hunt_bobby_2 - Super-Resolution of Imagery: Understanding...

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1 Super-Resolution of Imagery: Understanding the Basis for Recovery of Spatial Frequencies Beyond the Diffraction Limit B. R. Hunt Department of Electr. & Comp. Engr. and SAIC University of Arizona 101 N. Wilmot Rd. Tucson, AZ 85721 Tucson, AZ 85711 ( I ) Introduction The diffraction of electromagnetic waves causes an optical system to behave as a low-pass filter in the formation of an image. Fourier optics demonstrates that there exists a cut-off spatial frequency, which is directly determined by the shape and size of the limiting pupil in the optical system. Beyond the diffraction limit cut-off frequency no spatial frequency information about the object is passed into the image. Within the passband of the optical system, i.e., from DC to the optical cut-off spatial frequency, this alteration of the spatial frequency components of the object is governed by the optical transfer function ( OTF ). This description of the Fourier nature of the image formation process is valid for imaging in both coherent and incoherent light. In this paper we will consider only the case of imaging in incoherent light [1]. Since the formation of an image alters the recorded information content from that of the original object, there has been much interest and effort directed to processing the image so as to more closely match the original object. In recent years it has become clear that there are credible methods for the reconstruction of spatial frequencies of the object that are greater than the diffraction limit spatial frequencies in the image. Processes that achieve the recreation of frequencies beyond the image passband are usually referred to as super-resolution algorithms. The ability to achieve super-resolution of an image is controversial, with prominent literature proclaiming it as not possible [2]. However, the existence of algorithms that have demonstrated super- resolution in a number of different contexts has made it inescapable to conclude that super-resolution is possible. Understanding the basis of super-resolution also leads to understanding how the various algorithms for super-resolution function. ( II ) The Physical Basis of Super- Resolution Super-resolution has been criticized as being impossible because it seeks to recover information that, presumably, has been irretrievably lost. The formation of an image is given by the equation: - = ξ d f x h x g ) ( ) ( ) ( , (1) where g is the image, h is the point spread function ( the inverse Fourier transform of the OTF ) and f is the original object. From this equation we have the Fourier description:
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2 ) ( ) ( ) ( u F u H u G = . (2) Equation (2) implies a division by zero to compute F from the quotient of G and H. Since H is zero beyond the diffraction limit cut-off, the reconstruction of any information about F beyond the cut-off is impossible. On this basis, the argument goes, super-resolution can be dismissed as either a theoretical or practical concept. The argument against super-resolution
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hunt_bobby_2 - Super-Resolution of Imagery: Understanding...

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