Automatic sorting of point pattern sets using Minkowski functionals

Point pattern sets arise in many different areas of physical, biological, and applied research, representing many random realizations of underlying pattern formation mechanisms. These pattern sets can be heterogeneous with respect to underlying spatial processes, which may not be visually distiguishable. This heterogeneity can be elucidated by looking at statistical measures of the patterns sets and using these measures to divide the pattern sets into distinct groups representing like spatial processes. We introduce here a numerical procedure for sorting point pattern sets into spatially homogenous groups using functional principal component analysis (FPCA) applied to the approximated Minkowski functionals of each pattern. We demonstrate that this procedure correctly sorts pattern sets into similar groups both when the patterns are drawn from similar processes and when the second-order characteristics of the pattern are identical. We highlight this routine for distinguishing the molecular patterning of fluorescently labeled cell membrane proteins, a subject of much interest in studies investigating complex spatial signaling patterns involved in the human immune response.

Figure 1

Using photoactived localization microscopy, the fluorescently labeled proteins are localized by fitting a point spread function to the stochastically photoactived molecules; the final pattern represents thousands of fluorescent images.

Figure 2

(a) The Minkowski functionals are calculated by imposing disks on the point pattern. This new secondary structure can be characterized using topological measures, which vary for different radii. (b) The three reduced Minkowski functionals for a 2D Poisson (random) process. These functionals are unitless due to the normalization by the same measure one would expect for a set of nonoverlapping disks.

Figure 3

(a) The regularity of a Strauss process is completely determined by $\gamma$, the interaction parameter. For $\gamma =0$, no overlaps are allowed. For $\gamma =1$, all overlaps are allowed. (b) The leftmost panes display the $g\left(r\right)$ and $\chi \left(r\right)$ for the 40 simulated Strauss processes. On the right are the results of using FPCA scores to divide the pattern set into two groups. As can be seen, both $g\left(r\right)$ and the Minkowski functionals can perfectly separate the set into two groups corresponding to different values of $\gamma$.

Figure 4

(a) A Baddeley-Silverman (left) process side-by-side with a Poisson process (right). Despite the visible differences, the pairwise correlation functions are identical. (b) The leftmost panes display the $g\left(r\right)$ and $\chi \left(r\right)$ for the 58 patterns simulated. On the right are the results of using FPCA scores to divide the pattern set into two groups. As can be seen, FPCA sorting with $g\left(r\right)$ creates two perfectly heterogenous groups, while FPCA sorting with the Minkowski functionals groups the patterns correctly.

Figure 5

(a) The two proteins are both dispersed in the cell membrane. LAT (blue) and TAC (red) proteins are separately tagged, allowing them to be visualized separately. (b) The leftmost panes display the $g\left(r\right)$ and $\chi \left(r\right)$ for the 16 molecular patterns. On the right are the results of using FPCA scores to divide the pattern set into two groups. As can be seen, using Minkowski functionals with FPCA improves the differentiation of the two sets.

Figure 6

From left to right: Sorting with all three functionals, the area, the perimeter, and the Euler number. Considering the individual functionals (sorting using weighted modularity), the Euler number outperforms the other two, only misclassifying one pattern.