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4D Numerical Schemes for Cell Image Segmentation and Tracking. Mariana Remešíková Karol Mikula , Nadine Peyriéras , Michal Sm íšek. Motivation. The motivation for this work comes from developmental biology. zebrafish. Motivation. Motivation. Cell nuclei. Cell membranes. Motivation.
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4D Numerical Schemes for Cell Image Segmentation and Tracking Mariana Remešíková Karol Mikula, Nadine Peyriéras, Michal Smíšek
Motivation The motivation for this work comes from developmental biology zebrafish
Motivation Cell nuclei Cell membranes
Motivation Images of the embryo before and during organogenesis Question: Where do the organs come from?
Cell tracking Let us follow the time evolution of one cell nucleus
Cell tracking The celll nucleus evolution can be viewed as a tree-like structure
Cell tracking Let us pick one nucleus at the end of the evolution
Cell tracking We want to find the way down to the root of the tree
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
Segmentation of the cell evolution trees Generalized subjective surface model The function g(s) is the edge detector
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
Segmentation of the cell evolution trees • Approximation of the integrals • the time derivative term • the advection term – upwind principle
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
Segmentation of the cell evolution trees 4D volume (doxel)
Segmentation of the cell evolution trees Approximation of gradient on the doxel face
Segmentation of the cell evolution trees Approximation of gradient on the doxel face
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
Segmentation of the cell evolution trees Segmentation of zebrafish embryogenesis 4D data
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
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
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
Backtracking in the tree The time relaxed eikonal equation is discretized explicitly in time The space discretization is done by the Rouy-Tourin scheme
Backtracking in the tree The effect of the distance function d2
Backtracking in the tree Backtracking in artificial data
Backtracking in the tree Backtracking in zebrafish embryogenesis data
Backtracking in the tree Backtracking in zebrafish embryogenesis data