In this paper, we introduce a set of methods for processing and analyzing long time series of 3D images representing embryo evolution. The images are obtained by in vivo scanning using a confocal microscope where one of the channels represents the cell nuclei and the other one the cell membranes. Our image processing chain consists of three steps: image filtering, object counting (center detection) and segmentation. The corresponding methods are based on numerical solution of nonlinear PDEs, namely the geodesic mean curvature flow model, flux-based level set center detection and generalized subjective surface equation. All three models have a similar character and therefore can be solved using a common approach. We explain in details our semi-implicit time discretization and finite volume space discretization. This part is concluded by a short description of parallelization of the algorithms. In the part devoted to experiments, we provide the experimental order of convergence of the numerical scheme, the validation of the methods and numerous experiments with the data representing an early developmental stage of a zebrafish embryo.
Finite volume methods for solving hyperbolic systems on unstructured meshes are known for a long time. There are two basic formulations of the method: cell centered and vertex centered. For the cell centered method, the (finite) volumes used to satisfy
the integral form of the equation are the mesh elements itself. For the vertex centered approach, the finite volumes are elements of the mesh dual to the computational mesh. We present comparison of both formulations. The method is first evaluated on a scalar advection equation. Knowing the analytical solution of the problem,
convergence studies are performed. More complex test cases involve the 3D transonic flow past an Олега M6 airfoil. Discussion includes influence of the reconstruction and limiters on the solution. The results of the parallel implementation for a Linux PC cluster both with explicit and implicit time integration method are presented. and Obsahuje seznam literatury
The finite volume method is applied for solving the conservative Saint-Venant equations in case of a one-dimensional open channel flow with high temporal and spatial variability. A new shock-capturing discretisation scheme is proposed for computation of flow equations with source terms. The proposed scheme, called hybrid, combines adventages of flux-splitting and flux-difference-splitting schemes. Five benchmark tests are used to verify the hybrid scheme. The tests are: (1) Flow in a rectangular cross-section rough channel, (2) instantaneous dambreak over an horizontal, initially dry bed, (3) undercritical flow over a bump, (4) undercritical flow over a bump with change to supercritical flow, and (5) supercritical flow over a bump with hydraulic jump. The quality of the proposed scheme is evaluated and compared with that of well known flux-splitting and flux-difference-splitting schemes, on the base of the average percentual error. Results show that the hybrid scheme has a good precision for calculation of highly unsteady, varied flow and that the model is able to consider a partially dry bed. The average percentual errors of the computations ranged from 0.0 to 5.4 % for flow depth and from 0.0 to 6.3 % for specific discharge. and Metoda konečných objemů se používá při řešení Saint-Venantových rovnic pro jednorozměrné proudění v otevřeném kanále a při velkých časových a prostorových změnách. V práci je navrženo nové diskretizační schéma schopné zachytit rázové jevy pro řešení tokových rovnic se zdrojovými členy. Navržený postup nazýváme hybridním, neboť kombinuje výhody modelování pomocí dvou schémat, nazývaných fluxsplitting a flux-difference-splitting. Pro ověření hybridního schématu bylo testováno pět různých uspořádání. Byly zkoušeny: 1. proudění v drsném korytě s obdélníkovým průřezem, 2. okamžitý vtok do horizontálního původně suchého koryta, 3. podkritické proudění přes práh, 4. podkritické proudění přes práh s přechodem na nadkritické proudění a 5. nadkritické proudění přes práh s hydraulickým skokem. Kvalita navrženého postupu je vyhodnocena a porovnána s výsledky získanými pomocí známých výše uvedených schémat. Byly vyhodnoceny průměrné chyby v procentech. Výsledky naznačují, že hybridní schéma vykazuje dobrou přesnost při výpočtu neustáleného, proměnlivého proudění, a že tento model je schopen vzít v úvahu částečně suché koryto. Průměrné chyby v procentech se pohybovaly od nuly do 5,4 % pro hloubku a od nuly do 6,3 % pro specifický průtok.
In this paper, we describe an efficient method for 3D image segmentation. The method uses a PDE model - the so called generalized subjective surface equation which is an equation of advection-diffusion type. The main goal is to develop an efficient and stable numerical method for solving this problem. The numerical solution is based on semi-implicit time discretization and flux-based level set finite volume space discretization. The space discretization is discussed in details and we introduce three possible alternatives of the so called diamond cell finite volume scheme for this type of 3D nonlinear diffusion equation. We test the performance of the method and all its variants introduced in the paper by determining the experimental order of convergence. Finally we show a couple of practical applications of the method.
In this paper we deal with a problem of segmentation (including missing boundary completion) and subjective contour creation. For the corresponding models we apply the semi-implicit finite volume numerical schemes leading to methods which are robust, efficient and stable without any restriction to a time step. The finite volume discretization enables to use the spatial adaptivity and thus improve significantly the computational time. The computational results related to image segmentation with partly missing boundaries and subjective contour extraction are presented.
The aim of this article is a qualitative analysis of two modern finite volume (FVM) schemes. First one is the so called Modified Causon's scheme, which is based on the classical MacCormack FVM scheme in total variation diminishing (TVD) form, but is simplified in such a way that the demands on computational power are much smaller without loss of accuracy. Second one is implicit WLSQR (Weighted Least Square Reconstruction) scheme combined with various types of numerical fluxes (AUSMPW+ and HLLC). Two different test cases were chosen for the comparison −1) two-dimensional transonic inviscid nonstationary flow over an oscillating NACA 0012 profile and 2) three-dimensional transonic inviscid stationary flow around the Onera M6 wing. Nonstationary effects were simulated with the use of Arbitrary Lagrangian-Eulerian Method (ALE). Experimental results for these regimes of flow are easily available and so the numerical results are compared both in-between and with experimental data. The obtained numerical results in all considered cases (2D and 3D) are in a good agreement with experimental data.
Magnetic Resonance Diffusion Tensor Imaging (MR-DTI) is a noninvasive in vivo method capable of examining the structure of human brain, providing information about the position and orientation of the neural tracts. After a short introduction to the principles of MR-DTI, this paper describes the steps of the proposed neural tract visualization technique based on the DTI data. The cornerstone of the algorithm is a texture diffusion procedure modeled mathematically by the problem for the Allen-Cahn equation with diffusion anisotropy controlled by a tensor field. Focus is put on the issues of the numerical solution of the given problem, using the finite volume method for spatial domain discretization. Several numerical schemes are compared with the aim of reducing the artificial (numerical) isotropic diffusion. The remaining steps of the algorithm are commented on as well, including the acquisition of the tensor field before the actual computation begins and the postprocessing used to obtain the final images. Finally, the visualization results are presented.
The paper deals with a filter design for nonlinear continuous stochastic systems with discrete-time measurements. The general recursive solution is given by the Fokker-Planck equation (FPE) and by the Bayesian rule. The stress is laid on the computation of the predictive conditional probability density function from the FPE. The solution of the FPE and its integration into the estimation algorithm is the cornerstone for the whole recursive computation. A new usable numerical scheme for the FPE is designed. In the scheme, the separation technique based on the upwind volume method and the finite difference method for hyperbolic and parabolic part of the FPE is used. It is supposed that separation of the FPE and choice of a suitable numerical method for each part can achieve better estimation quality comparing to application of a single numerical method to the unseparated FPE. The approach is illustrated in some numerical examples.
In order to investigate effects of the dynamic capillary pressure-saturation relationship used in the modelling of a flow in porous media, a one-dimensional fully implicit numerical scheme is proposed. The numerical scheme is used to simulate an experimental procedure using a measured dataset for the sand and fluid properties. Results of simulations using different models for the dynamic effect term in capillary pressure-saturation relationship are presented and discussed.
A finite volume model for two-layer shallow water flow in microtidal salt-wedge estuaries is presented in this work. The governing equations are a coupled system of shallow water equations with source terms accounting for irregular channel geometry and shear stress at the bed and interface between the layers. To solve this system we applied the Qscheme of Roe with suitable treatment of source terms, coupling terms, and wet-dry fronts. The proposed numerical model is explicit in time, shock-capturing and it satisfies the extended conservation property for water at rest. The model was validated by comparing the steady-state solutions against a known arrested salt-wedge model and by comparing both steady-state and time-dependant solutions against field observations in Rječina Estuary in Croatia. When the interfacial friction factor was chosen correctly, the agreement between numerical results and field observations was satisfactory.