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.
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.