Multidimensional Systems and Signal Processing

, Volume 24, Issue 3, pp 503–542 | Cite as

A stochastic analysis of distance estimation approaches in single molecule microscopy: quantifying the resolution limits of photon-limited imaging systems

  • Sripad Ram
  • E. Sally Ward
  • Raimund J. OberEmail author
Open Access


Optical microscopy is an invaluable tool to visualize biological processes at the cellular scale. In the recent past, there has been significant interest in studying these processes at the single molecule level. An important question that arises in single molecule experiments concerns the estimation of the distance of separation between two closely spaced molecules. Presently, there exists different experimental approaches to estimate the distance between two single molecules. However, it is not clear as to which of these approaches provides the best accuracy for estimating the distance. Here, we address this problem rigorously by using tools of statistical estimation theory. We derive formulations of the Fisher information matrix for the underlying estimation problem of determining the distance of separation from the acquired data for the different approaches. Through the Cramer-Rao inequality, we derive a lower bound to the accuracy with which the distance of separation can be estimated. We show through Monte-Carlo simulations that the bound can be attained by the maximum likelihood estimator. Our analysis shows that the distance estimation problem is in fact related to the localization accuracy problem, the latter being a distinct problem that deals with how accurately the location of an object can be determined. We have carried out a detailed investigation of the relationship between the Fisher information matrices of the two problems for the different experimental approaches considered here. The paper also addresses the issue of a singular Fisher information matrix, which presents a significant complication when calculating the Cramer-Rao lower bound. Here, we show how experimental design can overcome the singularity. Throughout the paper, we illustrate our results by considering a specific image profile that describe the image of a single molecule.


Marked point process Photon statistics Performance bounds Fluorescence microscopy Resolution limits Rayleigh’s criterion 


  1. Apostol T. M. (1974) Mathematical analysis. Addison Wesley Publishing Company, Boston, USAzbMATHGoogle Scholar
  2. Betzig E., Patterson G. H., Sougrat R., Lindwasser O. W., Olenych S., Bonifacino J. S., Davidson M. W., Lippincott-Schwartz J., Hess H. F. (2006) Imaging intracellular fluorescent proteins at nanometer resolution. Science 313: 1642–1645CrossRefGoogle Scholar
  3. Born M., Wolf E. (1999) Principles of optics. Cambridge University Press, Cambridge, UKGoogle Scholar
  4. Chao J., Ram S., Abraham A., Ward E. S., Ober R. J. (2009a) Resolution in three-dimensional microscopy. Optics Communications 282: 1751–1761CrossRefGoogle Scholar
  5. Chao, J., Ram, S., Ward, E. S., & Ober, R. J. (2009b). A 3D resolution measure for optical microscopy. In IEEE international symposium on biomedical imaging (pp. 1115–1118).Google Scholar
  6. Chao J., Ram S., Ward E. S., Ober R. J. (2009c) A comparative study of high resolution microscopy imaging modalities using a three-dimensional resolution measure. Optics Express 17: 24,377–24,402Google Scholar
  7. Gordon M. P., Ha T., Selvin P. R. (2004) Single molecule high resolution imaging with photobleaching. Proceedings of the National Academy of Sciences USA 101: 6462–6465CrossRefGoogle Scholar
  8. Helstrom C. W. (1964) The detection and resolution of optical signals. IEEE Transactions on Information Theory 10: 275–287CrossRefGoogle Scholar
  9. Hess S. T., Girirajan T. P. K., Mason M. D. (2006) Ultra-high resolution imaging by fluorescence photoactivation localization microscopy. Biophysical Journal 91: 4258–4272CrossRefGoogle Scholar
  10. Kay S. M. (1993) Fundamentals of statistical signal processing. Prentice Hall PTR, New Jersey, USAzbMATHGoogle Scholar
  11. Lagerholm B. C., Averett L., Weinreb G. E., Jacobson K., Thompson N. L. (2006) Analysis method for measuring submicroscopic distances with blinking quantum dots. Biophysical Journal 91: 3050–3060CrossRefGoogle Scholar
  12. Lidke K. A., Rieger B., Jovin T. M., Heintzmann R. (2005) Superresolution by localization of quantum dots using blinking statistics. Optics Express 13: 7052–7062CrossRefGoogle Scholar
  13. Moerner W. E. (2007) New directions in single-molecule imaging and analysis. Proceedings of the National Academy of Sciences USA 104: 12,596–12,602CrossRefGoogle Scholar
  14. Ober R. J., Martinez C., Lai X., Zhou J., Ward E. S. (2004a) Exocytosis of IgG as mediated by the receptor, FcRn: an analysis at the single molecule level. Proceedings of the National Academy of Sciences USA 101: 11,076–11,081CrossRefGoogle Scholar
  15. Ober R. J., Ram S., Ward E. S. (2004b) Localization accuracy in single molecule microscopy. Bio- physical Journal 86: 1185–1200Google Scholar
  16. O’Sullivan J. A., Blahut R. E., Snyder D. L. (1998) Information-theoretic image formation. IEEE Transactions on Information Theory 44: 2094–2123MathSciNetzbMATHCrossRefGoogle Scholar
  17. Qu X., Wu D., Mets L., Scherer N. F. (2004) Nanometer-localized multiple single-molecule fluoresc- ence microscopy. Proceedings of the National Academy of Sciences USA 101: 11,298–11,303CrossRefGoogle Scholar
  18. Ram S., Ward E. S., Ober R. J. (2006a) Beyond Rayleigh’s criterion: A resolution measure with application to single-molecule microscopy. Proceedings of the National Academy of Sciences USA 103: 4457–4462CrossRefGoogle Scholar
  19. Ram S., Ward E. S., Ober R. J. (2006b) A stochastic analysis of performance limits for optical microscopes. Multidimensional Systems and Signal Processing 17: 27–58MathSciNetzbMATHCrossRefGoogle Scholar
  20. Rao C. R. (1965) Linear statistical inference and its applications. Wiley, New York, USAzbMATHGoogle Scholar
  21. Rohr K. (2007) Theoretical limits of localizing 3D landmarks and features. IEEE Transactions on Biomedical Engineering 54: 1613–1620CrossRefGoogle Scholar
  22. Rudin W. (1987) Real and complex analysis. McGraw Hill, New York, USAzbMATHGoogle Scholar
  23. Rust M. J., Bates M., Zhuang X. (2006) Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM). Nature Methods 4: 793–795CrossRefGoogle Scholar
  24. Santos A., Young I. T. (2000) Model-based resolution: Applying the theory in quantitative microscopy. Applied Optics 39: 2948–2958CrossRefGoogle Scholar
  25. Shahram M., Milanfar P. (2004) Imaging below the diffraction limit: A statistical analysis. IEEE Transactions on Image Processing 13: 677–689CrossRefGoogle Scholar
  26. Smith S. T. (2005) Statistical resolution limits and the complexified Cramer-Rao bound. IEEE Transactions on Signal Processing 53: 1597–1609MathSciNetCrossRefGoogle Scholar
  27. Stoica P., Marzetta T. L. (2001) Parameter estimation problems with singular Fisher information matrices. IEEE Transactions on Signal Processing 49: 87–90MathSciNetCrossRefGoogle Scholar
  28. Strauss W. A. (1992) Partial differential equations—an introduction. Wiley, New YorkzbMATHGoogle Scholar
  29. Van des Bos A. (2007) Parameter estimation for scientists and engineers. Wiley, New York, USACrossRefGoogle Scholar
  30. Wong Y., Lin Z., Ober R. J. (2011) Limit of the accuracy of parameter estimation for moving single molecules imaged by fluorescence microscopy. IEEE Transactions on Signal Processing 59: 895–908MathSciNetCrossRefGoogle Scholar
  31. Young I. T. (1996) Quantitative microscopy. IEEE Engineering in Medicine and Biology 15: 59–66CrossRefGoogle Scholar
  32. Zhang F. (1999) Matrix theory. Springer, New York, USAzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of ImmunologyUniversity of Texas Southwestern Medical CenterDallasUSA
  2. 2.Department of Electrical EngineeringUniversity of Texas at DallasRichardsonUSA

Personalised recommendations