Persistence diagrams as morphological signatures of cells: A method to measure and compare cells within a population
Fig 2
A) The input data: a cell contour and the centre of its nucleus marked; the latter serves as the base point for the radial distance function.
B) The radial distance function: The complete cell contour forms a graph G. The edges of this graph are measured relative to the cell centre by computing the largest Euclidean distance between the centre and the endpoints of the edge: the corresponding measure is the radial distance function with respect to the centre. Edges whose radial distance function is below a given cut-off value (or ‘time step’), illustrated as concentric circles around the centre, define a sub-graph of the whole contour. C) Graph filtration: Examples of sub-graphs for five different time steps. The different graphs obtained at increasing values of time form a filtration of the graph G. D) The persistence diagram captures the topological properties of the graph filtration. The points marked as ‘’ and ‘
’ indicate that the corresponding points have multiplicity 2 and 3, respectively, in the persistence diagram.