14 research outputs found
Two methods for optical flow estimation
summary:In this paper we describe two methods for optical flow estimation between two images. Both methods are based on the backward tracking of characteristics for advection equation and the difference is on the choice of advection vector field. We present numerical experiments on 2D data of cell nucleus
High resolution compact implicit numerical scheme for conservation laws
We present a novel implicit scheme for the numerical solution of
time-dependent conservation laws. The core idea of the presented method is to
exploit and approximate the mixed spatial-temporal derivative of the solution
that occurs naturally when deriving some second order accurate schemes in time.
Such an approach is introduced in the context of the Lax-Wendroff (or
Cauchy-Kowalevski) procedure when the second time derivative is not completely
replaced by space derivatives using the PDE, but the mixed derivative is kept.
If approximated in a suitable way, the resulting compact implicit scheme
produces algebraic systems that have a more convenient structure than the
systems derived by fully implicit schemes. We derive a high resolution TVD form
of the implicit scheme for some representative hyperbolic equations in the
one-dimensional case, including illustrative numerical experiments.Comment: Significantly revised version that is accepted to AMC journa
Unconditionally stable higher order semi-implicit level set method for advection equations
We present compact semi-implicit finite difference schemes on structured
grids for numerical solutions of the advection by an external velocity and by a
speed in normal direction that are applicable in level set methods. The most
involved numerical scheme is third order accurate for the linear advection with
a space dependent velocity and unconditionally stable in the sense of von
Neumann stability analysis. We also present a simple high-resolution scheme
that gives a TVD (Total Variation Diminishing) approximation of the spatial
derivative for the advected level set function. In the case of nonlinear
advection, the semi-implicit discretization is proposed to linearize the
problem. The compact form of implicit stencil in numerical schemes containing
unknowns only in the upwind direction allows applications of efficient
algebraic solvers like fast sweeping methods. Numerical tests to evolve a
smooth and non-smooth interface and an example with a large variation of
velocity confirm the good accuracy of the methods and fast convergence of the
algebraic solver even in the case of very large Courant numbers
Laplacian regularized eikonal equation with Soner boundary condition on polyhedral meshes
In this paper, we propose a numerical algorithm based on a cell-centered
finite volume method to compute a distance from given objects on a
three-dimensional computational domain discretized by polyhedral cells.
Inspired by the vanishing viscosity method, a Laplacian regularized eikonal
equation is solved and the Soner boundary condition is applied to the boundary
of the domain to avoid a non-viscosity solution. As the regularization
parameter depending on a characteristic length of the discretized domain is
reduced, a corresponding numerical solution is calculated. A convergence to the
viscosity solution is verified numerically as the characteristic length becomes
smaller and the regularization parameter accordingly becomes smaller. From the
numerical experiments, the second experimental order of convergence in the
norm error is confirmed for smooth solutions. Compared to solve a
time-dependent form of eikonal equation, the Laplacian regularized eikonal
equation has the advantage of reducing computational cost dramatically when a
more significant number of cells is used or a region of interest is far away
from the given objects. Moreover, the implementation of parallel computing
using domain decomposition with -ring face neighborhood structure can be
done straightforwardly by a standard cell-centered finite volume code
Counting number of cells and cell segmentation using advection-diffusion equations
summary:We develop a method for counting number of cells and extraction of approximate cell centers in 2D and 3D images of early stages of the zebra-fish embryogenesis. The approximate cell centers give us the starting points for the subjective surface based cell segmentation. We move in the inner normal direction all level sets of nuclei and membranes images by a constant speed with slight regularization of this flow by the (mean) curvature. Such multi- scale evolutionary process is represented by a geometrical advection-diffusion equation which gives us at a certain scale the desired information on the number of cells. For solving the problems computationally we use flux-based finite volume level set method developed by Frolkovič and Mikula in [FM1] and semi-implicit co-volume subjective surface method given in [CMSSg, MSSgCVS, MSSgchapter]. Computational experiments on testing and real 2D and 3D embryogenesis images are presented and the results are discussed
Semi-implicit methods for advection equations with explicit forms of numerical solution
We present a parametric family of semi-implicit second order accurate
numerical methods for non-conservative and conservative advection equation for
which the numerical solutions can be obtained in a fixed number of forward and
backward alternating substitutions. The methods use a novel combination of
implicit and explicit time discretizations for one-dimensional case and the
Strang splitting method in several dimensional case. The methods are described
for advection equations with a continuous variable velocity that can change its
sign inside of computational domain. The methods are unconditionally stable in
the non-conservative case for variable velocity and for variable numerical
parameter. Several numerical experiments confirm the advantages of presented
methods including an involvement of differential programming to find optimized
values of the variable numerical parameter
Mathematical modelling of the Germasogeia aquifer
Two challenges related to improving the management of the Germasogeia aquifer were presented to the Study Group by the Cyprus Water Development Department (WDD), the public organisation responsible for managing the wa- ter resources in Cyprus. The rst challenge was how to optimally recharge the aquifer in order to compensate for the extraction of drinking and irrigation water whilst preventing sea water intrusion. In order to address this challenge we developed model for the water in the aquifer. Note that by exploiting the long, thin nature of the aquifer we only develop two-dimensional models in this work. We rst develop a simple model based on Darcy ows for porous media which gives the water table height for given dam seepage rate, recharge and extraction rates; we neglect seawater intrusion. We then use the steady version of this model to develop an optimized recharge strategy with which we can identify minimal recharge required for a desired extracted water volume such that the minimum prescribed water table is respected. We explore 4 di erent scenarios and we nd that in certain cases there can be a considerable reduction in the amount of recharged water compared to the current empirical strategy the Water Development Department is employing, where water is recharged and extracted in equal proportions. To incorporate the e ects of seawater intrusion, which can be very damaging to the water quality, we next develop transient two- dimensional models of saturated-unsaturated groundwater ow and solve them numerically using the open source software SUTRASuite and the commercial package ANSYS FLUENT; the position of the water table and the seawater- freshwater interface are determined for various extraction/recharge strategies. Data from the WDD are used in some of the simulations. The second important challenge we were asked to look at was to predict the transport of pollutants in the aquifer in the case of an accidental leakage. An advection-difusion equation for the contaminant concentration is introduced and simulations are under- taken using the commercial package COMSOL. The concentration pro les of the contaminant are studied and we nd that the e ect of contamination varies depending on where the contamination site is; the closer the contamination site is to the dam, the larger the extent of contamination will be