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Image Reconstruction from the Oversampled Diffraction Pattern

Published online by Cambridge University Press:  02 July 2020

J. Miao ,
H. Chapman  and
D. Sayre
J. Miao
Affiliation:
Department of Physics, SUNY at Stony Brook, NY, 11794
H. Chapman
Affiliation:
Department of X-ray Optics, Lawrence Livermore National Laboratory, Livermore, CA94551
D. Sayre
Affiliation:
Department of Physics, SUNY at Stony Brook, NY, 11794
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In recording the intensity of the diffraction pattern from an arbitrary object, the phase is lost. This gives rise to the “phase problem”in reconstructing the object in crystallography, where the pattern is sampled at the Bragg peaks. It was shown by Bates and by Hayes that sampling at twice crystallographic density (2x oversampling) in each dimension is sufficient in principle to obtain a full reconstruction of the object. It was pointed out by Sayre in 1991 that the diffraction pattern of a non-crystalline object is continuous, therefore the pattern may be sampled on a finer scale, providing the additional information for the image reconstruction.

The loss of phase corresponds to a loss of half the information. It seems therefore that 2x oversampling in each dimension (8x in three dimension) may be unnecessary. We find that oversampling 1.67x in each dimension is sufficient to obtain a reconstruction, even in the presence of noise. However our algorithm fails when we reduce the oversampling factor to 1.58x.

Type
Computational Methods for Microscopy
Information
Microscopy and Microanalysis , Volume 3 , Issue S2: Proceedings: Microscopy & Microanalysis '97, Microscopy Society of America 55th Annual Meeting, Microbeam Analysis Society 31st Annual Meeting, Histochemical Society 48th Annual Meeting, Cleveland, Ohio, August 10-14, 1997 , August 1997 , pp. 1155 - 1156
Copyright
Copyright © Microscopy Society of America 1997

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References

1.Bates, R.H.T., Optik 61(1982)247.Google Scholar
2.Hayes, M.H., IEEE Trans. Acoust. Speech Signal Process., 30(1982)140.10.1109/TASSP.1982.1163863CrossRefGoogle Scholar
3.Sayre, D., in Schenk, H. Ed., Direct Methods of Solving Crystal Structure. Plenum Series B Physics vol. 274(1991)353.10.1007/978-1-4899-3692-9_33CrossRefGoogle Scholar
4.Fienup, J.R., Opt. Lett. 3(1978)27.10.1364/OL.3.000027CrossRefGoogle Scholar
5. The authors gratefully acknowledge Prof. Kirz, J. for many helpful discussions and Prof. Jacobsen, C. for the use of the computer facility.Google Scholar