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. 315316

In electron microscopy we obtain a twodimensional image – a projection of a threedimensional object (Fig. 2.5.6.1): The projection direction is defined by a unit vector and the projection is formed on the plane x perpendicular to The set of various projections may be assigned by a discrete or continuous set of points on a unit sphere (Fig. 2.5.6.2). The function reflects the structure of an object, but gives information only on coordinates of points of its projected density. However, a set of projections makes it possible to reconstruct from them the threedimensional (3D) distribution (Radon, 1917; DeRosier & Klug, 1968; Vainshtein et al., 1968; Crowther, DeRosier & Klug, 1970; Gordon et al., 1970; Vainshtein, 1971a; Ramachandran & Lakshminarayanan, 1971; Vainshtein & Orlov, 1972, 1974; Gilbert, 1972a; Herman, 1980). This is the task of the threedimensional reconstruction of the structure of an object:

The projection sphere and projection of along τ onto the plane . The case represents orthoaxial projection. Points indicate a random distribution of τ. 
Besides electron microscopy, the methods of reconstruction of a structure from its projections are also widely used in various fields, e.g. in Xray and NMR tomography, in radioastronomy, and in various other investigations of objects with the aid of penetrating, backscattered or their own radiations (Bracewell, 1956; Deans, 1983; Mersereau & Oppenheim, 1974).
In the general case, the function (2.5.6.1) (the subscript indicates dimension) means the distribution of a certain scattering density in the object. The function is the twodimensional projection density; one can also consider onedimensional projections of two (or three) dimensional distributions. In electron microscopy, under certain experimental conditions, by functions and we mean the potential and the projection of the potential, respectively [the electron absorption function μ (see Section 2.5.4) may also be considered as `density']. Owing to a very large depth of focus and practical parallelism of the electron beam passing through an object, in electron microscopy the vector τ is the same over the whole area of the irradiated specimen – this is the case of parallel projection.
The 3D reconstruction (2.5.6.2) can be made in the real space of an object – the corresponding methods are called the methods of direct threedimensional reconstruction (Radon, 1917; Vainshtein et al., 1968; Gordon et al., 1970; Vainshtein, 1971a; Ramachandran & Lakshminarayanan, 1971; Vainshtein & Orlov, 1972, 1974; Gilbert, 1972a).
On the other hand, threedimensional reconstruction can be carried out using the Fourier transformation, i.e. by transition to reciprocal space. The Fourier reconstruction is based on the well known theorem: the Fourier transformation of projection of a threedimensional object is the central (i.e. passing through the origin of reciprocal space) twodimensional plane cross section of a threedimensional transform perpendicular to the projection vector (DeRosier & Klug, 1968; Crowther, DeRosier & Klug, 1970; Bracewell, 1956). In Cartesian coordinates a threedimensional transform is The transform of projection along z is In the general case (2.5.6.1) of projecting the plane along the vector τ Reconstruction with Fourier transformation involves transition from projections at various to cross sections , then to construction of the threedimensional transform by means of interpolation between in reciprocal space, and transition by the inverse Fourier transformation to the threedimensional distribution : Transition (2.5.6.2) or (2.5.6.6) from twodimensional electronmicroscope images (projections) to a threedimensional structure allows one to consider the complex of methods of 3D reconstruction as threedimensional electron microscopy. In this sense, electron microscopy is an analogue of methods of structure analysis of crystals and molecules providing their threedimensional spatial structure. But in structure analysis with the use of Xrays, electrons, or neutrons the initial data are the data in reciprocal space in (2.5.6.6), while in electron microscopy this role is played by twodimensional images [(2.5.6.2), (2.5.6.6)] in real space.
In electron microscopy the 3D reconstruction methods are, mainly, used for studying biological structures (symmetric or asymmetric associations of biomacromolecules), the quaternary structure of proteins, the structures of muscles, spherical and rodlike viruses, bacteriophages, and ribosomes.
An exact reconstruction is possible if there is a continuous set of projections corresponding to the motion of the vector over any continuous line connecting the opposite points on the unit sphere (Fig. 2.5.6.2). This is evidenced by the fact that, in this case, the cross sections which are perpendicular to τ in Fourier space (2.5.6.4) continuously fill the whole of its volume, i.e. give (2.5.6.3) and thereby determine .
In reality, we always have a discrete (but not continuous) set of projections . The set of is, practically, obtained by the rotation of the specimen under the beam through various angles (Hoppe & Typke, 1979) or by imaging of the objects which are randomly oriented on the substrate at different angles (Kam, 1980; Van Heel, 1984). If the object has symmetry, one of its projections is equivalent to a certain number of different projections.
The object is finite in space. For function and any of its projections there holds the normalization condition where Ω is the total `weight' of the object described by the density distribution . If one assumes that the density of an object is constant and that inside the object = constant = 1, and outside it , then Ω is the volume of an object. The volume of an object, say, of molecules, viruses and so on, is usually known from data on the density or molecular mass.
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