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What is superresolution microscopy?
2. L. Rayleigh, “ On the theory of optical images, with special reference to the microscope,” The London, Edinburgh, Dublin Philos. Mag. J. Sci. 42(XV), 167–195 (1896).
5. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. ( Roberts and Company Publishers, Greenwood Village, CO, 2005). The first edition was published in 1968.
6. F. M. Huang and N. I. Zheludev, “ Super-resolution without evanescent waves,” Nano Lett. 9, 1249–1254 (2009). The authors give a modern implementation of the aperture schemes pioneered by Toraldo Di Francia (Ref. 4).
7. “Superresolution” is also sometimes used to describe sub-pixel resolution in an imaging detector. Since pixels are not necessarily related to intrinsic resolution, we do not consider such techniques here.
9. Superresolution fluorescence microscopy was the 2008 “Method of the Year” for Nature Methods, and its January 2009 issue contains commentary and interviews with scientists playing a principal role in its development. This is a good “cultural” reference.
10. B. O. Leung and K. C. Chou, “ Review of superresolution fluorescence microscopy for biology,” Appl. Spectrosc. 65, 967–980 (2011).
11. For example, a STED microscope is sold by the Leica Corporation.
12. C. J. R. Sheppard, “ Fundamentals of superresolution,” Micron 38, 165–169 (2007). Sheppard introduces three classes rather than two: Improved superresolution boosts spatial frequency response but leaves the cutoff frequency unchanged. Restricted superresolution includes tricks that increase the cut-off by up to a factor of two. We use “pseudo” superresolution for both cases. Finally, unrestricted superresolution refers to what we term “true” superresolution.
13. J. Mertz, Introduction to Optical Microscopy ( Roberts and Co., Greenwood Village, CO, 2010), Chap. 18. Mertz follows Sheppard's classification, giving a simple but broad overview.
15.An updated treatment of the one-point-source-or-two decision problem is given by A. R. Small, “ Theoretical limits on errors and acquisition rates in localizing switchable fluorophores,” Biophys. J. 96, L16–L18 (2009).
16.For a more formal Bayesian treatment, see S. Prasad, “ Asymptotics of Bayesian error probability and source super-localization in three dimensions,” Opt. Express 22, 16008–16028 (2014).
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22. A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. ( Cambridge U.P., Cambridge, UK, 2011), Chap. 12. This edition of a well-established text adds a section on superresolution techniques, with a view that complements the one presented here.
23. Equation (1) gives the lateral resolution. The resolution along the optical axis is poorer: .
24. However, the magnification of an objective does not determine its resolution.
25. P. M. Duffieux, The Fourier Transform and Its Applications to Optics, 2nd ed. ( John Wiley & Sons, Hoboken, NJ, 1983). The first edition, in French, was published in 1946. Duffieux formulated the idea of the optical transfer function in the 1930s.
27. A subtle point: The modulation transfer function is zero beyond a finite spatial frequency; yet the response in Fig. 3 is non-zero at all frequencies. The explanation is that an object of finite extent has a Fraunhofer diffraction pattern (Fourier transform) that is analytic, neglecting noise. Analytic functions are determined by any finite interval (analytic continuation), meaning that one can, in principle, extrapolate the bandwidth and deduce the exact behavior beyond the cutoff from that inside the cutoff. In practice, noise cuts off the information (Fig. 3). See Lucy (Ref. 28) for a brief discussion and Goodman's book (Ref. 5) for more detail.
28. L. B. Lucy, “ Statistical limits to superresolution,” Astron. Astrophys. 261, 706–710 (1992). Lucy does not assume the PSF width to be known and thus reaches the more pessimistic conclusion that Δx ∼ N−1∕8 . Since the second moments are then matched, one has to use the variance of the fourth moment to distinguish the images.
29. K. Piché, J. Leach, A. S. Johnson, J. Z. Salvail, M. I. Kolobov, and R. W. Boyd, “ Experimental realization of optical eigenmode superresolution,” Opt. Express 20, 26424 (2012). Instruments with finite aperture sizes have discrete eigenmodes (that are not simple sines and cosines), which should be used for more accurate image restoration.
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45. One should set the errors to be the square root of the smooth distribution value deduced from the initial fit and then iterate the fitting process (Ref. 46). The conclusions however, would not change, in this case.
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In this paper, we discuss what is, what is not, and what is only sort of
superresolution microscopy. We begin by considering optical resolution, first in terms of diffraction
theory, then in terms of linear-systems theory, and finally in terms of techniques that use prior information, nonlinearity, and other tricks to improve resolution. This discussion reveals two classes of superresolution microscopy, “pseudo” and “true.” The former improves images up to the diffraction limit, whereas the latter allows for substantial improvements beyond the diffraction limit. The two classes are distinguished by their scaling of resolution with photon counts. Understanding the limits to imaging resolution involves concepts that pertain to almost any measurement problem, implying a framework with applications beyond optics.
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