Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.Comment: 28 pages, 12 figure

Multi-Stencil Streamline Fast Marching: a general 3D Framework to determine Myocardial Thickness and Transmurality in Late Enhancement Images

We propose a fully three-dimensional methodology for the computation of myocardial non-viable tissue transmurality in contrast enhanced magnetic resonance images. The outcome is a continuous map defined within the myocardium where not only current state-of-the-art measures of transmurality can be calculated, but also information on the location of non-viable tissue is preserved. The computation is done by means of a partial differential equation framework we have called Multi- Stencil Streamline Fast Marching (MSSFM). Using it, the myocardial and scarred tissue thickness is simultaneously computed. Experimental results show that the proposed 3D method allows for the computation of transmurality in myocardial regions where current 2D methods are not able to as conceived, and it also provides more robust and accurate results in situations where the assumptions on which current 2D methods are based —i.e., there is a visible endocardial contour and its corresponding epicardial points lie on the same slice—, are not met

Compressed absorbing boundary conditions via matrix probing

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. The result is a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter.Comment: 29 pages with 25 figure

Travel times and ray paths for acoustic and elastic waves in generally anisotropic media

Wavefield travel time tomography is used for a variety of purposes in acoustics, geophysics and non-destructive testing. Since the problem is non-linear, assessing uncertainty in the results requires many forward evaluations. It is therefore important that the forward evaluation of travel times and ray paths is efficient, which is challenging in generally anisotropic media. Given a computed travel time field, ray tracing can be performed to obtain the fastest ray path from any point in the medium to the source of the travel time field. These rays can then be used to speed up gradient based inversion methods. We present a forward modeller for calculating travel time fields by localised estimation of wavefronts, and a novel approach to ray tracing through those travel time fields. These methods have been tested in a complex anisotropic weld and give travel times comparable to those obtained using finite element modelling while being computationally cheaper.<br/