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.