Optimized Amplitude Filtering for Superresolution Over a Restricted FieldII. Application of the Impulse-generating Filter

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In a previous paper [1] we showed that a certain passive amplitude filter φo (r) optimizes, with respect to the ratio of central irradiance to total energy, the diffraction pattern required for superresolution over a circular restricted field. Here we use Frieden's impulsive point-like pattern to determine suitable positions for the zeros giving rise to the restricted field. The impulse-generating filter, which involves the finite Hankel self-transforms, is expressed as a series of Zernike polynomials. There follows a simple expansion of the associated diffraction pattern in terms of Bessel functions, from which we derive an accurate iteration scheme for computing the pattern's zeros. A numerical study of the shapes of the impulse-generating filter and its pattern reveals that, under appropriate conditions, (1) the pattern behaves over a small central region as a compressed Airy pattern and (2) the filter exhibits a great resemblance, throughout the pupil, to a certain Zernike polynomial and, over a central portion of the pupil, to a J o function. An example is given of the application of the optimizing filter φo to the zeros of the point-impulse pattern; results of the scheme are discussed, and improvements are suggested.