Abstract
Traditionally, in computed tomography practiced in radiology, the resolution of the reconstruction is expressed in terms of the number of evenly spaced projections required for the faithful reconstruction of an object that has a given diameter (see equation (10) below).The tacit assumption is that projection data have a sufficient spectral signal-to-noise ratio (SSNR) in the whole frequency range in order to reproduce the object faithfully. In electron microscopy, the situation is dramatically different, as the electron dose limitations result in very low SSNR in the individual projections. The suppression of signal is particularly severe in high spatial frequencies, where the signal is affected by the envelope function of the microscope and the high amount of ambient noise, as well as in some low spatial frequency regions (due to the influence of the contrast transfer function (CTF) of the electron microscope). In single-particle reconstruction, a satisfactory level of the SSNR in the 3D reconstruction is achieved by including a large number of 2D projections (tens to hundreds of thousands) that are averaged during the reconstruction process. Except for rare cases (Boisset et al., 1998), the angular distribution of projections is not an issue, as the large number of molecules and the randomness of their orientations on the support grid all but guarantee uniform coverage of angular space. The concern is whether the number of projections per angular direction is sufficient to yield the desired SSNR or whether the angular distribution of projections is such that the oversampling of the 3D Fourier space achieved during the reconstruction process will yield the desired SSNR.
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References
Boisset, N., Penczek, P. A., Taveau, J. C., You, V., Dehaas, F. and Lamy, J. (1998). Overabundant single-particle electron microscope views induce a three-dimensional reconstruction artifact. Ultramicroscopy 74:201–207.
Böttcher, B., Wynne, S.A. and Crowther, R.A. (1997). Determination of the fold of the core protein of hepatitis B virus by electron cryomicroscopy. Nature 386:88–91.
Bracewell, R. N. and Riddle, A. C. (1967). Inversion of fan-beam scans in radio astronomy. Astrophys. J. 150:427–434.
Carazo, J. M. (1992). The fidelity of 3D reconstruction from incomplete data and the use of restoration methods. In Electron Tomography (J. Frank, ed.). Plenum, New York, pp. 117–166.
Carazo, J.M. and Carrascosa, J. L. (1986). Information recovery in missing angular data cases: an approach by the convex projections method in three dimensions. J. Microsc. 145:23–43.
Carazo, J. M. and Carrascosa, J. L. (1987). Restoration of direct Fourier three-dimensional reconstructions of crystalline specimens by the method of convex projections. J. Microsc. 145:159–177.
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.
Conway, J. F., Cheng, N., Zlotnick, A., Wingfield, P.T., Stahl, S. J. and Steven, A. C. (1997). Visualization of a 4-helix bundle in the hepatitis B virus capsid by cryo-electron microscopy. Nature 386:91–94.
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.
Faridani, A. (2003). Introduction to the mathematics of computed tomography. In Inside Out: Inverse Problems and Applications, vol. 47 (G. Uhlmann, ed.). Cambridge University Press, Cambridge, pp. 1–46.
Frank, J. (2006). Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Oxford University Press, New York.
Frank, J. and Al-Ali, L. (1975). Signal-to-noise ratio of electron micrographs obtained by cross correlation. Nature 256:376–379.
Keinert, F. (1989). Inversion of k-plane transforms and applications in computer-tomography. SIAM Rev. 31:273–298.
Lanzavecchia, S., Cantele, F., Bellon, P., Zampighi, L., Kreman, M., Wright, E. and Zampighi, G. (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.
Louis, A. K. (1984). Nonuniqueness in inverse Radon problems-the frequency-distribution of the ghosts. Math. Z. 185:429–440.
Maass, P. (1987). The X-ray transform—singular value decomposition and resolution. Inverse Probl. 3:729–741.
Natterer, F. (1986). The Mathematics of Computerized Tomography. John Wiley & Sons, New York.
Natterer, F. and Ritman, E. L. (2002). Past and future directions in x-ray computed tomography (CT). Int. J. Imaging Syst. Technol. 12:175–187.
Natterer, F. and Wübbeling, F. (2001). Mathematical Methods in Image Reconstruction. SIAM, Philadelphia.
Orlov, S. S. (1976). Theory of three-dimensional reconstruction 1. Conditions for a complete set of projections. Soviet Phys. Crystallogr. 20:312–314.
Penczek, P. (1998). Measures of resolution using Fourier shell correlation. J. Mol. Biol. 280:115–116.
Penczek, P., Marko, M., Buttle, K. and Frank, J. (1995). Double-tilt electron tomography. Ultramicroscopy 60:393–410.
Penczek, P. A. (2002). Three-dimensional spectral signal-to-noise ratio for a class of reconstruction algorithms. J. Struct. Biol. 138:34–46.
Penczek, P. A., Renka, R. and Schomberg, H. (2004). Gridding-based direct Fourier inversion of the three-dimensional ray transform. J. Opt. Soc. Am. A 21:499–509.
Radermacher, M. (1988). Three-dimensional reconstruction of single particles from random and nonrandom tilt series. J. Electron Microsc. Tech. 9:359–394.
Saxton, W.O. (1978). Computer Techniques for Image Processing of Electron Microscopy. Academic Press, New York.
Saxton, W. O. and Baumeister, W. (1982). The correlation averaging of a regularly arranged bacterial envelope protein. J. Microsc. 127:127–138.
Saxton, W. O., Baumeister, W. and Hahn, M. (1984). Three-dimensional reconstruction of imperfect two-dimensional crystals. Ultramicroscopy 13:57–70.
Sezan, M. I. (1992). An overview of convex projections theory and its application to image recovery problems. Ultramicroscopy 40:55–67.
Sezan, M. I. and Stark, H. (1982). Image restoration by the method of convex projections. II. Applications and numerical results. IEEE Trans. Med. Imaging 1:95–101.
Stewart, A. and Grigorieff, N. (2004). Noise bias in the refinement of structures derived from single particles. Ultramicroscopy 102:67–84.
Unser, M., Sorzano, C. O., Thevenaz, P., Jonic, S., El-Bez, C., De Carlo, S., Conway, J. F. and Trus, B. L. (2005). Spectral signal-to-noise ratio and resolution assessment of 3D reconstructions. J. Struct. Biol. 149:243–255.
Unser, M., Trus, B. L. and Steven, A. C. (1987). A new resolution criterion based on spectral signal-to-noise ratios. Ultramicroscopy 23:39–51.
Vainshtein, B. K. and Penczek, P. A. (2006). Three-dimensional reconstruction. In International Tables for Crystallography 3rd edn., vol. B Reciprocal Space (U. Shmueli, ed.).
van Heel, M. (1987). Similarity measures between images. Ultramicroscopy 21:95–100.
Wade, R.H. (1992). A brief look at imaging and contrast transfer. Ultramicroscopy 46:145–156.
Youla, D. C. and Webb, H. (1982). Image restoration by the method of convex projections. 1. Theory. IEEE Trans. Med. Imaging 1:81–94.
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Penczek, P.A., Frank, J. (2007). Resolution in Electron Tomography. In: Frank, J. (eds) Electron Tomography. Springer, New York, NY. https://doi.org/10.1007/978-0-387-69008-7_11
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DOI: https://doi.org/10.1007/978-0-387-69008-7_11
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