Successful adaptation of Besag’s boundary correction at non-binary fields of view.

The normalized and centered K-function (), as described by Lagache et al, was calculated at different particle intensities and radii using simulated square fields of view measuring 256 x 256 pixels, with a homogeneous Poisson distribution of particles (Complete Spatial Randomness). Densities are expressed in particles per pixel. Besag’s boundary correction was applied to all particles within the field of view and not just border pixels, as described in the text and in Fig 2. Panels A and B: The black line represents the mean and the dotted lines represent the upper and lower critical quantiles (Q01 and Q99). In order to ensure convergence, each mean/quantile was determined using a set of 25,000 simulations. The red dotted lines represent the Cornish-Fisher expansion used to estimate the critical quantiles, as validated by Lagache et al., while the green dotted lines represent the normal quantiles. Unlike the other unsuccessful implementations shown in Figs 3 and 4, mean remains centered on zero even at very high particle densities. Panel C: Quantifying relative error in the Cornish-Fisher expansion estimation of the empirically-determined critical quantiles.