Precision in iterative modulation enhanced single-molecule localization microscopy

Abstract

Super-resolution microscopy methods are able to image samples with improved resolution over the diffraction limit. Single-molecule localization microscopy (SMLM) methods break the diffraction limit by sparsely activating fluorescent emitters. The resulting sparsity in the emission signal can be exploited by estimation algorithms, which enables localization with improved precision. Modulation enhanced SMLM (meSMLM) methods further increase the localization precision by combining SMLM techniques with patterned illumination. By using prior information to refine emitter position estimates, iterative meSMLM (imeSMLM) methods such as iterative MINFLUX are able to locally improve the resolution. For (me)SMLM methods, the Cramér-Rao lower bound (CRLB) is often used to assess the localization precision. The CRLB bounds the variance of arbitrary unbiased estimators from below. As the CRLB treats estimands as deterministic unknowns, a prior distribution on the estimands cannot be incorporated into the bound. Therefore, the effect of prior information on the localization precision of imeSMLM methods is not captured by the CRLB. The Van Trees inequality (VTI) is a Bayesian variant of the CRLB. Because it treats estimands as random variables with a known prior distribution, it is able to account for the effect of prior information on the estimator precision. It is therefore able to accurately bound the localization precision of imeSMLM methods from below. An imeSMLM method is considered, in which the positions of sinusoidal intensity patterns are controlled over the course of multiple iterations. Intensity minima of sinusoidal patterns are placed symmetrically around the current estimate of the emitter position, at a distance based on the localization precision of the previous iteration. This strategy balances the information content of signal photons with the need for robustness to estimation errors. Using the VTI, we derive a fundamental limit on the localization precision of imeSMLM methods that make use of standing wave illumination patterns. This limit shows that in the absence of background, the information content of signal photons increases exponentially as a function of the iteration count. Using Monte Carlo simulations, the maximally achievable localization precision for different illumination pattern placement control strategies was evaluated. The VTI allows to assess the performance of pattern placement control strategies and is therefore a promising method for optimal control of imeSMLM methods.