Peter Frolkovič

Extraction of the Intercellular Skeleton from 2D Images of Embryogenesis Using Eikonal Equation and Advective Subjective Surface Method

We suggest an efficient method for automatic detection of the intercellular skeleton in microscope images of early embryogenesis. The method is based on the solution of two advective PDEs. First, we solve numerically the time relaxed eikonal equation in order to obtain the signed distance function to a given set --- a set of points representing cell centers or a set of closed curves representing segmented inner borders of cells. The second step is a segmentation process driven by the advective version of subjective surface equation where the velocity field is given by the gradient of the computed distance function. The first equation is discretized by Rouy-Tourin scheme and we suggest a fixing strategy that significantly improves the speed of the computation. The second equation is solved using a classical upwind strategy. We present several test examples and we show a practical application - the intercellular skeleton extracted from a 2D image of a zebrafish embryo.

Semi-implicit Level Set Method with Inflow-Based Gradient in a Polyhedron Mesh

In this paper , a semi-implicit method is proposed to solve a propagation in a normal direction with a cell-centered finite volume method. An inflow-based gradient is used to discretize the magnitude of the gradient and it brings the second order upwind difference in an evenly spaced one dimensional domain. In three dimensional domain, we numerically verify that the proposed scheme is second order. The implementation is straightforwardly combined with a conventional finite volume code and 1-ring face neighborhood for parallel computation. An experimental order of convergence and a comparison of wall clock time between semi-implicit and explicit method are illustrated by numerical examples.

Semi-implicit Second Order Accurate Finite Volume Method for Advection-Diffusion Level Set Equation

We present a second order accurate finite volume method for level set equation describing the motion in normal direction with the speed depending on external properties and curvature. A convenient combination of a Crank-Nicolson type of the time discretization for diffusion term [1] and an Inflow Implicit and Outflow Explicit scheme [6] for advection term is used. Numerical experiments for an example with the exact solution derived in this paper and for examples motivated by modeling of fire propagation in forests are presented.