  A variety of recent imaging techniques are able to beat the diffraction limit
in fluorescence microcopy by activating and localizing subsets of the
fluorescent molecules in the specimen, and repeating this process until all of
the molecules have been imaged. In these techniques there is a tradeoff between
speed (activating more molecules per imaging cycle) and error rates (activating
more molecules risks producing overlapping images that hide information on
molecular positions), and so intelligent image-processing approaches are needed
to identify and reject overlapping images. We introduce here a formalism for
defining error rates, derive a general relationship between error rates, image
acquisition rates, and the performance characteristics of the image processing
algorithms, and show that there is a minimum acquisition time irrespective of
algorithm performance. We also consider algorithms that can infer molecular
positions from images of overlapping blurs, and derive the dependence of the
minimimum acquisition time on algorithm performance.
