Technical Report MSC-2019-23

Title: Deep Eikonal Solvers
Authors: Moshe Lichtenstein
Supervisors: Ron Kimmel
PDFCurrently accessibly only within the Technion network
Abstract: Geodesic distance, a generalization of the Euclidean distance to surfaces with an effective curvature, is a ubiquitous tool used in three-dimensional data analysis. The equation that describes the geodesic distance locally over a surface is a PDE called the Eikonal equation. Typical to the numerical solution of PDEs, estimating the Eikonal solution on a discrete domain requires a trade-off between accuracy and computational complexity. In order to develop a quasilinear-time yet accurate algorithm, a deep learning approach to numerically approximate the solution to the Eikonal equation is introduced.

The proposed method is built on the fast marching scheme which comprises two components: a local numerical solver and an update scheme. We replace the formulaic local numerical solver with a trained neural network to provide highly accurate estimates of local distances for a variety of different geometries and sampling conditions. Our learning approach generalizes not only to flat Euclidean domains but also to curved surfaces enabled by the incorporation of certain invariant features in the neural network architecture. We show a considerable gain in performance, validated by smaller errors and higher orders of accuracy for the numerical solutions of the Eikonal equation computed on different surfaces.

The proposed approach leverages the approximation power of neural networks to enhance the performance of numerical algorithms, thereby, connecting the somewhat disparate themes of numerical geometry and learning.

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