A Stochastic Analysis of Performance Limits for Optical Microscopes

Open Access
Original Article


The optical microscope is a powerful instrument for observing cellular events. Recently, the increased use of microscopy in quantitative biological research, including single molecule microscopy, has generated significant interest in determining the performance limits of an optical microscope. Here, we formulate this problem in the context of a parameter estimation approach in which the acquired imaging data is modeled as a spatio-temporal stochastic process. We derive formulations of the Fisher information matrix for models that allow both stationary and moving objects. The effects of background signal, detector size, pixelation and noise sources are also considered. Further, formulations are given that allow the study of defocused objects. Applications are discussed for the special case of the estimation of the location of objects, especially single molecules. Specific emphasis is placed on the derivation of conditions that guarantee block diagonal or diagonal Fisher information matrices.


Spatio-temporal stochastic processes Fisher information matrix Cramer-Rao lower bound Parameter estimation Fluorescence microscopy Optical imaging Single Molecule Microscopy Localization accuracy 


  1. Bobroff, N. 1986Position measurement with a resolution and noise limited instrumentReview of Scientific Instruments.5711521157CrossRefGoogle Scholar
  2. Born, M., Wolf, E. 1999Principles of opticsCambridge University PressCambridge, UKGoogle Scholar
  3. Bowman, F. 1968Introduction to Bessel functionsDoverNew YorkGoogle Scholar
  4. Coleman, T., Branch, M.A., Grace, A. 1999MATLAB Optimization Toolbox user manualThe Mathworks, Inc. ver. 2NatickGoogle Scholar
  5. Coleman, T., Branch, M.A., Grace, A. 2000MATLAB programming reference manual, Version 6The Mathworks Inc.MAGoogle Scholar
  6. Goodman, J.W. 1996Introduction to Fourier optics2Mc Graw HillUSAGoogle Scholar
  7. Kay, S.M. 1993Fundamentals of statistical signal processingPrentice Hall PTRNew JerseyGoogle Scholar
  8. Kubitscheck, U., Kückmann, O., Kues, T., Peters, R. 2000Imaging and tracking single GFP molecules in solutionBiophysical Journal7821702179Google Scholar
  9. Michalet, X., Kapanidis, A.N., Laurence, T., Pinaud, F., Doose, S., Pflughoefft, M., Weiss, S. 2003The power and prospects of fluorescence microscopies and spectroscopiesAnnual Review of Biophysics and Biomolecular Structure32161182CrossRefGoogle Scholar
  10. Moerner, W.E., Fromm, D.P. 2003Methods of single-molecule fluorescence spectroscopy and microscopyReview of Scientific Instruments7435973619CrossRefGoogle Scholar
  11. Ober, R.J., Ram, S., Ward, E.S. 2004Localization accuracy in single molecule microscopyBiophysical Journal8611851200Google Scholar
  12. Ober, R.J., Martinez, C., Lai, X., Zhou, J., Ward, E. S. 2004Exocytosis of IgG as mediated by the receptor, FcRn: An analysis at the single molecule levelProceeding of the National Academy of Sciences1011107611081Google Scholar
  13. Papoulis, A., Pillai, S.U. 2002Probability, random variables and stochastic processes4McGraw HillNew YorkGoogle Scholar
  14. Ram, S. (in preparation). Ph.D. Dissertation, University of Texas at Arlington/University of Texas Southwestern Medical Center at DallasGoogle Scholar
  15. Ram, S., Ward, E.S., Ober, R.J. 2005How accurately can a single molecule be localized in three dimensions using a fluorescence microscope?Proceeding of SPIE5699426435Google Scholar
  16. Rudin, W. 1987Real and complex analysisMc Graw HillNew YorkGoogle Scholar
  17. Saleh, B. 1978Photoelectron statisticsSpringer VerlagBerlin, GermanyGoogle Scholar
  18. Santos, A., Young, I.T. 2000Model-based resolution: Applying the theory in quantitative microscopyApplied Optics3929482958Google Scholar
  19. Saxton, M.J., Jacobson, K. 1997Single particle tracking: Applications to membrane dynamicsAnnual Review of Biophysics and Biomolecular Structure26373399CrossRefGoogle Scholar
  20. Snyder, D.L., Miller, M.I. 1999Random point processes in time and space2Springer VerlagNew YorkGoogle Scholar
  21. Snyder, D.L., Helstrom, C.W., Lanterman, A.D., White, R.L. 1995Compensation for read out noise in charge coupled device imagesJournal of the Optical Society of America A- Optics Image Science and Vision12272283Google Scholar
  22. Thompson, R.E., Larson, D.R., Webb, W.W. 2002Precise nanometer localization analysis for individual fluorescent probesBiophysical Journal8227752783Google Scholar
  23. Watson, G.N. 1958A treatise on the theory of Bessel functionsCambridge University PressCambridge, UKGoogle Scholar
  24. Weiss, S. 1999Fluorescence spectroscopy of single biomoleculesScience28316761683CrossRefGoogle Scholar
  25. Winick, K. A. 1986Cramer-Rao lower bounds on the performance of charge coupled device optical position estimatorsJornal of the Optical Society America A-Optics Image Science and Vision318091815Google Scholar
  26. Zacks, S. 1971The theory of statistical inferenceJohn Wiley and SonsNew YorkGoogle Scholar
  27. Zhang, F. 1999Matrix theorySpringer VerlagNew YorkGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Center for Immunology, NB9.106University of Texas Southwestern Medical Center at DallasDallasUSA
  2. 2.Joint Biomedical Engineering Graduate ProgramUniversity of Texas at Arlington and University of Texas Southwestern Medical Center at DallasTexas
  3. 3.Department of Electrical EngineeringUniversity of Texas at Dallas RichardsonUSA

Personalised recommendations