Skip to main content

The Fidelity of 3D Reconstructions from Incomplete Data and the Use of Restoration Methods

  • Chapter
Electron Tomography

Abstract

During the last two decades it has become increasingly evident that electron microscopy images of typical thin biological specimens carry a large amount of information on the three-dimensional (3D) structure of the object. It has been shown many times how the information contained in a set of images (2D signals) can determine a useful estimate of the 3D structure of the specimen under study. Naturally, the mathematical methods used in these studies have become more and more elaborate as the complexity of the structural problems increased.

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 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

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

  • Agard, D. A. and Stroud, R. M. (1982). Linking regions between helices in bacteriorhodopsin revealed. Biophys. J. 37:589–602.

    PubMed  CAS  Google Scholar 

  • Andrews, H. C. and Hunt, B. R. (1977). Digital Image Restoration. Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Barth, M., Bryan, R. K., and Hegerl, R. (1989). Approximation of missing-cone data in 3D electron microscopy. Ultramicroscopy 31:365–378.

    Article  Google Scholar 

  • Barth, M., Bryan, R. K., Hegerl, R., and Baumeister, W. (1988). Estimation of missing data in threedimensional electron microscopv. Scanning Microsc. Suppl 2:277–284.

    Google Scholar 

  • Bricogne, G. (1988). A Bayesian statistical theory of the phase problem I. A multichannel maximumentropy formalism for constructing generalized joint probability distributions of structure factors. Acta Crystallogr. A 44:517–545.

    Article  Google Scholar 

  • Browder, F. E. (1965). Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Nat. Acad. Sci. USA 53:1272–1276.

    Article  PubMed  CAS  Google Scholar 

  • Carazo, J. M. and Carrascosa, J. L. (1987a). Information recovery in missing angular data cases: an approach by the convex projections method in three dimensions. J. Microsc. 145:23–43.

    Google Scholar 

  • Carazo, J. M. and Carrascosa, J. L. (1987b). Restoration of direct Fourier three-dimensional reconstructions of crystalline specimens by the method of convex projections. J. Microsscc. 145:159–177.

    Article  CAS  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 

  • Carazo, J. M., Wagenknecht, T., Radermacher, M., Mandiyan, V., Boublik, M., and Frank, J. (1988). Three-dimensional structure of 50S Escherichia coli ribosomal subunits depleted of proteins L7/L12. J. Mol. Biol. 201:393–404.

    Article  PubMed  CAS  Google Scholar 

  • Chalcroft, J. P. and Davey, C. L. (1984). A simply constructed extreme-tilt holder for the Philips eucentric stage. J. Microsc. 134:41–48.

    Article  Google Scholar 

  • Chiu, M. Y., Barrett, H. H., Simpson, R. G., Chou, C., Arendt, J. W., and Gindi, G. R. (1979). Threedimensional radiographic imaging with a restricted view angle. J. Opt. Soc. Am. 69(10):1323–1333.

    Article  Google Scholar 

  • Civanlar, M. R. and Trussell, H. J. (1986). Digital signal restoration using fuzzy sets. IEEE Trans. Acoust. Speech and Signal Process. ASSP-34(4):919–936.

    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. London A 317:319–340.

    Google Scholar 

  • DeRosier, D. J. and Moore, P. B. (1970). Reconstruction of three-dimensional images from electron micrographs of structures with helical symmetry. J. Mol. Biol. 52:355–369.

    Google Scholar 

  • Erickson, H. P. and Klug, F. R. S. (1971). Measurement and compensation of defocusing and aberrations in Fourier processing of electron micrographs. Phil. Trans. R. Soc. London B 261:105–113.

    Google Scholar 

  • Frank, J. and Radermacher, M. (1986). Three-dimensional reconstruction of nonperiodic macromolecular assemblies from electron micrographs, in Advanced Techniques in Biological Electron Microscopy (J. Koehler, ed.), Springer-Verlag, Berlin.

    Google Scholar 

  • Frieden, B. R. (1972). Restoring with maximum likelihood and maximum entropy. J. Opt. Soc. Am. G2:511–518.

    Article  Google Scholar 

  • Gerchberg, R. W. (1974). Super-resolution through error energy reduction. Opt. Acta 21(9):709–720.

    Article  Google Scholar 

  • Glaeser, R. M., Tom, L., and Kim, S. H. (1990). Three-dimensional reconstruction from incomplete data: Interpretability of density maps at “atomic” resolution. Ultramicroscopy 27:307–318.

    Article  Google Scholar 

  • Goldburg, M. and Marks II, R. J. (1985). Signal synthesis in the presence of an inconsistent set of constraints. IEEE Trans. Circuits Syst. CAS-32(7):647–663.

    Google Scholar 

  • Gordon, R. and Herman, G. T. (1971). Reconstruction of pictures from their projections. Comm. ACM 14:759–768.

    Article  Google Scholar 

  • Gulf, S. F. (1988). Developments in maximum entropy data analysis, in 8th MaxEnt Workshop, Cambridge, UK.

    Google Scholar 

  • Hanson, K. M. (1987). Bayesian and related methods in image reconstruction from incomplete data, in Image Recovery: Theory and Application (H. Stark, ed.), Academic Press, Orlando.

    Google Scholar 

  • Hanson, K. M. and Myers, K. J. (1990). Comparison of the algebraic reconstruction technique with the maximum entropy reconstruction technique for a variety of detection tasks. SPIE Proc. 1231.

    Google Scholar 

  • Hanson, K. M. and Wecksung, G. W. (1983). Bayesian approach to limited-angle reconstruction in computed tomography. J. Opt. Soc. Am. 73:1501–1509.

    Google Scholar 

  • Henderson, R. and Unwin, P. N. T. (1975). Three-dimensional model of purple membrane obtained by electron microscopy. Nature (London) 257:28–32.

    Google Scholar 

  • Herman, G. T. (1980). Image Reconstruction from Projections. Academic Press, New York.

    Google Scholar 

  • Howard, J. (1988). Tomography and reliable information. J. Opt. Soc. Am. A 5(7):999–1014.

    Article  CAS  Google Scholar 

  • Johnson, R. W. and Shore, J. E. (1983). Comments on and correction to “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory IT-29(6):942–943.

    Google Scholar 

  • Katz, M. B. (1978). Questions of uniqueness and resolution in reconstruction from projections, in Lectures in Biomathematics, Vol. 26, Springer-Verlag, Berlin.

    Google Scholar 

  • Klug, A. and Crowther, R. A. (1972). Three-dimensional image reconstruction from the viewpoint of information theory. Nature 238:435–440.

    Google Scholar 

  • Lawrence, M. C., Jaffer, M. A., and Sewell, B. T. (1989). The application of the maximum entropy method to electron microscopy tomography. Ultramicroscopy 31:285–302.

    Google Scholar 

  • Lepault, J. and Pitt, T. (1984). Projected structure of unstained frozen-hydrated T-layer of Bacillus brevis. EMBO J. 3(1):101–105.

    Google Scholar 

  • McCaughey, D. G. and Andrews, H. C. (1977). Degrees of freedom for projection imaging. IEEE Trans. Acoust., Speech and Signal Process. ASSP-25:63–73.

    Google Scholar 

  • Opial, Z. (1967). Weak convergence of the sequence of successive approximations for nonexpannsive mappings. Bull. Am. Math. Soc. 73:591–597.

    Article  Google Scholar 

  • Oskoui-Fard, P. and Stark, H. (1988). Tomographic image reconstruction using the theory of convex projections. IEEE Trans. Med. Imaging 7(1):45–58.

    Google Scholar 

  • Papoulis, A. (1975). A new algorithm in spectral analysis and band-limited extrapolations. IEEE Trans. Circuits Syst. CAS-22:735–742.

    Google Scholar 

  • Pavlovic, G. and Tekalp, A. M. (1990). Restoration in the presence of multiplicative noise with application to scanned photographic images, in Proc. Int. Conf. Acoust., Speech and Signal Processing.

    Google Scholar 

  • Penczek, P., Srivastava, S., and Frank, J. (1990). The structure of the 70S E. coliribosome in ice, in Proc. XIIth Int. Congr. Electron Microscopy Vol. 1. pp. 506–507.

    Google Scholar 

  • Peng, H. (1988). Fan-Beam Reconstruction in Computer Tomography from Full and Partial Projection Data. Ph.D. thesis, Rensselaer Polytechnique Institute, Troy, New York.

    Google Scholar 

  • Polak, E. (1971). Computational Methods in Optimization: A Unified Approach. Academic Press, New York.

    Google Scholar 

  • Radermacher, M. (1980). Dreidimensionale Rekon.struktion bei kegelförmiger Kippung im Elektronenmikroskop. Ph.D. thesis, Technische Universität München.

    Google Scholar 

  • Radermacher, M. (1988). Three-dimensional reconstruction of single particles from random and nonrandom tilt series. Elect. Micros. Tech. 9:359–394.

    Article  CAS  Google Scholar 

  • Radermacher, M. and Hoppe, W. (1980). Properties of 3-D reconstruction from projections by conical tilting compared to single axis tilting, in Proc. Seventh European Congr. Electron Microscopy, Vol. 1, pp. 132–133.

    Google Scholar 

  • Rushforth, C. K. (1987). Signal restoration, functional analysis and Fredholm integral equations of the first kind, in Image Recovery. Theory and Application (H. Stark, ed.), Academic Press, Orlando.

    Google Scholar 

  • Schafer, R. W., Mersereau, R. M., and Richards, M. A. (1981). Constrained iterative restoration algorithms. Proc. IEEE 69(4):432–450.

    Google Scholar 

  • Schelebusch, H.-J. and Splettstosser, W. (1985). On a conjecture of J. L. C. Sanz and T. S. Huang. IEEE Trans. Acoust. Speech and Signal Process. ASSP-33(6):1628–1630.

    Google Scholar 

  • Sezan, M. I. and Stark, H. (1982). Image restoration by the method of convex projections. II: Applications and numerical results. IEEE Trans. Med. Imaging MI-1:95–101.

    Google Scholar 

  • Sezan, M. I. and Stark, H. (1983). Image restoration by convex projections in the presence of noise. Appl. Opt. 22:2781–2789.

    Google Scholar 

  • Sezan, M. I. and Tekalp, A. M. (1990a). Adaptative image restoration with artifact suppression using the theory of convex projections. IEEE Trans. Acoust. Speech and Signal Process. 38(1):181–185.

    Google Scholar 

  • Sezan, M. I. and Tekalp, A. M. (1990b). A survey of recent developments in digital image restoration. Opt. Eng. 29(5):393–404.

    Google Scholar 

  • Sezan, M. I. and Trussell, H. J. (1991). Prototype image constraints for set-theoretic image restoration, in IEEE Trans. Acoust. Speech Signal Process. 39(10):2275–2285.

    Google Scholar 

  • Shore, J. E. and Johnson, R. W. (1980). Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory IT-26(1):26–37.

    Google Scholar 

  • Skilling, J. (1988). Classical MaxEnt data analysis, in 8th MaxEnt Workshop, Cambridge, UK.

    Google Scholar 

  • Skilling, J. and Bryan, R. K. (1984). Maximum entropy image reconstruction: general algorithm. Mon. Not. R. A.stronom. Soc. 211:111–214.

    Google Scholar 

  • Smith, K. T., Solmon, D. C., and Wagner, S. L. (1977). Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Am. Math. Soc. 83(6):1227–1268.

    Google Scholar 

  • Smith, P. R., Peters, T. H., and Bates, R. H. T. (1973). Image reconstruction from finite numbers of projections. J. Phys. A 6:361–382.

    Article  Google Scholar 

  • Tam, K. C. and Perez-Mendez, V. (1981a). Limited-angle three-dimensional reconstruction using Fourier transform iterations and Radon transform iterations. Opt. Eng. 20(4):586–589.

    Google Scholar 

  • Tam, K. C. and Perez-Mendez, V. (1981b). Tomographical imaging with limited-angle input. J. Opt. Soc. Am. 71:582–592.

    Google Scholar 

  • Tikochinsky, Y., Tishby, N. Z., and Levine, R. D. (1984). Consistent inference of probabilities for reproducible experiments. Phys. Rev. Lett. 52(16):1357–1360.

    Google Scholar 

  • Trussell, H. J. (1980). The relationship between image restoration by the maximum a posteriori method and the maximum entropy method. IEEE Trans. Acoust. Speech and Signal Process. ASSP-28(1):114–117.

    Google Scholar 

  • Trussell, H. P. (1984). A priori knowledge in algebraic reconstructions methods, in Advances in Computer Vision and Image Processing, Vol. 1, pp. 265–316, JAI Press.

    Google Scholar 

  • Trussell, H. J. and Civanlar, M. R. (1983). The initial estimate in constrained iterative restoration, in ICASSP’83, pp. 643–646.

    Google Scholar 

  • Trussel, H. P. and Civanlar, M. R. (1984). Feasible solution in signal restoration. IEEE Trans. Acoust. Speech and Signal Process. ASSP-32:201–212.

    Google Scholar 

  • Welton, T. A. (1979). A computational critique of an algorithm for image enhancement in bright field electron microscopy, in Advances in Electronics and Electron Physics(L. Maston, ed.), Vol. 48, pp. 37–101. Academic Press, New York.

    Google Scholar 

  • Whittaker, E. T. (1914). On the functions which are represented by expansion of the interpolation theory. Proc. Roy. Soc. Edinburg A35:181–194.

    Google Scholar 

  • Woods, J. W. (1981). Two-dimensional Kalman filtering, in Topics in Applied Physics (T. S. Huang, ed.), Vol. 2. Springer-Verlag, Berlin.

    Google Scholar 

  • Youla, D. C. (1987). Mathematical theory of image restoration by the method of convex projections, in Image Recovery: Theory and Applications (H. Stark, ed.), Academic Press, Orlando.

    Google Scholar 

  • Youla, D. C. and Velasco, V. (1986). Extensions of a result on the synthesis of signals in the presence of inconsistent constraints. IEEE Trans. Circuits Syst. CAS-33(4):465–468.

    Google Scholar 

  • Youla, D. C. and Webb, H. (1982). Image restoration by the method of convex projections. I: Theory. IEEE Trans. Med. Imaging MI-1:81–94.

    Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets. Inf. Vontrol 8:338–353.

    Google Scholar 

  • Zhou, X-W. and Xia, X.-G. (1989). The extrapolation of high-dimensionality band-limited functions. IEEE Trans. Acoust. Speech and Signal Process. ASSP-37(10):1576–1580.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Science+Business Media New York

About this chapter

Cite this chapter

Carazo, JM. (1992). The Fidelity of 3D Reconstructions from Incomplete Data and the Use of Restoration Methods. In: Frank, J. (eds) Electron Tomography. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2163-8_6

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-2163-8_6

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-2165-2

  • Online ISBN: 978-1-4757-2163-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics