An application of the Hilbert transform in electron microscopy: II. Non-iterative solution in bright-field microscopy and the dark-field problem

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Abstract
It is shown using the properties of the Hilbert transform that the phase distribution in a bright-field image may be recovered by a non-iterative scheme from a knowledge of the intensity distribution, provided that the image contrast is less than 70%. Application of the Hilbert transform to dark-field microscopy is subject to severe non-uniqueness problems in the solution for the phase distribution; although various methods for calculating the phase from dark-field intensity measurements are detailed, none of these methods give acceptable solutions. In relation to bright-field optics we outline the relation between single-sideband Fourier holography and the bright-field image recorded with a half-plane aperture in the Fourier (diffraction) plane of the object.