2D localization uncertainty

Let $\hat{\theta}_{\sigma}$ be the standard deviation of a fitted Gaussian PSF in $\mathrm{nm}$ , is the backprojected pixel size in $\mathrm{nm}$ , $\hat{\theta}_{N}$ is estimate of the number of photons detected for a given molecule, and $\hat{b}$ is the background signal level in photons calculated as the standard deviation of the residuals between the raw data and the fitted PSF model. The uncertainty of least-squares or maximum-likelihood estimate of lateral position of a molecule is estimated as

$\begin{array}[]{rcl}\left.(\Delta\hat{\theta}_{xy})^{2}\right|_{\mathrm{LSQ}}&% =&\frac{g\hat{\theta}_{\sigma^{2}}+a^{2}/12}{\hat{\theta}_{N}}\left(\frac{16}{% 9}+4\tau\right)\,,\\ \left.(\Delta\hat{\theta}_{xy})^{2}\right|_{\mathrm{MLE}}&=&\frac{g\hat{\theta% }_{\sigma^{2}}+a^{2}/12}{\hat{\theta}_{N}}\left(1+4\tau+\sqrt{\frac{2\tau}{1+4% \tau}}\right)\,,\end{array}$

(1)

respectively. Here

$\tau=\frac{2\pi(\hat{b}^{2}+r)(\hat{\theta}_{\sigma}^{2}+a^{2}/12)}{a^{2}\hat{% \theta}_{N}}\,.$

The uncertainty for least-squares estimate is also known as the Thompson-Larson-Webb formula [4], which has been altered with the correction factor of $\frac{16}{9}$ as suggested by [1]. The uncertainty for maximum-likelihood was derived in [3]. Finally, the compensation for readout noise and EM gain has been added by following [2], who suggested that when using EMCCD cameras, the correction factors should be set to , and when using CCD or sCMOS cameras the readout noise in electron counts should be set to .

3D localization uncertainty

The lateral uncertainty is calculated same as in the 1, but $\tau$ differs because of the axial defocus (PSF spreads and is never focused in both planes simultaneously, thus the uncertainty is worse). This has been derived in [3] as

$\tau=\frac{2\pi(\hat{b}^{2}+r)(\hat{\theta}_{\sigma_{1}}\hat{\theta}_{\sigma_{% 2}}(1+l^{2}/d^{2})+a^{2}/12)}{a^{2}\hat{\theta}_{N}}\,.$

Since the axial position is estimated from $\hat{\theta}_{\sigma_{1}}$ and $\hat{\theta}_{\sigma_{2}}$ , the axial uncertainty is calculated from uncertainty of these parameters

$\begin{array}[]{rcl}\left.(\Delta\hat{\theta}_{\sigma_{j}})^{2}\right|_{% \mathrm{LSQ}}&=&\frac{g\hat{\theta}_{\sigma_{j}^{2}}+a^{2}/12}{\hat{\theta}_{N% }}\left(1+8\tau\right)\,,\\ \left.(\Delta\hat{\theta}_{\sigma_{j}})^{2}\right|_{\mathrm{MLE}}&=&\frac{g% \hat{\theta}_{\sigma_{j}^{2}}+a^{2}/12}{\hat{\theta}_{N}}\left(1+8\tau+\sqrt{% \frac{9\tau}{1+4\tau}}\right)\,,\end{array}$

(2)

where can be substituted to calculate uncertainty of $\hat{\theta}_{\sigma_{1}}$ and $\hat{\theta}_{\sigma_{2}}$ . Then from error propagation follows

	$\displaystyle F^{2}=\frac{4l^{2}\hat{\theta}_{z}^{2}}{(l^{2}+d^{2}+\hat{\theta% }_{z}^{2})^{2}}\,,$		(3)
	$\displaystyle(\Delta F)^{2}=(1-F^{2})\left[\left(\frac{\Delta\hat{\theta}_{% \sigma_{1}}}{\hat{\theta}_{\sigma_{1}}}\right)^{2}+\left(\frac{\Delta\hat{% \theta}_{\sigma_{2}}}{\hat{\theta}_{\sigma_{2}}}\right)^{2}\right]\,,$		(4)
	$\displaystyle(\Delta\hat{\theta}_{z})^{2}=(\Delta F)^{2}\frac{(l^{2}+d^{2}+% \hat{\theta}_{z}^{2})^{4}}{4l^{2}(l^{2}+d^{2}-\hat{\theta}_{z}^{2})^{2}}\,,$		(5)

where is the distance between focal planes given by the astigmatic lens and the geometry of the setup and is a measure of focal depth. These quantities are already known during the 3D fitting process as $l^{2}\propto c_{1}c_{2}$ and $d^{2}\propto d_{1}d_{2}$ , where $c_{1},c_{2},d_{1},d_{2}$ are parameters of the defocus curves.

References

[1] K. I. Mortensen, L. S. Churchman, J. A. Spudich and H. Flyvbjerg(2010) Optimized localization analysis for single-molecule tracking and super-resolution microscopy. Nature Methods 7 (5), pp. 377–381. External Links: Document Cited by: 2D localization uncertainty.
[2] T. Quan, S. Zeng and Z.-L. Huang(2010) Localization capability and limitation of electron-multiplying charge-coupled, scientific complementary metal-oxide semiconductor, and charge-coupled devices for superresolution imaging. Journal of Biomedical Optics 15 (6), pp. 066005. External Links: Document Cited by: 2D localization uncertainty.
[3] B. Rieger and S. Stallinga(2014-03) The lateral and axial localization uncertainty in super-resolution light microscopy.. Chemphyschem: a European journal of chemical physics and physical chemistry 15 (4), pp. 664–70. External Links: Document, ISSN 1439-7641, Link Cited by: 2D localization uncertainty, 3D localization uncertainty.
[4] R. E. Thompson, D. R. Larson and W. W. Webb(2002) Precise nanometer localization analysis for individual fluorescent probes. Biophysical Journal 82 (5), pp. 2775–83. External Links: Document Cited by: 2D localization uncertainty.

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2D localization uncertainty

3D localization uncertainty

See also

References