Resolution Beyond the Diffraction Limit for Regularized Object Restoration

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Abstract
We propose a new formulation of Miller's regularization theory, which is particularly suitable for object restoration problems. By means of simple geometrical arguments, we obtain upper and lower bounds for the errors on regularized solutions. This leads to distinguish between ‘Hölder continuity’ which is quite good for practical computations and ‘logarithmic continuity’ which is very poor. However, in the latter case, one can reconstruct local weighted averages of the solution. This procedure allows for precise valuations of the resolution attainable in a given problem. Numerical computations, made for object restoration beyond the diffraction limit in Fourier optics, show that, when logarithmic continuity holds, the resolution is practically independent of the data noise level.