Let I : → be a given bounded image function, where is an open and bounded domain which belongs to n . Let us consider n = 2 for the purpose of illustration. Also, let S = {x i } i ∈ be a finite set of given points. We would like to find a contour ⊂ , such that is an object boundary interpolating the points from S. We combine the ideas of the geodesic active contour (cf. Caselles et al. [7,8]) and of interpolation of points (cf. Zhao et al. [40]) in a level set approach developed by Osher and Sethian [33]. We present modelling of the proposed method, both theoretical results (viscosity solution) and numerical results are given.

The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscillatory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.

In this paper, we propose a new scheme for both detection of boundaries and fitting of geometrical data based on a geometrical partial differential equation, which allows a rigorous mathematical analysis. The model is a geodesic-active-contour-based model, in which we are trying to determine a curve that best approaches the given geometrical conditions (for instance a set of points or curves to approach) while detecting the object under consideration. Formal results concerning existence, uniqueness (viscosity solution) and stability are presented as well. We give the discretization of the method using an additive operator splitting scheme which is very efficient for this kind of problem. We also give 2D and 3D numerical examples on real data sets.