Central slice theorem

Fig. 3 The discrete form of the central slice theorem in two dimensions. A projection p(r,cp) in real space (x,y) at angle p is a slice P(q,q>) at the same angle in Fourier space...

In practice, the reconstmction from projections is aided by an understanding of the relationship between an object and its projections in the Fourier space the central slice theorem states that the Fourier transform of an object s projection is a central plane in the Fourier transform of the object as shown in Figure 2. The Fourier transform of p(r, ff) is... 

The 3D reconstruction of an object is performed more conveniently in reciprocal (Fourier) space. The 2D Fourier transform of a projection of an object is identical to a plane of 3D Fourier transform of the original object normal to the projection direction (electron beam). The origin of each 2D Fourier transform of a projection is identical to the origin of the 3D Fourier transform of an object, provided that the projections are aligned so that they have the same (common) phase origin. This is known as the Fourier slice theorem or the central projection theorem. 

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