Weighted Back-Projection Methods

  • Michael Radermacher

Abstract

Traditionally, three-dimensional reconstruction methods have been classified into two major groups, Fourier reconstruction methods and direct methods (e.g., Crowther et al., 1970; Gilbert, 1972). Fourier methods are defined as algorithms that restore the Fourier transform of the object from the Fourier transforms of the projections and then obtain the real-space distribution of the object by inverse Fourier transformation. Included in this group are also equivalent reconstruction schemes that use expansions of object and projections into orthogonal function systems (e.g., Cormack, 1963, 1964; Smith et al., 1973; Zeitler, Chapter 4). In contrast, direct methods are defined as those that carry out all calculations in real space. These include the convolution back-projection algorithms (Bracewell and Riddle, 1967; Ramachandran and Lakshminarayanan, 1971; Gilbert, 1972) and iterative algorithms (Gordon et al., 1970; Colsher, 1977). Weighted back-projection methods are difficult to classify in this scheme, since they are equivalent to convolution back-projection algorithms, but work on the real-space data as well as the Fourier transform data of either the object or the projections. Both convolution back-projection and weighted back-projection algorithms are based on the same theory as Fourier reconstruction methods, whereas iterative methods normally do not take into account the Fourier relations between object transform and projection transforms. Thus, it seems justified to classify the reconstruction algorithms into three groups: Fourier reconstruction methods, modified back-projection methods, and iterative direct space methods, where the second group includes convolution backprojection as well as weighted back-projection methods.

Keywords

Weighting Function Inverse Fourier Transform Sinc Function Arbitrary Geometry Tilt Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Bracewell, R. N. and Riddle, A. C. (1967). Inversion of fan-beam scans in radio astronomy. Astrophys. J. 150:427–434.Google Scholar
  2. Carazo, J-M., Wagenknecht, T., and Frank, J. (1989). Variations of the three-dimensional structure of the Escherichia coli ribosome in the range of overlap views. Biophys. J. 55:465–477.Google Scholar
  3. Colsher, J. G. (1977). Iterative three-dimensional image reconstruction from tomographic projections. Comput. Graph. Image Proc. 6:513–537.Google Scholar
  4. Cormack, A. M. (1963). Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys. 34:2722–2727.Google Scholar
  5. Cormack, A. M. (1964). Representation of a function by its line integrals, with some radiological applications. II. J. Appl. Phys. 35:2908–2913.Google Scholar
  6. Crowther, R. A., DeRosier, D. J., and Klug, A. (1970). The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. Proc. R. Soc. London A 317:319–340.Google Scholar
  7. Deans, S. R. (1983). The Radon Transform and Some of Its Applications. Wiley, New York.Google Scholar
  8. Frank, J. and Goldfarb, W. (1980). Methods for averaging of single molecules and lattice fragments, in Electton Microscopy at Molecular Dimensions (W. Baumeister and W. Vogell, eds.), pp. 261–269. Springer-Verlag, Berlin.Google Scholar
  9. Frank, J., McEwen, B. F, Radermacher, M., Turner, J. N., and Rieder, C. L. (1987). Three-dimensional tomographic reconstruction in high-voltage electron microscopy. J. Electron Microsc. Technique 6:193–205.Google Scholar
  10. Frank, J., Carazo, J-M., and Radermacher, M. (1988). Refinement of the random conical reconstruction technique using multivariate statistical analysis and classification. Eur. J. Cell Biol. Suppl. 25 48:143–146.Google Scholar
  11. Gilbert, P. F. C. (1972). The reconstruction of a three-dimensional structure from projections and its application to electron microscopy. II:Direct methods. Proc. R. Soc. London B 182:89–102.Google Scholar
  12. Goodman, J. W. (1968). Introduction to Fourier Optics. McGraw-Hill, New York.Google Scholar
  13. Gordon, R., Bender, R., and Herman, G. T. (1970). Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography. J. Theor. Biol. 29:471–481.Google Scholar
  14. Harauz, G. and van Heel, M. (1986). Exact filters for general three-dimensional reconstruction. Optik 73:146–156.Google Scholar
  15. Hoppe, W., Schramm, H. J., Sturm, M., Hunsmann, N., and Gaßßmann, J. (1976). Three-dimensional electron microscopy of individual biological objects. I:Methods. Z. Naturforsch. 31a:645–655.Google Scholar
  16. Kwok, Y. S., Reed, I. S., and Truong, T. K. (1977). A generalized |ω-filter for 3D-reconstruction. IEEE Trans. Nucl. Sci. NS24:1990–2005.Google Scholar
  17. McEwen, B. F., Radermacher, M., Rieder, C. L., and Frank, J. (1986). Tomographic three-dimensional reconstruction of cilia ultrastructure from thick sections. Proc. Nat. Acad. Sci. USA 83:9040–9044.Google Scholar
  18. Papoulis, A. (1968). Systems and Transforms with Applications in Optics. McGraw-Hill, New York; reprint, Robert E. Krieger, Florida, 1986.Google Scholar
  19. Provencher, S. W. and Vogel, R. H. (1988). Three-dimensional reconstruction from electron micrographs of disordered specimes. I:Method. Ultramicroscopy 25:209–222.Google Scholar
  20. Radermacher, M. (1980). Dreidimensionale Rekonstruktion bei kegelförmiger Kippung im Elektronenmikroskop. Ph.D. thesis, Technische Universität München, Germany.Google Scholar
  21. Radermacher, M. (1988). Three-dimensional reconstruction of single particles from random and nonrandom tilt series. J. Electron. Microsc. Technique 9:359–394.Google Scholar
  22. Radermacher, M. and Hoppe, W. (1978). 3-D reconstruction from conically tilted projections, in Proc. 9th Int. Congr. Electron Microscopy, Vol. 1, pp. 218–219.Google Scholar
  23. Radermacher, M. and Hoppe, W. (1980). Properties of 3-D reconstructions from projections by conical tilting compared to single axis tilting, in Proc. 7th Europ. Congr. Electron Microscopy, Vol. 1, pp. 132–133.Google Scholar
  24. Radermacher, M., Wagenknecht, T., Verschoor, A., and Frank, J. (1986). A new 3-D reconstruction scheme applied to the 50S ribosomal subunit of E. coli. J. Microsc. 141:RP1–RP2.Google Scholar
  25. Radermacher, M., Wagenknecht, T., Verschoor, A., and Frank, J. (1987). Three-dimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal of Escherichia coli. J. Microsc. 146:113–136.PubMedCrossRefGoogle Scholar
  26. Radon, J. (1917). Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. König. Sächs. Ges. Wiss. Leipzig. Math. Phys. KI. 69:262–277.Google Scholar
  27. Ramachandran, G. N. and Lakshminarayanan, A. V. (1971). Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolution instead of Fourier transforms. Proc. Nat. Acad. Sci. USA 68:2236–2240.Google Scholar
  28. Shannon, C. E. (1949). Communication in the presence of noise. Proc. IRE 37:10–22.Google Scholar
  29. Shepp, L. A. (1980). Computerized tomography and nuclear magnetic resonance zeugmatography. J. Comput. Assist. Tomogr. 4:94–107.Google Scholar
  30. Smith, P. R., Peter, T. M., and Bates, R. H. T. (1973). Image reconstruction from a finite number of projections. J. Phys. A 6:361–382.CrossRefGoogle Scholar
  31. Suzuki, S. (1983). A study on the resemblance between a computed tomographic image and the original object, and the relationship to the filterfunction used in image reconstruction. Optik 66:61–71.Google Scholar
  32. Vainshtein, B. K. (1971). Finding the structure of objects from projections. Sov. Phys. Crystallogr. 15:781–787.Google Scholar
  33. Vainshtein, B. K. and Orlov, S. S. (1972). Theory of the recovery of functions from their projections. Sov. Phys. Crystallogr. 17:253–257.Google Scholar
  34. Vogel, R. W. and Provencher, S. W. (1988). Three-dimensional reconstruction from electron micrographs of disordered specimes. II:Implementation and results. Ultramicroscopy 25:223–240.Google Scholar
  35. Zwick, M. and Zeitler, E. (1973). Image reconstruction from projections. Optik 38:550–565.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Michael Radermacher
    • 1
  1. 1.Wadsworth Center for Laboratories and ResearchNew York State Department of HealthAlbanyUSA

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