4D Numerical Schemes for Cell Image Segmentation and Tracking

Nonlinear PDE based numerical methods for cell tracking in zebrafish embryogenesis

“…At the end of this section we shortly discuss an important difference between our new approach and previously suggested approaches [1,17]. In [1,17], only the distance function D 1 was used instead of combined distance function D. It means that we can extract only cell trajectories laying in simply connected components of 4D segmentation starting in the first time step of the image sequence.…”

“…Such approach using 4D generalized subjective surface (GSUBSURF) method was originally developed in [17] but used only locally in time (for short time sequence) and space (for a detail of the full 3D volume) and utilization of such type of method for long 3D+time image sequences remains a great challenge from modeling, discretization and (parallel) computation point of view and it will be objective of our further study. On the other hand, the simplified approach suggested and used in this paper also gives, thanks to the time overlap, the reliable 4D segmentation which can be used in further steps of cell trajectories extraction.…”

“…In order to overcome this difficulty we have to keep the extracted paths at a certain distance from the spatio-temporal cell boundaries or, in other words, they should be more centered inside the 4D segmentation. This can be achieved by using the constrained distance function D B (x 1 , x 2 , x 3 , θ) [15,1,17] values of which grow from boundaries to the center of the 4D spatio-temporal trees, see Fig. 8 right.…”

“…Such construction of the combined distance function D will help us to get all, also partial, cell trajectories contained in our 4D spatio-temporal tree structures. Its utilization represents a main important novelty with respect to [1,17] where just D = D 1 was used so only cell trajectories starting in the first frame could be detected. In order to construct the combined distance function D, first we define a set of values…”

“…In this paper we suggest new PDE based cell tracking method for such complex stages of zebrafish embryogenesis. In the present work we continue the effort started in [1,17] devoted to 2D+time and 3D+time cells tracking. In contrast to [13], in our approach we first extract all possible cell trajectories inside the 3D+time data and then the cell lineage tree is reconstructed by finding merging trajectories going backward in time.…”