first-order accuracy for both architectures. We also propose a novel triangle-based update scheme and its corresponding data structure for efficient irregular data mapping to parallel single-instruction multiple-data (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against state-of-the-art Eikonal solvers.

Fig. 2.1

A triangulation 𝒮_T of surface 𝒮 (left) and the local solver: update the value at vertex υ₃ in a triangle (right).

Fig. 2.2

Strategy to deal with obtuse triangles.

Fig. 2.3

Data structure: in this figure, T_i is a triangle, e_i,j represents the edge length, and f_i is the inverse of speed in a triangle. Φ_i means the value of the i_th vertex. I_i in NBH represents the data structure for the i_th vertex, each of which has q indices pointing (shown as arrows) to the value array.

Fig. 2.4

Virtual edge and virtual triangles.

Fig. 3.1

Sphere and Stanford dragon meshes.

Fig. 3.2

Laplacian experiment results.

Fig. 3.3

(a) The level sets of the solution of the Eikonal equation which represents distance to two circular boundaries. (b) The error as a function of resolution shows first-order convergence for smooth boundary conditions.