International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 318319

This method is based on the Fourier projection theorem [(2.5.6.3) –(2.5.6.5)]. The reconstruction is carried out according to scheme (2.5.6.6) (DeRosier & Klug, 1968; Crowther, DeRosier & Klug, 1970; Crowther, Amos et al. 1970; DeRosier & Moore, 1970; Orlov, 1975). The threedimensional Fourier transform is found from a set of twodimensional cross sections on the basis of the Whittaker–Shannon interpolation. If the object has helical symmetry (which often occurs in electron microscopy of biological objects, e.g. on investigating bacteriophage tails, muscle proteins) cylindrical coordinates are used. Diffraction from such structures with c periodicity and scattering density is defined by the Fourier–Bessel transform: The inverse transformation has the form so that and are the mutual Bessel transforms
Owing to helical symmetry, (2.5.6.22), (2.5.6.23) contain only those of the Bessel functions which satisfy the selection rule (Cochran et al., 1952) where N, q and p are the helix symmetry parameters, . Each layer l is practically determined by the single function with the lowest n; the contribution of other functions is neglected. Thus, the Fourier transformation of one projection of a helical structure, with an account of symmetry and phases, gives the threedimensional transform (2.5.6.23). We can introduce into this transform the function of temperaturefactor type filtering the `noise' from large spatial frequencies.
References
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