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. 319320

In the general case of 3D reconstruction from projections the projection vector τ occupies arbitrary positions on the projection sphere (Fig. 2.5.6.2). Then, as in (2.5.6.15), we can construct the threedimensional spatial synthesis. To do this, let us transform the twodimensional projections by extending them along τ as in (2.5.6.13) into threedimensional projection functions .
Analogously to (2.5.6.15), such a threedimensional synthesis is the integral over the hemisphere (Fig. 2.5.6.2) this is the convolution of the initial function with (Vainshtein, 1971b).
To obtain the exact reconstruction of we find, from each , the modified projection (Vainshtein & Orlov, 1974; Orlov, 1975)
By extending along τ we transform them into . Now the synthesis over the angles gives the threedimensional function
The approximation for a discrete set of angles is written on the right. In this case we are not bound by the coaxial projection condition which endows the experiment with greater possibilities; the use of object symmetry also profits from this. To carry out the 3D reconstruction (2.5.6.25) or (2.5.6.27) one should know all three Euler's angles ψ, θ, α (Fig. 2.5.6.2).
The projection vectors should be distributed more or less uniformly over the sphere (Fig. 2.5.6.2). This can be achieved by using special goniometric devices.
Another possibility is the investigation of particles which, during the specimen preparation, are randomly oriented on the substrate. This, in particular, refers to asymmetric ribosomal particles. In this case the problem of determining these orientations arises.
The method of spatial correlation functions may be applied if a large number of projections with uniformly distributed projection directions is available (Kam, 1980). The space correlation function is the averaged characteristic of projections over all possible directions which is calculated from the initial projections or the corresponding sections of the Fourier transform. It can be used to find the coefficients of the object density function expansion over spherical harmonics, as well as to carry out the 3D reconstruction in spherical coordinates.
Another method (Van Heel, 1984) involves the statistical analysis of image types, subdivision of images into several classes and image averaging inside the classes. Then, if the object is rotated around some axis, the 3D reconstruction is carried out by the iteration method.
If such a specimen is inclined at a certain angle with respect to the beam, then the images of particles in the preferred orientation make a series of projections inclined at an angle β and having a random azimuth. The azimuthal rotation is determined from the image having zero inclination.
If particles on the substrate have a characteristic shape, they may acquire a preferable orientation with respect to the substrate, their azimuthal orientation α being random (Radermacher et al., 1987).
In the general case, the problem of determining the spatial orientations of randomly distributed identical threedimensional particles with an unknown structure may be solved by measuring their twodimensional projections (Fig. 2.5.6.1) if the number i of such projections is not less than three, (Vainshtein & Goncharov, 1986a,b; Goncharov et al. 1987; Goncharov, 1987). The direction of the vector along which the projection is obtained is set by the angle (Fig. 2.5.6.2).
The method is based on the analysis of onedimensional projections of twodimensional projections where α is the angle of the rotation about vector τ in the p plane.
Lemma 1. Any two projections and (Fig. 2.5.6.6) have common (identical) onedimensional projections : Vectors and (Fig. 2.5.6.3) determine plane h in which they are both lying. Vector is normal to plane h and parallel to axis of the onedimensional projection ; both and axes along which the projections and are constructed are perpendicular to .
The corresponding lemma in the Fourier space states:
Lemma 2. Any two plane transforms, and intersect along the straight line (Fig. 2.5.6.7); the onedimensional transform is the transform of .
Thus in order to determine the orientations of a threedimensional particle it is necessary either to use projections in real space or else to pass to the Fourier space (2.5.6.5).
Now consider real space. The projections are known and can be measured but angles of their rotation about vector (Fig. 2.5.6.8) are unknown and should be determined. Let us choose any two projections and and construct a set of onedimensional projections and by varying angles and . In accordance with Lemma 1, there exists a onedimensional projection, common for both and , which determines angles and along which and should be projected for obtaining the identical projection (Fig. 2.5.6.5). Comparing and and using the minimizing function it is possible to find such a common projection . (A similar consideration in Fourier space yields .)

Plane projections of a threedimensional body. The systems of coordinates in planes (a) and (b) are chosen independently of one another. 
The mutual spatial orientations of any three noncoplanar projection vectors , , can be found from three different twodimensional projections , and by comparing the following pairs of projections: and , and , and and , and by determining the corresponding , and . The determination of angles , and reduces to the construction of a trihedral angle formed by planes , and . Then the projections with the known can be complemented with other projections and the corresponding values of ω can be determined. Having a sufficient number of projections and knowing the orientations , it is possible to carry out the 3D reconstruction of the object [see (2.5.6.27); Orlov, 1975; Vainshtein & Goncharov, 1986a; Goncharov et al., 1987].
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