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.

A massively parallel Eikonal solver on unstructured meshes

“…An upwind scheme is used to compute an unknown solution φ 4 , assuming that the given values φ 1 , φ 2 , and φ 3 comply with the causality property of the Eikonal solutions [8]. Since we assume a constant speed function within each tetrahedron the arrival time φ 4 is determined by the time associated with that segment from x 5 to x 4 that minimizes the solution value at x 4 , see Fig.…”

“…φ 5 is rewritten in the same way. If we plug φ 5 into equation (4) then we get the following expression for φ 4 :…”

“…We use the fast iterative method (FIM) [4,5] in its description by Fu, Kirby and Whitaker [1] as baseline Eikonal solver for our improvements in this paper. The original FIM is introduced in Sect.…”

“…In our reference case we are solving again for vertex 4. We observe from matrix M in (10) that vertex 3 (marked in red) is the vertex that is part of all the possible scalar products needed to solve for x 4 . It is not only part of all possible scalar products but it is also contained in both edges of each product.…”

“…We addressed the implementation of an Eikonal solver for Shared Memory (OpenMP) and for streaming architectures in CUDA with a low memory footprint in our previous work [2] wherein we achieved already a very short execution time of the algorithm and good quality results. We use the fast iterative method [1,4,5] for solving the Eikonal equation, especially the version by Fu, Kirby and Whitaker [1] wherein the memory footprint has been reduced by storing temporarily the inner products needed to compute the 3D solution of one vertex. This step is performed in the wave front computations and some data have to be transferred from main memory to compute the solution for each tetrahedron.…”