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Licensed Unlicensed Requires Authentication Published by De Gruyter February 14, 2023

Computational resolution in single molecule localization – impact of noise level and emitter density

  • Mathias Hockmann EMAIL logo , Stefan Kunis ORCID logo EMAIL logo and Rainer Kurre EMAIL logo
From the journal Biological Chemistry


Classical fluorescence microscopy is a powerful technique to image biological specimen under close-to-native conditions, but light diffraction limits its optical resolution to 200–300 nm-two orders of magnitude worse than the size of biomolecules. Assuming single fluorescent emitters, the final image of the optical system can be described by a convolution with the point spread function (PSF) smearing out details below the size of the PSF. In mathematical terms, fluorescence microscopy produces bandlimited space-continuous images that can be recovered from their spatial samples under the conditions of the classical Shannon-Nyquist theorem. During the past two decades, several single molecule localization techniques have been established and these allow for the determination of molecular positions with sub-pixel accuracy. Without noise, single emitter positions can be recovered precisely – no matter how close they are. We review recent work on the computational resolution limit with a sharp phase transition between two scenarios: 1) where emitters are well-separated with respect to the bandlimit and can be recovered up to the noise level and 2) closely distributed emitters which results in a strong noise amplification in the worst case. We close by discussing additional pitfalls using single molecule localization techniques based on structured illumination.

Corresponding authors: Mathias Hockmann, Stefan Kunis and Rainer Kurre, Department of Mathematics and Center for Cellular Nanoanalytics, Osnabrück University, Barbarastrasse 113, D-49076 Osnabruck, Germany, E-mail: (M. Hockmann), (S. Kunis), (R. Kurre)

Funding source: Volkswagen Foundation

Award Identifier / Grant number: Stability of Moment Problems and Super-Resolution

Award Identifier / Grant number: CRC944

  1. Author contributions: All authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was funded within the Collaborative Research Center 944 “Physiology and Dynamics of Cellular Microcompartments” (Project Z: Advanced imaging techniques) and by the Volkswagen Foundation project “Stability of Moment Problems and Super-Resolution Imaging”.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.


Batenkov, D., Goldman, G., and Yomdin, Y. (2021). Super-resolution of near-colliding point sources. Inf. Inference 10: 515–572.10.1093/imaiai/iaaa005Search in Google Scholar

Batenkov, D. and Yomdin, Y. (2014). Geometry and singularities of the prony mapping. J. Singularities 10: 1–25.10.5427/jsing.2014.10aSearch in Google Scholar

Chen, S. and Moitra, A. (2021). Algorithmic foundations for the diffraction limit. In: Proceedings of the 53rd annual ACM symposium on theory of computing, pp. 490–503.10.1145/3406325.3451078Search in Google Scholar

Cnossen, J., Hinsdale, T., Thorsen, R.Ø., Siemons, M., Schueder, F., Jungmann, R., Smith, C.S., Rieger, B., and Stallinga, S. (2019). Localization microscopy at doubled precision with patterned illumination. Nat. Methods 17: 59–63.10.1038/s41592-019-0657-7Search in Google Scholar PubMed PubMed Central

Demanet, L. and Nguyen, N. (2015). The recoverability limit for superresolution via sparsity. ArXiv Preprint, in Google Scholar

Diederichs, B. (2018). Sparse frequency estimation: stability and algorithms, Ph.D. thesis, University of Hamburg.Search in Google Scholar

Donoho, D.L. (1992). Superresolution via sparsity constraints. SIAM J. Math. Anal. 23: 1309–1331.10.1137/0523074Search in Google Scholar

Ehler, M., Kunis, S., Peter, T., and Richter, C. (2019). A randomized multivariate matrix pencil method for superresolution microscopy. Electron. Trans. Numer. Anal. 51: 63–74.10.1553/etna_vol51s63Search in Google Scholar

Fan, Z. and Li, J. (2022). Efficient algorithms for sparse moment problems without separation. ArXiv Preprint, in Google Scholar

Gu, L., Li, Y., Zhang, S., Xue, Y., Li, W., Li, D., Xu, T., and Ji, W. (2019). Molecular resolution imaging by repetitive optical selective exposure. Nat. Methods 16: 1114–1118.10.1038/s41592-019-0544-2Search in Google Scholar PubMed

Gustafsson, M.G.L. (2000). Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy. J. Microsc. 198: 82–87.10.1046/j.1365-2818.2000.00710.xSearch in Google Scholar PubMed

Heintzmann, R. and Cremer, C. (1998). Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating. Proc. SPIE 3568: 185–196.10.1117/12.336833Search in Google Scholar

Heintzmann, R. and Huser, T. (2017). Super-resolution structured illumination microscopy. Chem. Rev. 117: 13890–13908.10.1021/acs.chemrev.7b00218Search in Google Scholar PubMed

Hockmann, M., Kunis, S., and Kurre, R. (2021). Towards a mathematical model for single molecule structured illumination microscopy. Proc. Appl. Math. Mech. 20: e202000075.10.1002/pamm.202000075Search in Google Scholar

Hockmann, M. and Kunis, S. (2022). Weak sparse super resolution is well-conditioned. SIAM J. Imag. Sci., in press.10.1137/22M1521353Search in Google Scholar

Holden, S.J., Uphoff, S., and Kapanidis, A.N. (2011). DAOSTORM: an algorithm for high-density super-resolution microscopy. Nat. Methods 8: 279–280.10.1038/nmeth0411-279Search in Google Scholar PubMed

Hua, Y. and Sarkar, T.K. (1990). Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Signal Process. 38: 814–824.10.1109/29.56027Search in Google Scholar

Huang, F., Schwartz, S.L., Byars, J.M., and Lidke, K.A. (2011). Simultaneous multiple-emitter fitting or single molecule super-resolution imaging. Biomed. Opt. Express 2: 1377–1393.10.1364/BOE.2.001377Search in Google Scholar PubMed PubMed Central

Kunis, S., Nagel, D., and Strotmann, A. (2022). Multivariate Vandermonde matrices with separated nodes on the unit circle are stable. Appl. Comput. Harmon. Anal. 58: 50–59.10.1016/j.acha.2022.01.001Search in Google Scholar

Kunis, S. and Nagel, D. (2020). On the smallest singular value of multivariate Vandermonde matrices with clustered nodes. Linear Algebra Appl. 604: 1–20.10.1016/j.laa.2020.06.003Search in Google Scholar

Kusumi, A., Nakada, C., Ritchie, K., Murase, K., Suzuki, K., Murakoshi, H., Kasai, R.S., Kondo, J., and Fujiwara, T. (2005). Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecules. Annu. Rev. Biophys. Biomol. Struct. 34: 351–378.10.1146/annurev.biophys.34.040204.144637Search in Google Scholar PubMed

Li, W. and Liao, W. (2021). Stable super-resolution limit and smallest singular value of restricted Fourier matrices. Appl. Comput. Harmon. Anal. 51: 118–156.10.1016/j.acha.2020.10.004Search in Google Scholar

Lelek, M., Gyparaki, M.T., Beliu, G., Schueder, F., Griffié, J., Manley, S., Jungmann, R., Sauer, M., Lakadamyali, M., and Zimmer, C. (2021). Single-molecule localization microscopy. Nat. Rev. Methods Primers 1: 39.10.1038/s43586-021-00038-xSearch in Google Scholar PubMed PubMed Central

Liu, P. and Zhang, H. (2021). A mathematical theory of computational resolution limit in multi-dimensional spaces. Inverse Probl. 37: 104001.10.1088/1361-6420/ac245bSearch in Google Scholar

Möckl, L. and Moerner, W.E. (2020). Super-resolution microscopy with single molecules in biology and beyond–essentials: current trends and future challenges. J. Am. Chem. Soc. 142: 17828–17844.10.1021/jacs.0c08178Search in Google Scholar PubMed PubMed Central

Moitra, A. (2015). Super-resolution, extremal functions and the condition number of Van-dermonde matrices. In: Proceedings of the 47th annual ACM symposium on theory of computing, pp. 821–830.10.1145/2746539.2746561Search in Google Scholar

Mortensen, K I., Stirling Churchman, L., Spudich, J.A., and Flyvbjerg, H. (2010). Optimized localization analysis for single-molecule tracking and super-resolution microscopy. Nat. Methods 7: 377–381.10.1038/nmeth.1447Search in Google Scholar PubMed PubMed Central

Mukamel, E.A., Babcock, H., and Zhuang, X. (2012). Statistical deconvolution for superresolution fluorescene microscopy. Biophys. J. 102: 2391–2400.10.1016/j.bpj.2012.03.070Search in Google Scholar PubMed PubMed Central

Nehme, E., Weiss, L.E., Michaeli, T., and Shechtman, Y. (2018). Deep-storm: super-resolution single molecule microscopy by deep learning. Optica 5: 458–464.10.1364/OPTICA.5.000458Search in Google Scholar

Quan, T., Zhu, H., Liu, X., Liu, Y., Ding, J., Zeng, S., and Huang, Z. (2011). High-density localization of active molecules using structured sparse model and bayesian information criterion. Opt. Express 19: 16963–16974.10.1364/OE.19.016963Search in Google Scholar PubMed

Reymond, L., Ziegler, J., Knapp, C., Wang, F., Huser, T., Ruprecht, V., and Wieser, S. (2019). SIMPLE: structured illumination based point localization estimator with enhanced precision. Opt. Express 27: 24578–24590.10.1364/OE.27.024578Search in Google Scholar PubMed

Roy, R. and Kailath, T. (1989). Esprit – estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process. 37: 984–995.10.1109/29.32276Search in Google Scholar

Sage, D., Pham, T., Babcock, H., Lukes, T., Pengo, T., Chao, J., Velmurugan, R., Herbert, A., Agrawal, A., Colabrese, S., et al.. (2019). Super-resolution fight club: assessment of 2D and 3D single-molecule localization microscopy software. Nat. Methods 16: 387–395.10.1038/s41592-019-0364-4Search in Google Scholar PubMed PubMed Central

Schmidt, R. (1986). Multiple emitter location and signal parameter estimation. IEEE Trans. Antenn. Propag. 34: 276–280.10.1109/TAP.1986.1143830Search in Google Scholar

Shen, H., Tauzin, L.J., Baiyasi, R., Wang, W., Moringo, N., Shuang, B., and Landes, C.F. (2017). Single particle tracking: from theory to biophysical applications. Chem. Rev. 117: 7331–7376.10.1021/acs.chemrev.6b00815Search in Google Scholar PubMed

Smith, C.S., Joseph, N., Rieger, B., and Lidke, K.A. (2010). Fast, single-molecule localization that achieves theoretically minimum uncertainty. Nat. Methods 7: 373–375.10.1038/nmeth.1449Search in Google Scholar PubMed PubMed Central

Speiser, A., Müller, L.R., Hoess, P., Matti, U., Obara, C.J., Legant, W.R., Kreshuk, A., Macke, J.H., Ries, J., and Turaga, S.C. (2021). Deep learning enables fast and dense single-molecule localization with high accuracy. Nat. Methods 18: 1082–1090.10.1038/s41592-021-01236-xSearch in Google Scholar PubMed PubMed Central

Sotolongo Bellón, J., Birkholz, O., Richter, C.P., Eull, F., Kenneweg, H., Wilmes, S., Rothbauer, U., You, C., Walter, M.R., Kurre, R., et al.. (2022). Four-color single-molecule imaging with engineered tags resolves the molecular architecture of signaling complexes in the plasma membrane. Cell Rep. Methods 2: 100165.10.1016/j.crmeth.2022.100165Search in Google Scholar PubMed PubMed Central

Stone, M.B., Shelby, S.A., and Veatch, S.L. (2017). Super-resolution microscopy: shedding light on the cellular plasma membrane. Chem. Rev. 117: 7457–7477.10.1021/acs.chemrev.6b00716Search in Google Scholar PubMed PubMed Central

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Royal Stat. Soc. B 58: 267–288.10.1111/j.2517-6161.1996.tb02080.xSearch in Google Scholar

Wang, L., Lu, J., Ji, W., Wan, L., and Gu, L. (2022). Interferometrical single-molecule localization based on dynamic PSF engineering. Opt. Lett. 47: 1770–1773.10.1364/OL.453113Search in Google Scholar PubMed

Wang, Z., Wang, X., Zhang, Y., Xu, W., and Han, X. (2021). Principles and applications of single particle tracking in cell research. Small 17: e2005133.10.1002/smll.202005133Search in Google Scholar PubMed

Zhu, L., Zhang, W., Elnatan, D., and Huang, B. (2012). Faster STORM using compressed sensing. Nat. Methods 9: 721–723.10.1038/nmeth.1978Search in Google Scholar PubMed PubMed Central

Received: 2022-09-30
Accepted: 2023-01-24
Published Online: 2023-02-14
Published in Print: 2023-04-25

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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