  We present an analytical method to quantify clustering in super-resolution
localization images of static surfaces in two dimensions. The method also
describes how over-counting of labeled molecules contributes to apparent
self-clustering and how the effective lateral resolution of an image can be
determined. This treatment applies to clustering of proteins and lipids in
membranes, where there is significant interest in using super-resolution
localization techniques to probe membrane heterogeneity. When images are
quantified using pair correlation functions, the magnitude of apparent
clustering due to over-counting will vary inversely with the surface density of
labeled molecules and does not depend on the number of times an average
molecule is counted. Over-counting does not yield apparent co-clustering in
double label experiments when pair cross-correlation functions are measured. We
apply our analytical method to quantify the distribution of the IgE receptor
(Fc{\epsilon}RI) on the plasma membranes of chemically fixed RBL-2H3 mast cells
from images acquired using stochastic optical reconstruction microscopy (STORM)
and scanning electron microscopy (SEM). We find that apparent clustering of
labeled IgE bound to Fc{\epsilon}RI detected with both methods arises from
over-counting of individual complexes. Thus our results indicate that these
receptors are randomly distributed within the resolution and sensitivity limits
of these experiments.
