Skip to main content

Application of the HLSVD Technique to the Filtering of X-Ray Diffraction Data


A filter based on the Hankel-Lanczos singular value decomposition (HLSVD) technique is presented and applied for the first time to X-ray diffraction (XRD) data. Synthetic and real powder XRD intensity profiles of nanocrystals are used to study the filter performances with different noise levels. Results show the robustness of the HLSVD filter and its capability to extract easily and effciently the useful crystallographic information. These characteristics make the filter an interesting and user-friendly tool for processing of XRD data.


  1. Mierzwa B, Pielaszek J: Smoothing of low-intensity noisy X-ray diffraction data by Fourier filtering: application to supported metal catalyst studies. Journal of Applied Crystallography 1997,30(5):544-546. 10.1107/S0021889897000198

    Article  Google Scholar 

  2. Hieke A, Dörfler H-D: Methodical developments for X-ray diffraction measurements and data analysis on lyotropic liquid crystals applied to K-soap/glycerol systems. Colloid and Polymer Science 1999,277(8):762-776. 10.1007/s003960050450

    Article  Google Scholar 

  3. Schmidt M, Rajagopal S, Ren Z, Moffat K: Application of singular value decomposition to the analysis of time-resolved macromolecular X-ray data. Biophysical Journal 2003,84(3):2112-2129. 10.1016/S0006-3495(03)75018-8

    Article  Google Scholar 

  4. Rajagopal S, Schmidt M, Anderson S, Ihee H, Moffat K: Analysis of experimental time-resolved crystallographic data by singular value decomposition. Acta Crystallographica Section D 2004,60(5):860-871.

    Article  Google Scholar 

  5. Aubanel EE, Oldham KB: Fourier smoothing without the fast Fourier transform. Byte 1985,10(2):207-222.

    Google Scholar 

  6. Wooff C: Smoothing of data by least squares fitting. Computer Physics Communications 1986,42(2):249-251. 10.1016/0010-4655(86)90040-8

    Article  Google Scholar 

  7. Barkhuijsen H, de Beer R, van Ormondt D: Improved algorithm for noniterative time-domain model fitting to exponentially damped magnetic resonance signals. Journal of Magnetic Resonance 1987,73(3):553-557.

    Google Scholar 

  8. Laudadio T, Mastronardi N, Vanhamme L, van Hecke P, van Huffel S: Improved Lanczos algorithms for blackbox MRS data quantitation. Journal of Magnetic Resonance 2002,157(2):292-297. 10.1006/jmre.2002.2593

    Article  Google Scholar 

  9. Wales DJ: Structure, dynamics, and thermodynamics of clusters: tales from topographic potential surfaces. Science 1996,271(5251):925-929. 10.1126/science.271.5251.925

    Article  Google Scholar 

  10. Siegel RW, Hu E, Cox DM, et al.: Nanostructure Science and Technolgy. A Worldwide Study. The Interagency Working Group on NanoScience, Engineering and Technolgy,

  11. Zanchet D, Hall MBD, Ugarte D: Structure population in thioi-passivated gold nanoparticles. Journal of Physical Chemistry B 2000,104(47):11013-11018.

    Article  Google Scholar 

  12. Golub GH, Reinsch C: Singular value decomposition and least squares solutions. Numerische Mathematik 1970,14(5):403-420. 10.1007/BF02163027

    MathSciNet  Article  Google Scholar 

  13. Anderson E, Bai Z, Bischof C, et al.: LAPACK Users' Guide. SIAM, Philadelphia, Pa, USA; 1995.

    MATH  Google Scholar 

  14. Young RA: The Rietvel Method. Oxford University Press, New York, NY, USA; 1993.

    Google Scholar 

  15. Cervellino A, Giannini C, Guagliardi A: Determination of nanoparticle structure type, size and strain distribution from X-ray data for monatomic f.c.c.-derived non-crystallographic nanoclusters. Journal of Applied Crystallography 2003,36(5):1148-1158. 10.1107/S0021889803013542

    Article  Google Scholar 

  16. Taylor JR: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Scientific Books, Sausalito, Calif, USA; 1997.

    Google Scholar 

  17. Stoica P, Moses R: Introduction to Spectral Analysis. Prentice-Hall, Upper Saddle River, NJ, USA; 1997.

    MATH  Google Scholar 

  18. Simon HD: The Lanczos algorithm with partial reorthogonalization. Mathematics of Computation 1984,42(165):115-142. 10.1090/S0025-5718-1984-0725988-X

    MathSciNet  Article  Google Scholar 

  19. Marple SL: Digital Spectral Analysis with Applications. Prentice-Hall, Englewood Cliffs, NJ, USA; 1987.

    Google Scholar 

  20. Golub G, Pereyra V: Separable nonlinear least squares: the variable projection method and its applications. Inverse Problems 2003,19(2):R1-R26. 10.1088/0266-5611/19/2/201

    MathSciNet  Article  Google Scholar 

  21. Baxter BJC, Iserles A: On approximation by exponentials. Annals of Numerical Mathematics 1997, 4: 39–54. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, hskip 1em plus 0.5em minus 0.4em

    MathSciNet  MATH  Google Scholar 

  22. Beylkin G, Monzón L: On approximation of functions by exponential sums. Applied and Computational Harmonic Analysis 2005,19(1):17-48. 10.1016/j.acha.2005.01.003

    MathSciNet  Article  Google Scholar 

  23. Bjoirck A: Numerical Methods for Least Squares Problems. SIAM, Philadelphia, Pa, USA; 1996.

    Book  Google Scholar 

  24. Kung SY, Arun KS, Bhaskar Rao DV: State-space and singular-value decomposition-based approximation methods for the harmonic retrieval problem. Journal of the Optical Society of America 1983,73(12):1799-1811. 10.1364/JOSA.73.001799

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to M. Ladisa.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Ladisa, M., Lamura, A., Laudadio, T. et al. Application of the HLSVD Technique to the Filtering of X-Ray Diffraction Data. EURASIP J. Adv. Signal Process. 2007, 039575 (2007).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI:


  • Information Technology
  • Noise Level
  • Diffraction Data
  • Quantum Information
  • Intensity Profile