ABSTRACT
Imaging beyond the diffraction limit via a set of techniques nowadays termed localization microscopy has seen a sharp rise after the initial works around 2006; the most notable methods introduced were (uorescence) photoactivated localization microscopy (PALM)1,2 and stochastic optical reconstruction microscopy.3 The common idea to achieve imaging below the diffraction limit in the optical far eld is to localize single stochastically activated uorescent molecules. These molecules are
18.1 Introduction ..................................................................................................409 18.2 Resolution ..................................................................................................... 410
18.2.1 Resolution Measures ......................................................................... 410 18.2.1.1 Localization Uncertainty ................................................... 411 18.2.1.2 Full Width at Half Maximum ............................................ 412 18.2.1.3 Two-Point Resolution ......................................................... 412 18.2.1.4 Density of Localizations .................................................... 412 18.2.1.5 Kernel Density Estimation ................................................. 413 18.2.1.6 Information Transfer Function ........................................... 413
18.2.2 Fourier Ring Correlation .................................................................. 414 18.2.3 Local and Anisotropic (2D and 3D) Resolution ............................... 416
18.3 Visualization ................................................................................................. 416 18.3.1 Description of Visualization Methods .............................................. 417 18.3.2 Comparison of Visualization Methods ............................................. 418
18.3.2.1 Simulations ........................................................................ 419 18.3.2.2 Setup .................................................................................. 420 18.3.2.3 Results ................................................................................ 420 18.3.2.4 Theoretical Considerations ................................................ 422
18.4 Discussion and Conclusion ........................................................................... 426 References .............................................................................................................. 426
switched between a uorescent on-state and a nonuorescent off-state. The on-state molecules form a sparse subset of all molecules such that only one is active in a region the size on the order of the diffraction limit. The positions of these emitting molecules are estimated, after which they return to the off-state and other molecules are activated and localized until all molecules have been imaged. Essential to this process is the localization of single uorescent molecules, hence the common name for the techniques. The high-resolution capability of these techniques follows from the precision with which the positions of the molecules can be estimated, which is much better than the diffraction limit.4,5 This precision is on the order of σpsf / n
, where n is the number of recorded emission photons and σpsf is the width of the point spread function (PSF).6,7 Typically, hundreds or thousands of photons can be recorded and with σpsf ≈ 250 nm, this results in commonly achieved localization precisions on the order of tens of nanometers, although smaller values in the range of nanometers have been reported.8-10 In comparison, Abbe’s diffraction limit is given by λ/(2NA) ≈ 200 nm, where λ is the wavelength of light and NA is the numerical aperture of the imaging system. This superior precision is what makes localization microscopy images crisper and sharper than wideeld images and explains the widespread use of the technique nowadays. Even now, more and more avors of localization-based microscopy techniques are introduced; we give by no means an exhaustive list.11-18
As the family of localization microscopy techniques came of age and sharper and sharper images were recorded, the question “what is the resolution for these types of images?” arose. In other techniques for super-resolution imaging, such as stimulated emission depletion (STED) microscopy,19,20 structured illumination microscopy (SIM),21,22 or image scanning microscopy (ISM),23,24 the system can be identied as having a smaller effective PSF. Once the width of this PSF is measured or calculated, the resolution in the Abbe sense can be given. Where Abbe and Nyquist dened resolution as the inverse of the spatial bandwidth of the imaging system,25,26 Rayleigh and Sparrow captured resolution empirically. Rayleigh found a limit of 0.61λ/NA and Sparrow found a limit of 0.47λ/NA, which is very similar to Abbe’s diffraction limit of 0.5λ/NA for incoherent light. For localization microscopy, there is no natural extension of the PSF methodology as the position estimation of a single emitter from a PSF image is the key concept.
