Superresolution capabilities of the Gerchberg Method in the band-pass case: an eingevalue analysis

This article first provides a general introduction to the thus uniquely determined by its values over any nonzero measure Gerchberg superresolution algorithm. Some specific properties of this subset of the Fourier space. In principle, an entire function known algorithm, when applied to 2D band-pass images, are then studied on a bounded domain can be extrapolated anywhere by analytical by means of an eigenvalue analysis of the imaging operator. The main continuation. However, the presence of noise limits the extrapofeature derived is the capability to recover the dc component of the lating ability of any superresolving technique. The problem of unknown object that has to be reconstructed from the noisy image available. This aspect is important with band-pass images of strictly analytical continuation is ill-posed, and regularization techniques positive objects, in that recovering the low-frequency and dc compo- are needed to obtain a unique and stable solution. For an outline nents in this case is tantamount to suppressing intolerable artifacts. of research on superresolution, see [2] or [1] and the references A set of eigenpairs of the imaging operator was calculated numeri- therein. cally. From the dominant eigenvalues, the spectrum extrapolation A simple iterative extrapolation algorithm, usually referred to capabilities of the method can be derived. From the behavior of the as the Gerchberg method (GM) , was proposed in [3,4] . It works eigenfunctions, the capability of the method to recover the dc compo- by cyclically exploiting the available information in the objectnent of the original object can be evaluated. Some of the calculated and the Fourier domains. The compact-support information is eigenfunctions are shown as examples

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