Fourier slice theorem

Typical tomographic 2D-reconstruction, like the filtered backprojection teelinique in Fan-Beam geometry, are based on the Radon transform and the Fourier slice theorem [6]. 

Favorable properties of the Fourier transform itself provide general means either to split the general problem of data analysis into sub-problems or even to obtain structure parameters without much modeling work. In this respect the Fourier slice theorem must be pointed out because of its superior impact on scattering (Bonart [16] ... 

Anisotropic Particle Scattering Varying Intensity Decay in Different Directions. In case of anisotropy the decay of the scattering intensity 7 (s) is a function of the direction chosen. The intensity extending from s = 0 outward in a deliberately chosen direction i is mathematically the deAnition of a slice (cf. Sect. 2.7.1, p. 22). Thus, the Fourier-Slice theorem, Eq. (2.38), turns the particle density function Ap (r) into a projection Ap (r) j (r,) and the scattering intensity is related to structure by... 

In Q the non-topological structure parameters of the material s nanostructure are combined. For multiphase systems this fact can be deduced by application of the Fourier-slice theorem and the considerations which lead to Porod s law. In particular, for a two-phase system it follows64... 

As pointed out by Stribeck [139,171] g (x) is, as well, suitable for the study of oriented microfibrillar structures and, generally, for the study of ID slices in deliberately chosen directions of the correlation function. This follows from the Fourier-slice theorem and its impact on structure determination in anisotropic materials, as discussed in a fundamental paper by Bonart [16]. 

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. 

The Fourier Slice Theorem. The two-dimensional Fourier transform of the object function, according to Eq. (26.37), is given by... 

This is the Fourier slice theorem, which states that the Fourier transform of a parallel projection of an object taken at angle 0 to the a axis in physical space is equivalent to a slice of the two-dimensional transform F(u, v) of the object function f(x, y), inclined at an angle 0 to the u axis in frequency space (Fig. 26.16). 

While the Fourier slice theorem implies that given a sufficient number of projections, an estimate of the two-dimensional transform of the object could be assembled and by inversion an estimate of the object obtained, this simple conceptual model of tomography is not implemented in practice. The approach that is usually adopted for straight ray tomography is that known as the filtered back-projection algorithm. This method has the advantage that the reconstruction can be started as soon as the first projection has been taken. Also, if numerical interpolation is necessary to compute the contribution of each projection to an image point, it is usually more accinate to conduct this in physical space rather than in frequency space. 

Then it follows from the slice theorem Eq. (2.38) for the integral breadth of the Fourier transformed function H (5)... 

Fig. 1 The Bracewell slice rojection theorem. The Fourier transform of a slice through the evolution dimension at an inclination a (left) is the projection of the corresponding frequency-domain spectrum at the same angle a (right)...

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...

The question we have not yet addressed is how to determine the value of 4 s 4 lo, the phase dilference between the signal and local oscillator fields. This relative phase factor will determine how the total signal is split into real and imaginary components and currently an arbitrary number has been chosen. To determine the relative phase, we use the projection slice theorem, which was originally used in 2D nuclear magnetic resonance (NMR) spectroscopy for the same purpose. The theorem deals with a 2D spectrum in frequency, 5(mi, coi) which is a Fourier transform of a 2D spectrum in time, 5(fi, 12)- It states that... 

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... 

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