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Stable Desmearing of Slit-Collimated SAXS Patterns by Adequate Numerical Conditioning

Stable Desmearing of Slit-Collimated SAXS Patterns by Adequate Numerical Conditioning Small-angle X-ray scattering (SAXS) patterns from slit cameras (`Kratky cameras') require a subsequent desmearing procedure in order to obtain the pinhole scattering curve that is suitable for subsequent structure analysis. Since the corresponding integral equation contains a singularity, its solutions are usually unstable and fail if large noise is present. It is demonstrated how analytical stability can be achieved by physically reliable conditioning of the experimental data, introduction of the Moore-Penrose pseudoinverse of the equation's discretized integral operator and solving the equation by a FFT algorithm. This ensures the consistency of the solution as well as its stability, and hence its convergence. This solution can account for arbitrarily nonsymmetrical primary-beam profiles. The algorithm does not require antecedent smoothing of the scattering curve. It allows on the contrary low-pass filter smoothing during desmearing but remains stable despite large noise contributions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A: Foundations of Crystallography International Union of Crystallography

Stable Desmearing of Slit-Collimated SAXS Patterns by Adequate Numerical Conditioning

Stable Desmearing of Slit-Collimated SAXS Patterns by Adequate Numerical Conditioning


Abstract

Small-angle X-ray scattering (SAXS) patterns from slit cameras (`Kratky cameras') require a subsequent desmearing procedure in order to obtain the pinhole scattering curve that is suitable for subsequent structure analysis. Since the corresponding integral equation contains a singularity, its solutions are usually unstable and fail if large noise is present. It is demonstrated how analytical stability can be achieved by physically reliable conditioning of the experimental data, introduction of the Moore-Penrose pseudoinverse of the equation's discretized integral operator and solving the equation by a FFT algorithm. This ensures the consistency of the solution as well as its stability, and hence its convergence. This solution can account for arbitrarily nonsymmetrical primary-beam profiles. The algorithm does not require antecedent smoothing of the scattering curve. It allows on the contrary low-pass filter smoothing during desmearing but remains stable despite large noise contributions.

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References (4)

Publisher
International Union of Crystallography
Copyright
Copyright (c) 1998 International Union of Crystallography
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S010876739800573X
Publisher site
See Article on Publisher Site

Abstract

Small-angle X-ray scattering (SAXS) patterns from slit cameras (`Kratky cameras') require a subsequent desmearing procedure in order to obtain the pinhole scattering curve that is suitable for subsequent structure analysis. Since the corresponding integral equation contains a singularity, its solutions are usually unstable and fail if large noise is present. It is demonstrated how analytical stability can be achieved by physically reliable conditioning of the experimental data, introduction of the Moore-Penrose pseudoinverse of the equation's discretized integral operator and solving the equation by a FFT algorithm. This ensures the consistency of the solution as well as its stability, and hence its convergence. This solution can account for arbitrarily nonsymmetrical primary-beam profiles. The algorithm does not require antecedent smoothing of the scattering curve. It allows on the contrary low-pass filter smoothing during desmearing but remains stable despite large noise contributions.

Journal

Acta Crystallographica Section A: Foundations of CrystallographyInternational Union of Crystallography

Published: Sep 1, 1998

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