We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the aperture. This allows us to use results about the invertibility of the kernel mean map to provide a statement about the invertibility of Fraunhofer diffraction, showing that imaging processes with arbitrarily small apertures can in principle be invertible, i.e., do not lose information, provided the objects to be imaged satisfy a generic condition. A real world experiment shows that we can super-resolve beyond the Rayleigh limit.
Barnes @cite_5 proposed a reconstruction procedure for coherent illumination. He uses the assumption of bounded support to write the convolution operator in the imaging equation in such a way that it can be decomposed into prolate spheroidal wave functions @cite_14 . This allows inversion of that operator, similar to division in Fourier space. Rushforth and Harris @cite_6 study the influence of noise on reconstruction methods to overcome the diffraction limit. Their conclusion is that the Rayleigh criterion is an approximate measure of the resolution which can be achieved easily.''
@cite_5 @cite_14 @cite_6
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This paper presents a formal solution to the problem of object restoration in a one-dimensional, diffraction-limited imaging system. It is found that if the illumination in the object space is confined to a finite region, then the imaging equation can be solved for the object in terms of the image. The solution can be expressed as a series expansion on the eigenfunctions of the imaging operator. This paper treats the problem of restoring the detail to an optical image which has been degraded by diffraction and noise. The particular contribution of the paper is a more complete analysis of the effects of various types of noise on system performance than has been given previously. Background noise, measurement noise, and computer roundoff error are considered, and the errors in the reconstructed image caused by these noise processes are evaluated. Numerical results for the special case of a perfect one-dimensional slit aperture are obtained. A general conclusion is that the reconstruction technique described here is most useful when the smoothing is severe and when a modest improvement of resolution may be worthwhile.