Journal of Mathematical Imaging and Vision

, Volume 44, Issue 1, pp 38–51 | Cite as

A Derivative-Based Fast Autofocus Method in Electron Microscopy

  • M. E. RudnayaEmail author
  • H. G. ter Morsche
  • J. M. L. Maubach
  • R. M. M. Mattheij
Open Access


Most automatic focusing methods are based on a sharpness function, which delivers a real-valued estimate of an image quality. In this paper, we study an L 2-norm derivative-based sharpness function, which has been used before based on heuristic consideration. We give a more solid mathematical foundation for this function and get a better insight into its analytical properties. Moreover an efficient autofocus method is presented, in which an artificial blur variable plays an important role.

We show that for a specific choice of the artificial blur control variable, the function is approximately a quadratic polynomial, which implies that after the recording of at least three images one can find the approximate position of the optimal defocus. This provides the speed improvement in comparison with existing approaches, which usually require recording of more than ten images for autofocus. The new autofocus method is employed for the scanning transmission electron microscopy. To be more specific, it has been implemented in the FEI scanning transmission electron microscope and its performance has been tested as a part of a particle analysis application.


Autofocus/focusing Sharpness function Scanning transmission electron microscopy 


  1. 1.
    Brenner, J.F., Dew, B.S., Horton, J.B., King, T., Neurath, P.W., Selles, W.D.: An automated microscope for cytologic research a preliminary evaluation. J. Histochem. Cytochem. 24(1), 100–111 (1976) CrossRefGoogle Scholar
  2. 2.
    Carasso, A.S.: The APEX method in image sharpening and the use of low exponent levy stable laws. SIAM J. Appl. Math. 63(2), 593–618 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Carasso, A.S., Bright, D.S., Vladar, A.E.: APEX method and real-time blind deconvolution of scanning electron microscopy imagery. Opt. Eng. 41(10), 2499–2514 (2002) CrossRefGoogle Scholar
  4. 4.
    Erasmus, S.J., Smith, K.C.A.: An automatic focusing and astigmatism correction system for the SEM and CTEM. J. Microsc. 127(2), 185–199 (1982) CrossRefGoogle Scholar
  5. 5.
    Erteza, A.: Sharpness index and its application to focus control. Appl. Opt. 15(4), 877–881 (1976) CrossRefGoogle Scholar
  6. 6.
    Goodhew, P.J., Humphreys, J., Beanland, R.: Electron Microscopy and Analysis, 3rd edn. Taylor & Francis, London (2001) Google Scholar
  7. 7.
    Goodman, J.W.: Fourier Optics, 3rd edn. Roberts & Company, Greenwood Village (2006) Google Scholar
  8. 8.
    Haider, M., Muller, H., Uhlemann, S.: Advances in imaging and electron physics. In: Present and Future Hexapole Aberration Correctors for High-Resolution Electron Microscopy, vol. 153, pp. 43–120. Academic Press, Amsterdam (2008) Google Scholar
  9. 9.
    Hilsenstein, V.: Robust autofocusing for automated microscopy imaging of fluorescently labelled bacteria. In: Proc. International Conference on Digital Image Computing: Techniques and Applications (2005) Google Scholar
  10. 10.
    Johnson, C.B.: A method for characterizing electro-optical device modulation transfer function. Photogr. Sci. Eng. 14, 413–415 (1970) Google Scholar
  11. 11.
    Jutamulia, S., Asakura, T., Bahuguna, R.D., De Guzman, C.: Autofocusing based on power-spectra analysis. Appl. Opt. 33(26), 6210–6212 (1994) CrossRefGoogle Scholar
  12. 12.
    Kautsky, J., Flusser, J., Zitova, B., Simberova, S.: A new wavelet-based measure of image focus. Pattern Recognit. Lett. 23, 1785–1794 (2002) zbMATHCrossRefGoogle Scholar
  13. 13.
    Kirkland, E.J.: Advanced Computing in Electron Microscopy. Plenum Press, New York (1998) Google Scholar
  14. 14.
    Krotkov Focusing, E.: Int. J. Comput. Vis. 1, 223–237 (1987) CrossRefGoogle Scholar
  15. 15.
    Kumar, K., Pisarenco, M., Rudnaya, M.E., Savcenco, V., Srivastava, S.: Shape reconstruction techniques for optical sectioning of arbitrary objects. Math. Ind. Case Stud. J. 3, 19–36 (2011) Google Scholar
  16. 16.
    Le, T., Chartrand, R., Asaki, T.J.: A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imaging Vis. 27(3), 257–263 (2007) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Levin, A., Weiss, Y., Durand, F., Freeman, W.T.: Understanding and evaluating blind deconvolution algorithms. In: Proc. of IEEE Conf. on Computer Vision and Pattern Recognition (CVPR) (2009) Google Scholar
  18. 18.
    Liu, X.Y., Wang, W.H., Sun, Y.: Dynamic evaluation of autofocusing for automated microscopic analysis of blood smear and pap smear. J. Microsc. 227, 15–23 (2007) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lupini, A.R., Pennycook, S.J.: Rapid autotuning for crystalline specimens from an inline hologram. J. Electron Microsc. 57(6), 195–201 (2008) CrossRefGoogle Scholar
  20. 20.
    Morigi, S., Reichel, L., Sgallari, F., Shyshkov, A.: Cascadic multiresolution methods for image deblurring. SIAM J. Imaging Sci. 1(1), 51–74 (2007) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Muller, R.A., Buffington, A.: Real-time correction of atmospherically degraded telescope images through image sharpening. J. Opt. Soc. Am. 64(9), 1200–1210 (1974) CrossRefGoogle Scholar
  22. 22.
    Nakamae, K., Chikahisa, M., Fujioka, H.: Estimation of electron probe profile from SEM image through wavelet multiresolution analysis for inline SEM inspection. Image Vis. Comput. 25, 1117–1123 (2007) CrossRefGoogle Scholar
  23. 23.
    Nayar, S.K., Nakagawa, Y.: Shape from focus. IEEE Trans. Pattern Anal. Mach. Intell. 16(8), 824–831 (1994) CrossRefGoogle Scholar
  24. 24.
    Ong, K.H., Phang, J.C.H., Thong, J.T.L.: A robust focusing and astigmatism correction method for the scanning electron microscope-part II: autocorrelation-based coarse focusing method. Scanning 20, 324–334 (1997) CrossRefGoogle Scholar
  25. 25.
    Papoulis, A.: Signal Analysis. McGraw-Hill, New York (1977) zbMATHGoogle Scholar
  26. 26.
    Rudnaya, M.E., Kho, S.C., Mattheij, R.M.M., Maubach, J.M.L.: Derivative-free optimization for autofocus and astigmatism correction in electron microscopy. In: Proc. 2nd International Conference on Engineering Optimization, Lisbon, Portugal (2010) Google Scholar
  27. 27.
    Rudnaya, M.E., Mattheij, R.M.M., Maubach, J.M.L.: Iterative autofocus algorithms for scanning electron microscopy. Microsc. Microanal. 15(2), 1108–1109 (2009) CrossRefGoogle Scholar
  28. 28.
    Rudnaya, M.E., Mattheij, R.M.M., Maubach, J.M.L.: Evaluating sharpness functions for automated scanning electron microscopy. J. Microsc. 240, 38–49 (2010) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rudnaya, M.E., Mattheij, R.M.M., Maubach, J.M.L., ter Morsche, H.: Autocorrelation-based sharpness functions. In: Proc. 3rd IEEE Int. Conf. on Sign. Proc. Syst., Yantai, China (2011) Google Scholar
  30. 30.
    Rudnaya, M.E., Mattheij, R.M.M., Maubach, J.M.L., ter Morsche, H.: Gradient-based sharpness function. In: Proc. International Conference of Applied and Engineering Mathematics, London, UK (2011) Google Scholar
  31. 31.
    Rudnaya, M.E., Van den Broek, W., Doornbos, R.M.P., Mattheij, R.M.M., Maubach, J.M.L.: Autofocus and twofold astigmatism correction in HAADF-STEM. Ultramicroscopy 111, 1043–1054, (2011) CrossRefGoogle Scholar
  32. 32.
    Santos, A., De Solórzano, C.O., Vaquero, J.J., Peña, J.M., Malpica, N., Del Pozo, F.: Evaluation of autofocus functions in molecular cytogenetic analysis. J. Microsc. 188, 264–272 (1997) CrossRefGoogle Scholar
  33. 33.
    Subbarao, M., Tyan, J.: Selecting the optimal focus measure for autofocusing and depth-from-focus. IEEE Trans. Pattern Anal. Mach. Intell. 20(8), 864–870 (1998) CrossRefGoogle Scholar
  34. 34.
    Sun, Y., Duthaler, S., Nelson, B.J.: Autofocusing in computer microscopy: selecting the optimal focus algorithm. Microsc. Res. Tech. 65, 139–149 (2004) CrossRefGoogle Scholar
  35. 35.
    Tanaka, N., Hu, J.J., Baba, N.: An on-linea correction method of defocus and astigmatism in HAADF-STEM. Ultramicroscopy 78, 103–110 (1999) CrossRefGoogle Scholar
  36. 36.
    Tejada, A., Van Den Broek, W., van der Hoeven, S., den Dekker, A.J.: Towards STEM control: modeling framework and development of a sensor for defocus control. In: Proc. 48th IEEE Conference on Decision and Control, pp. 8310–8315 (2009) Google Scholar
  37. 37.
    Van den Broek, W.: Advanced focus methods in electron microscopy: tomographic reconstruction of the EELS data cube autofocus of HAADF-STEM images. PhD thesis, University of Antwerp, 2007 Google Scholar
  38. 38.
    Vollath, D.: Automatic focusing by correlative methods. J. Microsc. 147, 279–288 (1987) CrossRefGoogle Scholar
  39. 39.
    Yang, G., Nelson, B.J.: Wavelet-based autofocusing and unsupervised segmentation of microscopic images. In: Proc. of the Intl. Conference on Intelligent Robots and Systems (2003) Google Scholar
  40. 40.
    Yeo, T.T.E., Ong, S.H., Jayasooriah, Sinniah, R.: Autofocusing for tissue microscopy. Image Vis. Comput. 11(10), 629–639 (1993) CrossRefGoogle Scholar
  41. 41.
    Zemlin, F., Weiss, K., Schiske, W., Kunath, W., Herrmann, K.H.: Coma-free alignment of high resolution electron microscopes with the aid of optical diffractograms. Ultramicroscopy 3, 49–60 (1978) CrossRefGoogle Scholar
  42. 42.
    Zhang, Y., Zhang, Y., Changyun, W.: A new focus measure method using moments. Image Vis. Comput. 18, 959–965 (2000) CrossRefGoogle Scholar

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© The Author(s) 2011

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • M. E. Rudnaya
    • 1
    Email author
  • H. G. ter Morsche
    • 1
  • J. M. L. Maubach
    • 1
  • R. M. M. Mattheij
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

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