Editing Super-resolution imaging (section)

====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>