Skip to main content

Weighted Back-projection Methods

  • Chapter
Electron Tomography

Abstract

Traditionally, 3D 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 their 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; Chapter 9 of this volume). 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; Gilbert, 1972; Ramachandran and Lakshminarayanan, 1971) 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 relationships between object transform and projection transforms.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Barth, M., Bryan, R. K. and Hegerl, R. (1989). Approximation of missing-cone data in electron microscopy. Ultramicroscopy 31:365–378.

    Article  Google Scholar 

  • Bellon, P. L., Cantele, F., Kreman, M., Lanzavecchia, S., Wright, E., Zampighi, G. A. and Zampighi, L. (2005). Conical tomography of freeze-fracture replicas: a method for the study of integral membrane proteins inserted in phospholipid bilayers. J. Struct. Biol. 149:87–98.

    Article  PubMed  CAS  Google Scholar 

  • Bellon, P. L. and Lanzanvacchia, S. (1997). Fast direct Fourier methods, based on one-and two-pass coordinate transformations, yield accurate reconstructions of X-ray CT clinical images. Phys. Med. Biol. 42:443–463.

    Article  PubMed  CAS  Google Scholar 

  • Bellon, P. L., Lanzavecchia, S. and Radermacher, M. (1999). Fast and accurate three-dimensional reconstruction from projections with random orientations via Radon Transforms. J. Struct. Biol. 128:152–164.

    Article  PubMed  Google Scholar 

  • Bracewell, R. N. and Riddle, A. C. (1967). Inversion of fan-beam scans in radio astronomy. Astrophys. J. 150:427–434.

    Article  Google Scholar 

  • Carazo, J.M. and Carrascosa, J. L. (1987). Information recovery in missing angular data cases: an approach by the convex projections method in three dimensions. J. Microsc. 45:23–43.

    Google Scholar 

  • 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.

    Article  PubMed  CAS  Google Scholar 

  • Cardone, G., Grünewald, K. and Steven, A.C. (2005). A resolution criterion for electron tomography based on cross-validation. J. Struct. Biol. 151:117–129.

    Article  PubMed  Google Scholar 

  • Colsher, J.G. (1977). Iterative three-dimensional image reconstruction from tomographic projections. Comput. Graph. Image Proc. 6:513–537.

    Article  Google Scholar 

  • Cormack, A. M. (1963). Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys. 34:2722–2727.

    Article  Google Scholar 

  • Cormack, A. M. (1964). Representation of a function by its line integrals, with some radiological applications. II. J. Appl. Phys. 35:2908–2913.

    Article  Google Scholar 

  • 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. A 317:319–340.

    Article  Google Scholar 

  • Deans, S. R. (1983). The Radon Transform and Some of Its Applications. Wiley, New York.

    Google Scholar 

  • 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 

  • Frank, J. and Goldfarb, W. (1980). Methods for averaging of single molecules and lattice fragments. in Electron Microscopy at Molecular Dimensions (W. Baumeister and W. Vogell, eds). Springer-Verlag, Berlin, pp. 261–269.

    Google Scholar 

  • 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. Tech. 6:193–205.

    Article  Google Scholar 

  • 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. B 182:89–102.

    CAS  Google Scholar 

  • Goodman, J.W. (1968). Introduction to Fourier Optics. McGraw-Hill, New York.

    Google Scholar 

  • 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.

    Article  PubMed  CAS  Google Scholar 

  • Harauz, G. and van Heel, M. (1986). Exact filters for general three-dimensional reconstruction. Optik 73:146–156.

    Google Scholar 

  • Hoppe, W., Schramm, H. J., Sturm, M., Hunsmann, N. and GaBmann, J. (1976). Threedimensional electron microscopy of individual biological objects. I: Methods. Z. Naturforsch. 31a:645–655.

    Google Scholar 

  • Hsieh, C.-E., Marko, M., Frank, J. and Mannella, C. A. (2002). Electron tomographic analysis of frozen-hydrated tissue sections. J. Struct. Biol. 138:63–73.

    Article  PubMed  Google Scholar 

  • 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 

  • Lanzavecchia, S. and Bellon P. L. (1998). Fast computation of 3D Radon transform via a direct Fourier method. Bioinformatics 14:212–216.

    Article  PubMed  CAS  Google Scholar 

  • Levi, A. and Stark, H. (1983). Signal restoration from phase by projections onto convex sets. J. Opt. Soc. Am. 73:810–822.

    Google Scholar 

  • 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.

    Article  PubMed  CAS  Google Scholar 

  • Papoulis, A. (1968). Systems and Transforms with Applications in Optics. McGraw-Hill, New York; reprint, Robert E. Krieger, Florida, 1986.

    Google Scholar 

  • Provencher, S. W. and Vogel, R. H. (1988). Three-dimensional reconstruction from electron micrographs of disordered specimes. I: Method. Ultramicroscopy 25:209–222.

    Article  PubMed  CAS  Google Scholar 

  • Radermacher, M. (1980). Dreidimensionale Rekonstruktion bei kegelförmiger Kippung im Elektronenmikroskop. PhD thesis, Technische Universität München, Germany.

    Google Scholar 

  • Radermacher, M. (1988). Three-dimensional reconstruction of single particles from random and non-random tilt series. J. Electron. Microsc. Tech. 9:359–394.

    Article  PubMed  CAS  Google Scholar 

  • Radermacher, M. (1994). Three-dimensional reconstruction from random projections: orientational alignment via Radon transforms. Ultramicroscopy 53:121–136.

    Article  PubMed  CAS  Google Scholar 

  • Radermacher, M. (1997). Radon transform techniques for alignment and three-dimensional reconstruction from random projections. Scanning Microsc. 11:171–177.

    Google Scholar 

  • 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 

  • Radermacher, M. and Hoppe, W. (1980). Properties of 3-D reconstructions from projections by conical tilting compared to single axis tilting. In Proceeding of the 7th European Congress on Electron Microscopy Vol. 1, pp. 132–133.

    Google Scholar 

  • 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.

    PubMed  CAS  Google Scholar 

  • Radermacher, M., Wagenknecht, T., Verschoor, A. and Frank, J. (1987). Threedimensional reconstruction from a single-exposure random conical tilt series applied to the 50S ribosomal of Escherichia coli. J. Microsc. 146:113–136.

    PubMed  CAS  Google Scholar 

  • Radon, J. (1917). Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. König. Sächs. Ges. Wiss. Leipzig, Math. Phys. Kl. 69:262–277.

    Google Scholar 

  • Ramachandran, G. N. and Lakshminarayanan, A.V. (1971). Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolution instead of Fourier transforms. Proc. Natl. Acad. Sci. USA 68:2236–2240.

    Article  PubMed  CAS  Google Scholar 

  • Shannon, C. E. (1949). Communication in the presence of noise. Proc. IRE 37:10–22.

    Article  Google Scholar 

  • Shepp, L.A. (1980). Computerized tomography and nuclear magnetic resonance zeugmatography. J. Comput. Assist. Tomogr. 4:94–107.

    Article  PubMed  CAS  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Stöffler, D., Feja, B., Fahrenkrog, B., Walz, J., Typke, D. and Aebi, U. (2003). Cryoelectron tomography provides novel insights into nuclear pore architecture: implications for nucleocytoplasmic transport. J. Mol. Biol. 328:119–130.

    Article  PubMed  CAS  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  PubMed  CAS  Google Scholar 

  • Zampighi, G. A., Zampighi, L., Fain, N., Wright, E. M., Cantele, F. and Lanzavecchia, S. (2005). Conical tomography II: A method for the study of cellular organelles in thin sections. J. Struct. Biol. 151:263–274.

    Article  PubMed  CAS  Google Scholar 

  • Zwick, M. and Zeitler, E. (1973). Image reconstruction from projections. Optik 38:550–565.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Radermacher, M. (2007). Weighted Back-projection Methods. In: Frank, J. (eds) Electron Tomography. Springer, New York, NY. https://doi.org/10.1007/978-0-387-69008-7_9

Download citation

Publish with us

Policies and ethics