A stochastic analysis of distance estimation approaches in single molecule microscopy: quantifying the resolution limits of photonlimited imaging systems
 Sripad Ram,
 E. Sally Ward,
 Raimund J. Ober
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 CramerRao inequality, we derive a lower bound to the accuracy with which the distance of separation can be estimated. We show through MonteCarlo 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 CramerRao 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.
 Apostol, T. M. (1974) Mathematical analysis. Addison Wesley Publishing Company, Boston, USA
 Betzig, E., Patterson, G. H., Sougrat, R., Lindwasser, O. W., Olenych, S., Bonifacino, J. S., Davidson, M. W., LippincottSchwartz, J., Hess, H. F. (2006) Imaging intracellular fluorescent proteins at nanometer resolution. Science 313: pp. 16421645 CrossRef
 Born, M., Wolf, E. (1999) Principles of optics. Cambridge University Press, Cambridge, UK
 Chao, J., Ram, S., Abraham, A., Ward, E. S., Ober, R. J. (2009) Resolution in threedimensional microscopy. Optics Communications 282: pp. 17511761 CrossRef
 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).
 Chao, J., Ram, S., Ward, E. S., Ober, R. J. (2009) A comparative study of high resolution microscopy imaging modalities using a threedimensional resolution measure. Optics Express 17: pp. 24,37724,402
 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: pp. 64626465 CrossRef
 Helstrom, C. W. (1964) The detection and resolution of optical signals. IEEE Transactions on Information Theory 10: pp. 275287 CrossRef
 Hess, S. T., Girirajan, T. P. K., Mason, M. D. (2006) Ultrahigh resolution imaging by fluorescence photoactivation localization microscopy. Biophysical Journal 91: pp. 42584272 CrossRef
 Kay, S. M. (1993) Fundamentals of statistical signal processing. Prentice Hall PTR, New Jersey, USA
 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: pp. 30503060 CrossRef
 Lidke, K. A., Rieger, B., Jovin, T. M., Heintzmann, R. (2005) Superresolution by localization of quantum dots using blinking statistics. Optics Express 13: pp. 70527062 CrossRef
 Moerner, W. E. (2007) New directions in singlemolecule imaging and analysis. Proceedings of the National Academy of Sciences USA 104: pp. 12,59612,602 CrossRef
 Ober, R. J., Martinez, C., Lai, X., Zhou, J., Ward, E. S. (2004) 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: pp. 11,07611,081 CrossRef
 Ober, R. J., Ram, S., Ward, E. S. (2004) Localization accuracy in single molecule microscopy. Bio physical Journal 86: pp. 11851200
 O’Sullivan, J. A., Blahut, R. E., Snyder, D. L. (1998) Informationtheoretic image formation. IEEE Transactions on Information Theory 44: pp. 20942123 CrossRef
 Qu, X., Wu, D., Mets, L., Scherer, N. F. (2004) Nanometerlocalized multiple singlemolecule fluoresc ence microscopy. Proceedings of the National Academy of Sciences USA 101: pp. 11,29811,303 CrossRef
 Ram, S., Ward, E. S., Ober, R. J. (2006) Beyond Rayleigh’s criterion: A resolution measure with application to singlemolecule microscopy. Proceedings of the National Academy of Sciences USA 103: pp. 44574462 CrossRef
 Ram, S., Ward, E. S., Ober, R. J. (2006) A stochastic analysis of performance limits for optical microscopes. Multidimensional Systems and Signal Processing 17: pp. 2758 CrossRef
 Rao, C. R. (1965) Linear statistical inference and its applications. Wiley, New York, USA
 Rohr, K. (2007) Theoretical limits of localizing 3D landmarks and features. IEEE Transactions on Biomedical Engineering 54: pp. 16131620 CrossRef
 Rudin, W. (1987) Real and complex analysis. McGraw Hill, New York, USA
 Rust, M. J., Bates, M., Zhuang, X. (2006) Subdiffractionlimit imaging by stochastic optical reconstruction microscopy (STORM). Nature Methods 4: pp. 793795 CrossRef
 Santos, A., Young, I. T. (2000) Modelbased resolution: Applying the theory in quantitative microscopy. Applied Optics 39: pp. 29482958 CrossRef
 Shahram, M., Milanfar, P. (2004) Imaging below the diffraction limit: A statistical analysis. IEEE Transactions on Image Processing 13: pp. 677689 CrossRef
 Smith, S. T. (2005) Statistical resolution limits and the complexified CramerRao bound. IEEE Transactions on Signal Processing 53: pp. 15971609 CrossRef
 Stoica, P., Marzetta, T. L. (2001) Parameter estimation problems with singular Fisher information matrices. IEEE Transactions on Signal Processing 49: pp. 8790 CrossRef
 Strauss, W. A. (1992) Partial differential equations—an introduction. Wiley, New York
 Van des Bos, A. (2007) Parameter estimation for scientists and engineers. Wiley, New York, USA CrossRef
 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: pp. 895908 CrossRef
 Young, I. T. (1996) Quantitative microscopy. IEEE Engineering in Medicine and Biology 15: pp. 5966 CrossRef
 Zhang, F. (1999) Matrix theory. Springer, New York, USA CrossRef
 Title
 A stochastic analysis of distance estimation approaches in single molecule microscopy: quantifying the resolution limits of photonlimited imaging systems
 Journal

Multidimensional Systems and Signal Processing
Volume 24, Issue 3 , pp 503542
 Cover Date
 20130901
 DOI
 10.1007/s1104501201756
 Print ISSN
 09236082
 Online ISSN
 15730824
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

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

 Sripad Ram ^{(1)}
 E. Sally Ward ^{(1)}
 Raimund J. Ober ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Immunology, University of Texas Southwestern Medical Center, Dallas, TX, USA
 2. Department of Electrical Engineering, University of Texas at Dallas, Richardson, TX, USA