Science China Information Sciences

, Volume 56, Issue 9, pp 1–12 | Cite as

A fast calculation strategy of density function in ISAF reconstruction algorithm

  • GongMing Wang
  • Fa Zhang
  • Qi Chu
  • LiYa Fan
  • Fei Sun
  • ZhiYong Liu
Research Paper


The ISAF reconstruction algorithm is a new method for reconstructing icosahedral molecules from their projections. This algorithm works in spherical coordinate system and can achieve higher resolution than the traditional Fourier-Bessel algorithm in cylindrical coordinate system; however this method needs huge computations, which limits its application in reality. The main bottleneck lies in the calculation of density function as it occupies 90% running time of the whole algorithm. A fast calculation strategy of density function is proposed to solve this problem. This strategy is composed of three components: the fast calculation method of density function of mesh point in spherical coordinate system, the transformation method of density function of mesh point from spherical coordinate system to Cartesian coordinate system and the fast two-phase mapping method. The time complexity of calculating density function is decreased from O[(L M )8] to O[(L M )7] in our strategy. The experimental results on Psv-F simulated data indicate that the speed of calculating density function is increased almost two orders of magnitude and the speedup of the whole algorithm could reach 30 times. In addition, the speedup could go up with the increase in the number of images and the requirement of accuracy.


ISAF 3D reconstruction density function spherical coordinate system quaternion interpolation 


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  1. 1.
    Shannon D D J, Klug A. Reconstruction of three-dimensional structures from electron micrographs. Nature, 1968, 217: 130–134CrossRefGoogle Scholar
  2. 2.
    Colsher J G. Iterative three-dimensional image reconstruction from tomographic projections. Comput Graph Image Process, 1977, 6: 513–537CrossRefGoogle Scholar
  3. 3.
    Frank J, Goldfarb W, Eisenberg D, et al. Reconstruction of glutamine synthetase using computer averaging. Ultramicroscopy, 1978, 3: 283–290CrossRefGoogle Scholar
  4. 4.
    Dubochet J, Adrian M, Chang J J, et al. Cryo-electron microscopy of vitrified specimens. Q Rev Biophys, 1988, 21: 129–228CrossRefGoogle Scholar
  5. 5.
    Chiu W. Electron microscopy of frozen, hydrated biological specimens. Annu Rev Biophys Biophys Chem, 1986, 15: 237–257CrossRefGoogle Scholar
  6. 6.
    Crowther R A. Procedures for three-dimensional reconstruction of spherical viruses by Fourier synthesis from Electron Micrographs. Philos Trans Roy Soc B, 1971, 261: 221–230CrossRefGoogle Scholar
  7. 7.
    Crowther R A, DeRosier D J, Klug A. The reconstruction of a three-dimensional atructure from projections and its application to electron microscopy. P Roy Soc Lond A Mat, 1970, 317: 319–340CrossRefGoogle Scholar
  8. 8.
    Navaza J. On the three-dimensional reconstruction of icosahedral particles. J Struct Biol, 2003, 144: 13–23CrossRefGoogle Scholar
  9. 9.
    Liu H R, Cheng L P, Zeng S J, et al. Symmetry-adapted spherical harmonics method for high-resolution 3D single-particle reconstructions. J Struct Biol, 2008, 161: 64–73CrossRefGoogle Scholar
  10. 10.
    Liu H R. The new method for high-resolution 3D reconstrcution of virus. PhD Thesis. Xiangtan: Xiangtan University. 2008. 2Google Scholar
  11. 11.
    Baker T S, Olson N H, Fuller S D. Adding the third dimension to virus life cycles: three dimensional reconstruction of icosahedral viruses from cryo-electron micrographs. Microbiol Mol Biol R, 1999, 63: 862–922Google Scholar
  12. 12.
    Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. New York: Dover Press, 1965Google Scholar
  13. 13.
    Bradley C J, Cracknell A P. The Mathematical Theory of Symmetry in Solids. New York: Oxford University Press, 1972MATHGoogle Scholar
  14. 14.
    Prandl W, Schiebel P, Wulf K. A recursive algorithm for the generation of symmetry-adapted functions: principles and applications to the icosahedral group. Acta Crystallogr A, 1996, 52: 171–175CrossRefGoogle Scholar
  15. 15.
    Jackson J D. Classical Electrondynamics. 3rd ed. New York: Wiley Press, 1999Google Scholar
  16. 16.
    Pan J, Dong L, Lin L, et al. Atomic structure reveals the unique capsid organization of a dsRNA virus. P Natl Acad Sci USA, 2009, 106: 4225–4230CrossRefGoogle Scholar
  17. 17.
    Ludtke S J, Baldwin P R, Chiu W. EMAN: semi-automated software for high-resolution single-particle reconstructions. J Struct Biol, 1999, 128: 82–97CrossRefGoogle Scholar
  18. 18.
    Liang Y R, Ke E Y, Zhou Z H. IMIRS: a high-resolution 3D reconstruction package integrated with a relational image database. J Struct Biol, 2002, 137: 292–304CrossRefGoogle Scholar
  19. 19.
    Willett P. Chemical similarity searching. J Chem Inf Comp Science, 1998, 38: 983–996CrossRefGoogle Scholar
  20. 20.
    Harauz G, van Heel M. Exact filters for general geometry three dimensional reconstruction. Optik, 1986, 73: 146–156Google Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Computing TechnologyChinese Academy of SciencesBeijingChina
  2. 2.Graduate University of Chinese Academy of SciencesBeijingChina
  3. 3.IBM China Research LabBeijingChina
  4. 4.Institute of Biophysics, Chinese Academy of SciencesBeijingChina

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