Bayesian MINFLUX localization microscopy

Abstract

MINFLUX microscopy allows for localization of fluorophores with nanometer precision using targeted scanning with an illumination profile with a minimum. However, current scanning patterns and the overall procedure are based on heuristics, and may therefore be suboptimal. Here we present a rigorous Bayesian that offers maximal resolutions from either minimal detected photons or minimal exposures. We estimate using simulated localization runs that this approach should reduce the number of photons required for 1 nm resolution by a factor of about four.

Figures (5)

• Figure 1: Selected Bayesian MINFLUX localization. Shown are the posterior densities $P_k$ (blue), the likelihoods $P(n_k \mid \mathrm{\mathbf{x}}, \mathrm{\mathbf{r}}_k, \eta_k)$ (orange), the expected information gain $\mathrm{EIG}(\mathrm{\mathbf{r}}, P_{k-1})$ (green), and the observed photon counts $n_k$. Localization was performed from an isotropic Gaussian prior $P_0$ with standard deviation $\sigma = 150\,\mathrm{nm}$ with an expected photon count per step of $\mu = 2$ and minimum positions (red dots) chosen to maximize the expected information gain.
• Figure 2: Current maximum a posteriori (MAP) estimates $\operatorname{argmax}_{\mathrm{\mathbf{x}}} P_k(\mathrm{\mathbf{x}})$ and donut minimum positions $\mathrm{\mathbf{r}}_k$ for a selected Bayesian MINFLUX localization run performed with $\mu = 0.5$ for $150$ exposures $k$. Radial scale is linear below $1\,\mathrm{nm}$ and logarithmic above.
• Figure 3: Median localization accuracy $\sigma_\mathrm{1D}$ for various values of $\mu$ as function of a the cumulative detected photon count and b the number of exposures for the Bayesian approach using maximized EIG (solid colored lines) and semi-heuristic placement (dotted lines) as well as the conventional approach using a hexagonal pattern (black line) with varying photon counts for the final stage (diamond markers).
• Figure 4: Optimization of $\mathrm{\mathbf{r}}_k$ for $P_{k-1} = \mathcal{N(\sigma)}$ at $\mu = 0.1$ photons per step. a Expected information gain as a function of $\lVert \mathrm{\mathbf{r}}\rVert$ for various values of $\sigma$. b Distance $D(\sigma) = \lVert \operatorname{argmax}_{\mathrm{\mathbf{r}}} \mathrm{EIG}(\mathrm{\mathbf{r}}, \mathcal{N}(\sigma))\rVert$ at which the expected information gain is maximized as a function of $\sigma$ for various values of $\mu$.
• Figure 5: Median localization accuracy $\sigma_\mathrm{1D}$ for various background levels $b$ as function of the cumulative detected photon count for both optimized EIG (solid lines) and radial heuristic placement (dashed lines).