Estimation Theoretic Measure of Resolution for Stochastic Localization Microscopy

One approach to super-resolution fluorescence microscopy, termed stochastic localization microscopy, relies on the nanometer scale spatial localization of individual fluorescent emitters that stochastically label specific features of the specimen. The precision of emitter localization is an important determinant of the resulting image resolution but is insufficient to specify how well the derived images capture the structure of the specimen. We address this deficiency by considering the inference of specimen structure based on the estimated emitter locations. By using estimation theory, we develop a measure of spatial resolution that jointly depends on the density of the emitter labels, the precision of emitter localization, and prior information regarding the spatial frequency content of the labeled object. The Nyquist criterion does not set the scaling of this measure with emitter number. Given prior information and a fixed emitter labeling density, our resolution measure asymptotes to a finite value as the precision of emitter localization improves. By considering the present experimental capabilities, this asymptotic behavior implies that further resolution improvements require increases in labeling density above typical current values. Our treatment also yields algorithms to enhance reliable image features. Overall, our formalism facilitates the rigorous statistical interpretation of the data produced by stochastic localization imaging techniques.

Figure 1

The optimal linear estimator outperforms any unbiased estimator. (A) We consider how the error associated with the optimal linear estimator compares to the theoretically optimal unbiased estimator (${ϵ}_{\mathrm{unbiased}}^2=1/(\overline{M}|H_{\mathrm{eff}}(k){|}^2)$) as a function of the ratio between the expected and true spectral densities (plotted logarithmically on the $x$ axis) and the mean SNR ($y$ axis). When this ratio is near unity, the optimal linear estimator is superior. At high SNR, the role of the bias decreases, but unbiased estimation is only superior for spatial frequencies present in the specimen but not in the expected spectral density. (B) The expected estimation error as a function of the ratio between the expected and true spectral densities ($x$ axis) and the mean SNR ($y$ axis).

Figure 2

Labeling density more strongly affects the resolution limit in stochastic localization microscopy than conventional microscopy. We assume that ${\alpha }^2={\sigma }^2/100$ to approximate an object of radius 15 nm and set $\beta =3$. Resolution depends both on the average number of emitters that label the object ($x$ axis) and on the average number of photons collected per emitter ($y$ axis). (A) Over the range of relevant labeling densities, the effective resolution in conventional imaging only changes modestly. (B) In stochastic localization microscopy, the achievable resolution varies substantially over the same range of labeling densities. Note the different color scales in (A) and (B).

Figure 3

Optimally filtered data approximate axonal cross sections more accurately than standard methods. We thresholded and rescaled each image in the data set [18] to have $2\mathrm{nm}\times 2\mathrm{nm}$ pixels and simulated stochastic localization microscopy with a Gaussian effective PSF whose standard deviation is 14 nm. Rows show different peak labeling densities: (i) $500\mu {\mathrm{m}}^{-2}$, (ii) $5\times {10}^3\mu {\mathrm{m}}^{-2}$, (iii) $5\times {10}^4\mu {\mathrm{m}}^{-2}$, (iv) $5\times {10}^5\mu {\mathrm{m}}^{-2}$. Columns show different reconstruction methods: (A) true specimen structure (200 nm scale bar) with estimated emitter locations shown for (Ai)-(Aiii); (B) estimate from the optimal linear filter; (C) estimate formed by reducing the number of pixels, which smooths the image; (D) estimate from Gaussian smoothing; (E) estimate from Eq. (8); (F) estimate from Eq. (9). Numbers in the lower left specify the average error over 1000 sessions. We report errors in multiples of the error from the optimal linear filter at the same labeling density. Numbers in the upper left represent method parameters: (A) mean number of emitters that label the displayed image; (C) number of pixels per side; (D) standard deviation of the Gaussian kernel; (E)–(F) number of iterations. All method parameters were optimized for each labeling density to minimize the method’s error. For each labeling density, we write the smallest error in red, the second smallest in violet, and the third smallest in blue. We chose the displayed axonal specimen randomly.