  In this paper, we introduce and study a novel segmentation method for 4D
images based on surface evolution governed by a nonlinear partial differential
equation, the generalized subjective surface equation. The new method uses 4D
digital image information and information from a thresholded 4D image in a
local neighborhood. Thus, the 4D image segmentation is accomplished by defining
the edge detector function's input as the weighted sum of the norm of gradients
of presmoothed 4D image and norm of presmoothed thresholded 4D image in a local
neighborhood. Additionally, we design and study a numerical method based on the
finite volume approach for solving the new model. The reduced diamond cell
approach is used for approximating the gradient of the solution. We use a
semi-implicit finite volume scheme for the numerical discretization and show
that our numerical scheme is unconditionally stable. The new 4D method was
tested on artificial data and applied to real data representing 3D+time
microscopy images of cell nuclei within the zebrafish pectoral fin and
hind-brain. In a real application, processing 3D+time microscopy images amounts
to solving a linear system with several billion unknowns and requires over
$1000$ GB of memory; thus, it may not be possible to process these images on a
serial machine without parallel implementation utilizing the MPI. Consequently,
we develop and present in the paper OpenMP and MPI parallel implementation of
designed algorithms. Finally, we include cell tracking results to show how our
new method serves as a basis for finding trajectories of cells during
embryogenesis.
