Introduction
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).
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, Cs 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, Csand
D
should be found by image deconvolution. Of these three factors,
Cs
and D can be determined experimentally without much difficulties.
Further more, in contrast to Df , Csand
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 Cs 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
Fijes,
P. L. (1977) Approximations for the calculation of high-resolution electron-microscopy
images of thin films. Acta Cryst., A33: 109-113.
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.
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.