The following materials are quoted from the paper by Fan Hai-fu, "Direct Methods in Electron Crystallography", Microscopy Research and Technique, 46, 104-116 (1999). pdf file

Image Deconvolution
Resolution enhancement

        The goal of image deconvolution is to retrieve the structure image from one or a series of blurred electron-microscopy images (EMs), or equivalently, to extract a set of structure factors from them. Different procedures have been proposed. Most of them use a series of EMs with different defocus. Uyeda and Ishizuka (1974, 1975) first proposed a method for the deconvolution of a single EM under the weak-phase-object approximation. Inspired by this work, direct methods in X-ray crystallography were introduced into high resolution electron microscopy for the image deconvolution using a single EM (Li and Fan, 1979; Han, Fan and Li, 1986; Liu, Xiang, Fan, Tang, Li, Pan, Uyeda and Fujiyoshi, 1990).

With the weak-phase-object approximation, in which dynamical diffraction effects are neglected, the Fourier transform of an EM can be expressed as

,     (1)

which can be rearranged to give

,      (2)

Here s = p/lU, l is the electron wavelength and U the accelerating voltage. h is the reciprocal lattice vector within the resolution limit. F(h) is the structure factor of electron diffraction, which is the Fourier transform of the potential distribution j(r) of the object. sinc₁(h)exp[-c₂(h)] is the contrast transfer function, in which

,

.

Here Df  is the defocus value, C_s is the spherical aberration coefficient and D is the standard deviation of the Gaussian distribution of defocus due to the chromatic aberration (Fijes,1977). The values of Df, C_sand D should be found by image deconvolution. Of these three factors, C_s and D can be determined experimentally without much difficulties. Further more, in contrast to Df , C_sand D do not change much from one image to another. This means that the main problem is the evaluation of Df. With the estimated values of C_s and D, a set of F(h) can be calculated from Equation (2) for a given value of Df. If the Df value is correct then the corresponding set of F(h) should obey the Sayre equation (Sayre, 1952)

,      (3)

where q is the atomic form factor and V is the volume of the unit cell. Hence the true Df can be found by a systemic change of the trial Df so as to improve the consistency with the Sayre equation. For the evaluation of the quality of each trial, figures of merit used for direct methods in X-ray crystallography (see Woolfson and Fan, 1995) were introduced.
 

References

         Han, F. S., Fan H. F. and Li, F. H. (1986) Image processing in high resolution electron microscopy using the direct method II. Image deconvolution. Acta Cryst., A42: 353-356.
        Fijes, P. L. (1977) Approximations for the calculation of high-resolution electron-microscopy images of thin films. Acta Cryst., A33: 109-113.
        Li, F. H. and Fan, H. F. (1979) Image deconvolution in high resolution electron microscopy by making use of Sayre's equation. Acta Phys. Sin., 28: 276-278. (in Chinese)
        Liu, Y. W., Xiang, S. B., Fan, H. F., Tang, D., Li, F. H. Pan, Q., Uyeda, N. and Fujiyoshi, Y. (1990) Image deconvolution of a single high resolution electron micrograph. Acta Cryst., A46: 459-463.
        Sayre, D. (1952) The squaring method: a new method for phase determination. Acta Cryst., 5: 60-65.
        Uyeda, N. and Ishizuka, K. (1974) Correct molecular image seeking in the arbitrary defocus series. In: Sanders, J. V. and Goodchild, D. J. Eds. Eighth Int. Congr. Electron Microscopy, Vol.1, pp. 322-323.
        Uyeda, N. and Ishizuka, K. (1975) Molecular image reconstruction in high resolution electron microscopy. J. Electron Microscopy, 24: 65-72.
        Woolfson, M. M. and Fan, H. F. (1995) Physical and Non-Physical Methods of Solving Crystal Structures, Cambridge Univ. Press, Cambridge, pp.106-107.