4D Numerical Schemes for Cell Image Segmentation and Tracking

1. 4D Numerical Schemes for Cell Image Segmentation and Tracking Mariana Remešíková Karol Mikula, Nadine Peyriéras, Michal Smíšek

2. Motivation The motivation for this work comes from developmental biology zebrafish

3. Motivation

4. Motivation Cell nuclei Cell membranes

5. Motivation Images of the embryo before and during organogenesis Question: Where do the organs come from?

6. Cell tracking Let us follow the time evolution of one cell nucleus

7. Cell tracking The celll nucleus evolution can be viewed as a tree-like structure

8. Cell tracking Let us pick one nucleus at the end of the evolution

9. Cell tracking We want to find the way down to the root of the tree

10. Cell tracking • Our cell tracking method consists of three steps: • detection of approximate cell center positions Flux based level set center detection (FBLSCD) P.Frolkovič, K.Mikula, N.Peyriéras, A.Sarti:A counting number of cells and cell segmentation using advection-diffusion equations. Kybernetika 43 (6) (2007) • segmentation of the cell evolution trees • descend to the roots of the trees from the cell center positions in the last time frame

11. Segmentation of the cell evolution trees Generalized subjective surface model The function g(s) is the edge detector

12. Segmentation of the cell evolution trees Semi-implicit time approximation • Finite volume space discretization • we consider a rectangular domain Rn and a uniform grid where the grid elements are n-dimensional cubes (pixels, voxels, doxels…) • we integrate over the grid element Vi, i=i1e1+i2e2+..+inen

13. Segmentation of the cell evolution trees • Approximation of the integrals • the time derivative term • the advection term – upwind principle

14. Segmentation of the cell evolution trees Fi-2 The curvature term Fi-2 Fi-3 Fi-1 Vi Fi+1 Fi-1 Fi+1 Fi+3 Fi+2 Fi+2

15. Segmentation of the cell evolution trees 2D case

16. Segmentation of the cell evolution trees 3D case

17. Segmentation of the cell evolution trees 4D volume (doxel)

18. Segmentation of the cell evolution trees Approximation of gradient on the doxel face

19. Segmentation of the cell evolution trees Approximation of gradient on the doxel face

20. Segmentation of the cell evolution trees Segmentation of artificial 4D data K=1.0 h=1.0, S=0.1, TS=30 wa=5.0, wc=0.1

21. Segmentation of the cell evolution trees Segmentation of zebrafish embryogenesis 4D data

22. Backtracking in the tree starting positions for tracking In order to descend to the root of the tree, we compute the distance function to the root cell position The distance function d1 is computed only inside the tree We move from the top of the tree in the direction of decreasing distance function d1 This might not be enough – we can get into a wrong tree root cell positions

23. Backtracking in the tree In order to prevent dropping into a wrong tree, we compute the distance function to the border of the segmented tree (d2) Keeping the distance function d2maximized in each step of the tracking, we move along the center line of the tubes

24. Backtracking in the tree The distance function is computed by solving the time relaxed eikonal equation with Dirichlet type condition For the distance function d1, 0 is the set of cell center positions in the first time step For the distance function d2, 0 are the borders of the segmented cell evolution trees

25. Backtracking in the tree The time relaxed eikonal equation is discretized explicitly in time The space discretization is done by the Rouy-Tourin scheme

26. Backtracking in the tree The effect of the distance function d2

27. Backtracking in the tree Backtracking in artificial data

28. Backtracking in the tree Backtracking in zebrafish embryogenesis data

29. Backtracking in the tree Backtracking in zebrafish embryogenesis data

30. Thank you for attention!