The article is concerned with the development and comparison of three di erent algorithms for the computation of rst-arrival traveltimes: the fast marching method (FMM), the group marching method (GMM), and a secondorder nite di erence eikonal solver. GMM is introduced as a variant of FMM. It proceeds the solution by advancing a selected group of grid points at a time, rather than sorting the s...

As Planck’s constant ~ (treated as a free parameter) tends to zero, the solution to the eikonal equation |∇S(X)| = f(X) can be increasingly closely approximated by the solution to the corresponding Schrödinger equation. When the forcing function f(X) is set to one, we get the Euclidean distance function problem. We show that the corresponding Schrödinger equation has a closed form solution whic...

Abstract We propose a new methodology for carrying out eikonal-based traveltime tomography arising from important applications such as seismic imaging and medical imaging. The new method formulates the traveltime tomography problem as a variational problem for a certain cost functional explicitly with respect to both traveltime and sound speed. Furthermore, the cost functional is penalized to e...

Fast sweeping methods are a class of efficient iterative methods for solving steady state hyperbolic PDEs. They utilize the Gauss-Seidel iterations and alternating sweeping strategy to cover a family of characteristics of the hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The first order fast sweeping method for solving Eikonal equations (Zhao in Math Comput 74:60...

We study the origin of subleading soft and collinear poles of form factors and amplitudes in dimensionally-regulated massless gauge theories. In the case of form factors of fundamental fields, these poles originate from a single function of the coupling, denoted G(αs), depending on both the spin and gauge quantum numbers of the field. We relate G(αs) to gauge-theory matrix elements involving th...

The calculation of threshold corrections to hard scattering cross sections beyond leading logarithms requires the calculation of loop diagrams in the eikonal approximation [1]. One-loop calculations have been performed for all 2 → 2 partonic processes in heavy quark [2] and jet [3] production. The soft anomalous dimension matrix ΓS at one-loop allows the resummation of soft-gluon corrections at...

The Eikonal equation is well established for modelling wave propagation in the field of cardiac electrophysiology[?Franzone]. It offers highly tempting computation speeds in contrast to the bidomain or monodomain models. Consequently, developing effective algorithmic solution techniques to the Eikonal equation is an immediate and valuable task. In this paper, we build on the methods existing in...

Mathematical morphology (MM) offers a wide range of operators to address various image processing problems. These operators can be defined in terms of algebraic (discrete) sets or as partial differential equations (PDEs). In this paper, we introduce a novel formulation of MM formalized as a framework of partial difference equations (PdEs) over weighted graphs of arbitrary topology. Then, we pre...

The evolution of a wavefront propagating in a two dimensional inhomogeneous isotropic medium with sound speed F = F (x, y) is governed by the Eikonal equation (EE) |∇T | = 1

Transient elastography and supersonic imaging are promising new techniques for characterizing the elasticity of soft tissues. Using this method, an ‘ultrafast imaging’ system (up to 10 000 frames s−1) follows in real time the propagation of a low frequency shear wave. The displacement of the propagating shear wave is measured as a function of time and space. The objective of this paper is to de...