Unveiling the limits of precision in iterative MINFLUX.

In single-molecule microscopy, a big question is how precisely we can estimate the location of a single molecule. Our research shows that by using iterative localisation microscopy and factoring in the prior information, we can boost precision and reduce the number of photons needed. Leveraging the Van Trees inequality aids in determining the optimal precision achievable. Our approach holds promise for wider application in discerning the optimal precision across diverse imaging scenarios, encompassing various illumination strategies, point spread functions and overarching control methodologies.

Image scanning microscopy: a vectorial physical optics analysis.

Image scanning microscopy (ISM) achieves resolution beyond the diffraction limit by a factor of 2. However, prior ISM research predominantly employs scalar diffraction theory, neglecting critical physical effects such as polarization, aberrations, and Stokes shift. This paper presents a comprehensive vectorial ISM point spread function (PSF) model that accounts for these phenomena. By considering the effect of polarization in emission and excitation paths, as well as aberrations and Stokes shift, our model provides a more accurate representation of ISM. We analyze the differences between scalar and vectorial theories in ISM and investigate the impact of pinhole size and aberration strength on resolution. At a numerical aperture of 1.2, the full width half maximum (FWHM) discrepancy between scalar and vectorial ISM PSFs can reach 45 nm, representing a 30% deviation from the vectorial model. Additionally, we explore multiphoton excitation in ISM and observe increased FWHM for 2-photon and 3-photon excitation compared to 1-photon excitation. The FWHM of the 2-photon excitation ISM PSF increases by 20% and the FWHM of the 3-photon excitation ISM PSF increases by 28% compared to the 1-photon excitation ISM. In addition, we found that the optimal sweep factor for 2-photon ISM is 1.22, and the optimal sweep factor of 3-photon ISM is 1.12 instead of the 2 predicted by the one-photon scalar ISM theory. Our work improves the understanding of ISM and contributes to its advancement as a high-resolution imaging technique.

Quantifying the minimum localization uncertainty of image scanning localization microscopy.

Modulation enhanced single-molecule localization microscopy (meSMLM), where emitters are sparsely activated with sequentially applied patterned illumination, increases the localization precision over single-molecule localization microscopy (SMLM). The precision improvement of modulation enhanced SMLM is derived from retrieving the position of an emitter relative to individual illumination patterns, which adds to existing point spread function information from SMLM. Here, we introduce SpinFlux: modulation enhanced localization for spinning disk confocal microscopy. SpinFlux uses a spinning disk with pinholes in its illumination and emission paths, to sequentially illuminate regions in the sample during each measurement. The resulting intensity-modulated emission signal is analyzed for each individual pattern to localize emitters with improved precision. We derive a statistical image formation model for SpinFlux and we quantify the theoretical minimum localization uncertainty in terms of the Cramér-Rao lower bound. Using the theoretical minimum uncertainty, we compare SpinFlux to localization on Fourier reweighted image scanning microscopy reconstructions. We find that localization on image scanning microscopy reconstructions with Fourier reweighting ideally results in a global precision improvement of 2.1 over SMLM. When SpinFlux is used for sequential illumination with three patterns around the emitter position, the localization precision improvement over SMLM is twofold when patterns are focused around the emitter position. If four donut-shaped illumination patterns are used for SpinFlux, the maximum local precision improvement over SMLM is increased to 3.5. Localization of image scanning microscopy reconstructions thus has the largest potential for global improvements of the localization precision, where SpinFlux is the method of choice for local refinements.

Bayesian posterior density estimation reveals degeneracy in three-dimensional multiple emitter localization.

Single-molecule localization microscopy requires sparse activation of emitters to circumvent the diffraction limit. In densely labeled or thick samples, overlap of emitter images is inevitable. Single-molecule localization of these samples results in a biased parameter estimate with a wrong model of the number of emitters. On the other hand, multiple emitter fitting suffers from point spread function degeneracy, which increases model and parameter uncertainty. To better estimate the model, parameters and uncertainties, a three-dimensional Bayesian multiple emitter fitting algorithm was constructed using Reversible Jump Markov Chain Monte Carlo. It reconstructs the posterior density of both the model and the parameters, namely the three-dimensional position and photon intensity, of overlapping emitters. The ability of the algorithm to separate two emitters at varying distance was evaluated using an astigmatic point spread function. We found that for astigmatic imaging, the posterior distribution of the emitter positions is multimodal when emitters are within two times the in-focus standard deviation of the point spread function. This multimodality describes the ambiguity in position that astigmatism introduces in localization microscopy. Biplane imaging was also tested, proving capable of separating emitters up to 0.75 times the in-focus standard deviation of the point spread function while staying free of multimodality. The posteriors seen in astigmatic and biplane imaging demonstrate how the algorithm can identify point spread function degeneracy and evaluate imaging techniques for three-dimensional multiple-emitter fitting performance.

Precision in iterative modulation enhanced single-molecule localization microscopy.

Modulation enhanced single-molecule localization microscopy (meSMLM) methods improve the localization precision by using patterned illumination to encode additional position information. Iterative meSMLM (imeSMLM) methods iteratively generate prior information on emitter positions, used to locally improve the localization precision during subsequent iterations. The Cramér-Rao lower bound cannot incorporate prior information to bound the best achievable localization precision because it requires estimators to be unbiased. By treating estimands as random variables with a known prior distribution, the Van Trees inequality (VTI) can be used to bound the best possible localization precision of imeSMLM methods. An imeSMLM method is considered, where the positions of in-plane standing-wave illumination patterns are controlled over the course of multiple iterations. Using the VTI, we analytically approximate a lower bound on the maximum localization precision of imeSMLM methods that make use of standing-wave illumination patterns. In addition, we evaluate the maximally achievable localization precision for different illumination pattern placement strategies using Monte Carlo simulations. We show that in the absence of background and under perfect modulation, the information content of signal photons increases exponentially as a function of the iteration count. However, the information increase is no longer exponential as a function of the iteration count under non-zero background, imperfect modulation, or limited mechanical resolution of the illumination positioning system. As a result, imeSMLM with two iterations reaches at most a fivefold improvement over SMLM at 8 expected background photons per pixel and 95% modulation contrast. Moreover, the information increase from imeSMLM is balanced by a reduced signal photon rate. Therefore, SMLM outperforms imeSMLM when considering an equal measurement time and illumination power per iteration. Finally, the VTI is an excellent tool for the assessment of the performance of illumination control and is therefore the method of choice for optimal design and control of imeSMLM methods.

Theoretical minimum uncertainty of single-molecule localizations using a single-photon avalanche diode array.

Single-photon avalanche diode (SPAD) arrays can be used for single-molecule localization microscopy (SMLM) because of their high frame rate and lack of readout noise. SPAD arrays have a binary frame output, which means photon arrivals should be described as a binomial process rather than a Poissonian process. Consequentially, the theoretical minimum uncertainty of the localizations is not accurately predicted by the Poissonian Cramér-Rao lower bound (CRLB). Here, we derive a binomial CRLB and benchmark it using simulated and experimental data. We show that if the expected photon count is larger than one for all pixels within one standard deviation of a Gaussian point spread function, the binomial CRLB gives a 46% higher theoretical uncertainty than the Poissonian CRLB. For typical SMLM photon fluxes, where no saturation occurs, the binomial CRLB predicts the same uncertainty as the Poissonian CRLB. Therefore, the binomial CRLB can be used to predict and benchmark localization uncertainty for SMLM with SPAD arrays for all practical emitter intensities.