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.
Gerchberg @cite_12 (and independently Papoulis @cite_0 ) proposed an algorithm analogous to Gerchberg and Saxton's phase retrieval method @cite_15 incorporating also positivity. As Jones @cite_24 points out, this algorithms converges under certain conditions only rather slowly.
@cite_0 @cite_15 @cite_24 @cite_12
2108943734 1484412996 2067298775 1982061533
If only a segment of a function f (t) is given, then its Fourier spectrum F( ) is estimated either as the transform of the product of f(t) with a time-limited window w(t) , or by certain techniques based on various a priori assumptions. In the following, a new algorithm is proposed for computing the transform of a band-limited function. The algorithm is a simple iteration involving only the fast Fourier transform (FFT). The effect of noise and the error due to aliasing are determined and it is shown that they can be controlled by early termination of the iteration. The proposed method can also be used to extrapolate bandlimited functions. An algorithm is presented for the rapid solution of the phase of the complete wave function whose intensity in the diffraction and imaging planes of an imaging system are known. A proof is given showing that a defined error between the estimated function and the correct function must decrease as the algorithm iterates. The problem of uniqueness is discussed and results are presented demonstrating the power of the method. The discrete version of the Gerchberg algorithm for iterative restoration of a time-constrained function from only partial knowledge of its spectrum (or vice versa) is analyzed. Although convergence is guaranteed, eigenvalues close to unity inhibit iteration to the limit. Identification of these large eigenvalues, allowing extrapolation to the limit, is described. A new view of the problem of continuing a given segment of the spectrum of a finite object is presented. Based on this, the problem is restated in terms of reducing a defined ‘error energy’ which is implicit in the truncated spectrum. A computational procedure, which is readily implemented on general purpose computers, is devised which must reduce this error. It is demonstrated that by so doing, resolution well beyond the diffraction limit is attained. The procedure is shown to be very effective against noisy data.