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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
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.
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.
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.
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.
Bracewell, R. N. and Riddle, A. C. (1967). Inversion of fan-beam scans in radio astronomy. Astrophys. J. 150:427–434.
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.
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.
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.
Colsher, J.G. (1977). Iterative three-dimensional image reconstruction from tomographic projections. Comput. Graph. Image Proc. 6:513–537.
Cormack, A. M. (1963). Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys. 34:2722–2727.
Cormack, A. M. (1964). Representation of a function by its line integrals, with some radiological applications. II. J. Appl. Phys. 35:2908–2913.
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.
Deans, S. R. (1983). The Radon Transform and Some of Its Applications. Wiley, New York.
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.
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.
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.
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.
Goodman, J.W. (1968). Introduction to Fourier Optics. McGraw-Hill, New York.
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.
Harauz, G. and van Heel, M. (1986). Exact filters for general three-dimensional reconstruction. Optik 73:146–156.
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.
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.
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.
Lanzavecchia, S. and Bellon P. L. (1998). Fast computation of 3D Radon transform via a direct Fourier method. Bioinformatics 14:212–216.
Levi, A. and Stark, H. (1983). Signal restoration from phase by projections onto convex sets. J. Opt. Soc. Am. 73:810–822.
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.
Papoulis, A. (1968). Systems and Transforms with Applications in Optics. McGraw-Hill, New York; reprint, Robert E. Krieger, Florida, 1986.
Provencher, S. W. and Vogel, R. H. (1988). Three-dimensional reconstruction from electron micrographs of disordered specimes. I: Method. Ultramicroscopy 25:209–222.
Radermacher, M. (1980). Dreidimensionale Rekonstruktion bei kegelförmiger Kippung im Elektronenmikroskop. PhD thesis, Technische Universität München, Germany.
Radermacher, M. (1988). Three-dimensional reconstruction of single particles from random and non-random tilt series. J. Electron. Microsc. Tech. 9:359–394.
Radermacher, M. (1994). Three-dimensional reconstruction from random projections: orientational alignment via Radon transforms. Ultramicroscopy 53:121–136.
Radermacher, M. (1997). Radon transform techniques for alignment and three-dimensional reconstruction from random projections. Scanning Microsc. 11:171–177.
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.
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.
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.
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.
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.
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.
Shannon, C. E. (1949). Communication in the presence of noise. Proc. IRE 37:10–22.
Shepp, L.A. (1980). Computerized tomography and nuclear magnetic resonance zeugmatography. J. Comput. Assist. Tomogr. 4:94–107.
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.
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.
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.
Vainshtein, B. K. and Orlov, S. S. (1972). Theory of the recovery of functions from their projections. Sov. Phys. Crystallogr. 17:253–257.
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.
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.
Zwick, M. and Zeitler, E. (1973). Image reconstruction from projections. Optik 38:550–565.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-0-387-69008-7_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-31234-7
Online ISBN: 978-0-387-69008-7
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)