On the equivalence of binary phase masks optimized for localization or detection in extended depth-of-field localization microscopy.

Binary annular masks have recently been proposed to extend the depth of field (DoF) of single-molecule localization microscopy. A strategy for designing optimal masks has been introduced based on maximizing the emitter localization accuracy, expressed in terms of Fisher information, over a targeted DoF range. However, the complete post-processing pipeline to localize a single emitter consists of two successive steps: detection, where the regions containing emitters are determined, and localization, where the sub-pixel position of each detected emitter is estimated. Phase masks usually optimize only this second step. The presence of a phase mask also affecting detection, the purpose of this paper is to quantify and mitigate this effect. Using a rigorous framework built from a detection-oriented information theoretical criterion (Bhattacharyya distance), we demonstrate that in most cases of practical significance, annular binary phase masks maximizing Fisher information also maximize the detection probability. This result supports the common design practice consisting of optimizing a phase mask by maximizing Fisher information only.