A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe the wave-front si...

We study a new formulation for the eikonal equation |∇u| = 1 on a bounded subset of R. Considering a field P of orthogonal projections onto 1-dimensional subspaces, with divP ∈ L, we prove existence and uniqueness for solutions of the equation P divP = 0. We give a geometric description, comparable with the classical case, and we prove that such solutions exist only if the domain is a tubular n...

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...

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...

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...

We demonstrate that a class of torus-shaped Hopf maps with arbitrary linking number obeys the static complex eikonal equation. Further , we explore the geometric structure behind these solutions, explaining thereby the reason for their existence. As this equation shows up as an integrability condition in certain non-linear field theories, the existence of such solutions is of some interest.

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

Single-particle dynamics in electron microscopes, ion or electron lithographic instruments, particle accelerators, and particle spectrographs is described by weakly nonlinear ordinary differential equations. Therefore, the linear part of the equation of motion is usually solved and the nonlinear effects are then found in successive order by iteration methods. When synchrotron radiation is not i...

It is known that the Hamiltonian of the eikonal equation for an anisotropic medium may be nonconvex, which excludes the application of Fermat’s minimum-time principle related to minimum-time control problems. The idea proposed in this paper consists in finding a conflict control problem (differential game) whose Hamiltonian coincides with the Hamiltonian of the eikonal equation. It turns out th...