Editing Super-resolution imaging (section)

===Geometrical or image-processing super-resolution===
[[File:Super-resolution example closeup.png|thumb|right|220px|Compared to a single image marred by noise during its acquisition or transmission (left), the [[Signal-to-noise ratio (imaging)|signal-to-noise ratio]] is improved by suitable combination of several separately-obtained images (right). This can be achieved only within the intrinsic resolution capability of the imaging process for revealing such detail.]]

====Multi-exposure image noise reduction====
When an image is degraded by noise, there can be more detail in the average of many exposures, even within the diffraction limit. See example on the right.

====Single-frame deblurring====
{{main|Deblurring}}
Known defects in a given imaging situation, such as [[defocus]] or [[optical aberration|aberration]]s, can sometimes be mitigated in whole or in part by suitable spatial-frequency filtering of even a single image. Such procedures all stay within the diffraction-mandated passband, and do not extend it.

[[File:Localization Resolution.png|thumb|left|220px|Both features extend over 3 pixels but in different amounts, enabling them to be localized with precision superior to pixel dimension.]]

====Sub-pixel image localization====
The location of a single source can be determined by computing the "center of gravity" ([[centroid]]) of the light distribution extending over several adjacent pixels (see figure on the left). Provided that there is enough light, this can be achieved with arbitrary precision, very much better than pixel width of the detecting apparatus and the resolution limit for the decision of whether the source is single or double.  This technique, which requires the presupposition that all the light comes from a single source, is at the basis of what has become known as [[super-resolution microscopy]], e.g. [[stochastic optical reconstruction microscopy]] (STORM),  where fluorescent probes attached to molecules give [[Nanoscopic scale|nanoscale]] distance information.  It is also the mechanism underlying visual [[hyperacuity]].<ref>{{cite journal | last1 = Westheimer | first1 = G | year = 2012 | title = Optical superresolution and visual hyperacuity | journal = Prog Retin Eye Res | volume = 31 | issue = 5| pages = 467–80 | doi=10.1016/j.preteyeres.2012.05.001| pmid = 22634484 | doi-access = free }}</ref>

====Bayesian induction beyond traditional diffraction limit====
{{Main|Bayesian inference}}
Some object features, though beyond the diffraction limit, may be known to be associated with other object features that are within the limits and hence contained in the image.  Then conclusions can be drawn, using statistical methods, from the available image data about the presence of the full object.<ref>Harris, J.L., 1964. Resolving power and decision making. J. opt. soc. Am. 54, 606–611.</ref>  The classical example is Toraldo di Francia's proposition<ref>Toraldo di Francia, G., 1955. Resolving power and information. J. opt. soc. Am. 45, 497–501.</ref> of judging whether an image is that of a single or double star by determining whether its width exceeds the spread from a single star.  This can be achieved at separations well below the classical resolution bounds, and requires the prior limitation to the choice "single or double?"

The approach can take the form of [[extrapolation|extrapolating]] the image in the frequency domain, by assuming that the object is an [[analytic function]], and that we can exactly know the [[Function (mathematics)|function]] values in some [[Interval (mathematics)|interval]]. This method is severely limited by the ever-present noise in digital imaging systems, but it can work for [[radar]], [[astronomy]],  [[microscope|microscopy]] or [[magnetic resonance imaging]].<ref>[[#refPoot12|D. Poot, B. Jeurissen, Y. Bastiaensen, J. Veraart, W. Van Hecke, P. M. Parizel, and J. Sijbers, "Super-Resolution for Multislice Diffusion Tensor Imaging", Magnetic Resonance in Medicine, (2012)]]</ref> More recently, a fast single image super-resolution algorithm based on a closed-form solution to ''<math>\ell_2-\ell_2</math>'' problems has been proposed and demonstrated to accelerate most of the existing Bayesian super-resolution methods significantly.<ref>N. Zhao, Q. Wei, A. Basarab, N. Dobigeon, D. Kouamé and J-Y. Tourneret, [https://arxiv.org/abs/1510.00143 "Fast single image super-resolution using a new analytical solution for ''<math>\ell_2-\ell_2</math>'' problems"], IEEE Trans. Image Process., 2016, to appear.</ref>