Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.Comment: 19 page

Free boundary problems describing two-dimensional pulse recycling and motion in semiconductors

An asymptotic analysis of the Gunn effect in two-dimensional samples of bulk n-GaAs with circular contacts is presented. A moving pulse far from contacts is approximated by a moving free boundary separating regions where the electric potential solves a Laplace equation with subsidiary boundary conditions. The dynamical condition for the motion of the free boundary is a Hamilton-Jacobi equation. We obtain the exact solution of the free boundary problem (FBP) in simple one-dimensional and axisymmetric geometries. The solution of the FBP is obtained numerically in the general case and compared with the numerical solution of the full system of equations. The agreement is excellent so that the FBP can be adopted as the basis for an asymptotic study of the multi-dimensional Gunn effect.Comment: 19 pages, 9 figures, Revtex. To appear in Phys. Rev.

Globally optimal 3D image reconstruction and segmentation via energy minimisation techniques

This paper provides an overview of a number of techniques developed within our group to perform 3D reconstruction and image segmentation based of the application of energy minimisation concepts. We begin with classical snake techniques and show how similar energy minimisation concepts can be extended to derive globally optimal segmentation methods. Then we discuss more recent work based on geodesic active contours that can lead to globally optimal segmentations and reconstructions in 2D. Finally we extend the work to 3D by introducing continuous flow globally minimal surfaces. Several applications are discussed to show the wide applicability and suitability of these techniques to several difficult image analysis problems

Some free boundary problems in potential flow regime usinga based level set method

Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover the fluid front will possibly be carrying some material substance which will diffuse in the front and be advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models

Integrating Surface Normal Vectors Using Fast Marching Method

Abstract. Integration of surface normal vectors is a vital component in many shape reconstruction algorithms that require integrating surface normals to produce their final outputs, the depth values. In this paper, we introduce a fast and efficient method for computing the depth val-ues from surface normal vectors. The method is based on solving the Eikonal equation using Fast Marching Method. We introduce two ideas. First, while it is not possible to solve for the depths Z directly using Fast Marching Method, we solve the Eikonal equation for a function W of the form W = Z+λf. With appropriately chosen values for λ, we can ensure that the Eikonal equation for W can be solved using Fast March-ing Method. Second, we solve for W in two stages with two different λ values, first in a small neighborhood of the given initial point with large λ, and then for the rest of the domain with a smaller λ. This step is needed because of the finite machine precision and rounding-off errors. The proposed method is very easy to implement, and we demonstrate experimentally that, with insignificant loss in precision, our method is considerably faster than the usual optimization method that uses conju-gate gradient to minimize an error function.

Aquatics reconstruction software: the design of a diagnostic tool based on computer vision algorithms

Computer vision methods can be applied to a variety of medical and surgical applications, and many techniques and algorithms are available that can be used to recover 3D shapes and information from images range and volume data. Complex practical applications, however, are rarely approachable with a single technique, and require detailed analysis on how they can be subdivided in subtasks that are computationally treatable and that, at the same time, allow for the appropriate level of user-interaction. In this paper we show an example of a complex application where, following criteria of efficiency, reliability and user friendliness, several computer vision techniques have been selected and customized to build a system able to support diagnosis and endovascular treatment of Abdominal Aortic Aneurysms. The system reconstructs the geometrical representation of four different structures related to the aorta (vessel lumen, thrombus, calcifications and skeleton) from CT angiography data. In this way it supports the three dimensional measurements required for a careful geometrical evaluation of the vessel, that is fundamental to decide if the treatment is necessary and to perform, in this case, its planning. The system has been realized within the European trial AQUATICS (IST-1999-20226 EUTIST-M WP 12), and it has been widely tested on clinical data

A new ghost cell/level set method for moving boundary problems:application to tumor growth

In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth

Some flows in shape optimization

Geometric flows related to shape optimization problems of Bernoulli type are investigated. The evolution law is the sum of a curvature term and a nonlocal term of Hele-Shaw type. We introduce generalized set solutions, the definition of which is widely inspired by viscosity solutions. The main result is an inclusion preservation principle for generalized solutions. As a consequence, we obtain existence, uniqueness and stability of solutions. Asymptotic behavior for the flow is discussed: we prove that the solutions converge to a generalized Bernoulli exterior free boundary problem