X-ray diffraction tomography: image filtering by singular value decomposition and 1D smoothing Whittaker-Eilers methods

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Abstract

Digital processing of the 2D noisy X-ray diffraction images (2D-XDI) of a single point defect in Si(111) crystal, registered at the level of dispersion of statistical Gaussian noise of the detector using filtering methods such as singular value decomposition and 1D-line-by-line smoothing of test 2D-XDIs, is carried out. The efficiency of digital filtering of 2D-XDI is evaluated and analyzed by means of control parameter FOM (figure-of-merit) value of reconstruction of the displacement field function of a point defect of Coulomb type fh(r–r0), (h – diffraction vector, r0 – radius-vector of the defect position in the sample). It is shown that the filtering technique using the singular value decomposition of 2D-XDI works significantly better than the 1D linear-by-line smoothing method of 2D-XDI, which, apparently, in relation to our problem requires further research on its improvement.

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About the authors

F. N. Chukhovskii

National Research Center “Kurchatov Institute”

Author for correspondence.
Email: f_chukhov@yahoo.ca

Shubnikov Institute of Crystallography of the Kurchatov Complex Crystallography and Photonics

Russian Federation, Moscow

P. V. Konarev

National Research Center “Kurchatov Institute”

Email: f_chukhov@yahoo.ca

Shubnikov Institute of Crystallography of the Kurchatov Complex Crystallography and Photonics

Russian Federation, Moscow

V. V. Volkov

National Research Center “Kurchatov Institute”

Email: f_chukhov@yahoo.ca

Shubnikov Institute of Crystallography of the Kurchatov Complex Crystallography and Photonics

Russian Federation, Moscow

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4. Fig. 1. Schematic diagram of 2D X-ray diffraction microtomography data acquisition. The data set is collected at a sample rotation angle Φ around the diffraction vector, θB is the Bragg angle.

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5. Fig. 2. Noise-free reference 2D X-ray diffraction data (181 × 181 pixels) (a). Dependence of 2D X-ray diffraction singular values on their ordinal number (b). The inset shows the region corresponding to the first 30 singular values of the SVD decomposition of the 2D X-ray diffraction data. The sharp change in the slope of the curve, which occurs in the region of the 24th SV, is highlighted with a circle.

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6. Fig. 3. Original 2D X-ray diffraction data (181 × 181 pixels) with the addition of 5% Gaussian noise (a). Dependence of 2D X-ray diffraction singular values on their ordinal number (b). The inset shows the region corresponding to the first 20 singular values of the SVD decomposition of the 2D-RDI data. The slope of the curve changes after the eighth SV. The circle highlights the SV range where the quality of reconstruction of the defect displacement field function fh(r–r0) improves in the process of solving the inverse Radon problem. 2D-RDI data calculated using the modified Whittaker–Eilers algorithm (c). 2D-RDI data calculated using the SVD decomposition over the first 11 SVs (d).

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