Fig. 3

Scheme for calculating PCC and MPCC for two representative projected images R and G arising from two random and independent distributions in 3D. a Heat maps of R and G with 200 nm pixels. Each image comprises ~ 10,000 molecules. Color scale indicates the number of molecules in each pixel. b Standard PCC calculation. Top: The 2D uniform reference distribution \( \overline{\mathrm{R}} \) or \( \overline{\mathrm{G}} \) that is subtracted from images R or G. Bottom: Normalized difference matrices \( \sim \left(\mathrm{R}-\overline{\mathrm{R}}\right) \) and \( \sim \left(\mathrm{G}-\overline{\mathrm{G}}\right) \) obtained after subtraction. c Modified PCC calculation. Top: Reference distribution \( {\overset{\sim }{\mathrm{U}}}^{\mathrm{R}} \) and \( {\overset{\sim }{\mathrm{U}}}^{\mathrm{G}} \), which are 2D projections of 3D random distributions of 100,000 molecules within the spherocylinder and normalized to have a total of 10,000 molecules. These are subtracted from images R and G, respectively. Bottom: Normalized difference matrices \( {\hat{\Delta}}^{\mathrm{R}} \) and \( {\hat{\Delta}}^{\mathrm{G}} \) obtained after subtraction. d Scatter plot of individual normalized difference matrix elements for PCC (Red) and for MPCC (Black). The MPCC elements are randomly distributed, while the PCC elements are positively correlated. The resulting MPCC and PCC values are + 0.10 and + 0.98, respectively