Maximum-entropy data restoration using both real- and Fourier-space analysis

Abstract

MULTIPLE REFLECTIONS IN A PLANE-PARALLEL MOSAIC CRYSTALand then look for those values of A other than 0 or which cause the right-hand side of (25) to vanish. In practice it has been observed that there is only one such value, ,4 0 , which is the root, when it exists, of a transcendental equation. As a consequence, there may be a dip at the centre of the reflection and a maximum at ,40. Notice that the existence of the dip is not confined to the approximate solution (22) in which case it could be a mathematical artefact.In conclusion, we can say that the present investigation has shown that, in the case of multiple scattering of electromagnetic radiation, in general there is coupling between the two states of polarization either through the amplitudes or through the intensities. Only in the coplanar case do the two X-ray components act independently and can be decoupled. The numerical examples which have been reported are the first exact or almost exact calculations of the effect of multiple reflections in a mosaic crystal within the limits of validity of the transfer equations. These calculations show that application of the kinematical approximation to crystals having reflectivities comparable to those of copper can be grossly in error (up to a factor of two) if the crystal thickness is of the order of what is used in practice (~ol-1 for X-rays and /xol-0.01 for neutrons). This discrepancy can be significantly reduced using the two-beam formulas; however, if one is interested in obtaining accurate structure factors, multiple reflections have to be taken into account. Equations (22), (23) and (24) AbstractAn extension of the maximum-entropy (ME) datarestoration method is presented that is sensitive to periodic correlations in data. The method takes advantage of the higher signal-to-noise ratio for periodic information in Fourier space, thus enhancing statistically significant frequencies in a manner which avoids the user bias inherent in conventional Fourier * Current address: Physical Chemistry 1, Chemical Center, University of Lund, PO Box 124, S-221 00 Lund, Sweden. filtering. This procedure incorporates concepts underlying new approaches in quantum mechanics that consider entropies in both position and momentum spaces, although the emphasis here is on data restoration rather than quantum physics. After a fast Fourier transform of the image, the phases are saved and the array of Fourier moduli are restored using the maximum-entropy criterion. A first-order continuation method is introduced that speeds convergence of the M E computation. The restored moduli together with the original phases are then Fourier inverted to yield a new image; traditional real-space ME restoration is applied to this new image completing one stage in the restoration process. In test cases with various types of added noise and in examples of normal and high-resolution electron-microscopy images, dramatic improvement can be obtained from 0108-7673/89/100686-13503.00