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, p. 318

This method is also called the synthesis of projection functions. Let us consider a twodimensional case and stretch along each onedimensional projection (Fig. 2.5.6.5) by a certain length b; thus, we obtain the projection function Let us now superimpose h functions The continuous sum over the angles of projection synthesis is this is the convolution of the initial function with a rapidly falling function (Vainshtein, 1971b). In (2.5.6.15), the approximation for a discrete set of h projections is also written. Since the function approaches infinity at , the convolution with it will reproduce the initial function , but with some background B decreasing around each point according to the law . At orthoaxial projection the superposition of cross sections arranged in a pile gives the threedimensional structure .
Radon operator. Radon (1917; see also Deans, 1983) gave the exact solution of the problem of reconstruction. However, his mathematical work was for a long time unknown to investigators engaged in reconstruction of a structure from images; only in the early 1970s did some authors obtain results analogous to Radon's (Ramachandran & Lakshminarayanan, 1971; Vainshtein & Orlov, 1972, 1974; Gilbert, 1972a).
The convolution in (2.5.6.15) may be eliminated using the Radon integral operator, which modifies projections by introducing around each point the negative values which annihilate on superposition the positive background values. The onedimensional projection modified with the aid of the Radon operator has the form Now is calculated analogously to (2.5.6.14), not from the initial projections L but from the modified projection :
The reconstruction of highsymmetry structures, in particular helical ones, by the direct method is carried out from one projection making use of its equivalence to many projections. The Radon formula in discrete form can be obtained using the double Fourier transformation and convolution (Ramachandran & Lakshminarayanan, 1971).
References
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