The Projection Slice Theorem

The Filtered Backprojection (FBP) method may be used to process by reconstructing the original image from its projections in two steps Filtering and Backprojection. 

Filtering the projection The first step of FB Preconstruction is to perform the frequency integration (the inner integration) of the above equation. This entails filtering each of the projections using a filter with frequency response of magnitude /.  

The filtering operation may be implemented by ascertaining the filter impulse response required and then performing convolution or a FFT/IFFT combination to correlate p(t) against the impulse response. 

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Figure 3. Tomographic reconstruction the Projection Slice Theorem.

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... 

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... 

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. 

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. 

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... 

For image slicers, contiguous slices of the sky are re-arranged end-to-end to form the pseudo-slit. In that case it is obvious that the sky can be correctly sampled (according to the Nyquist sampling theorem) by the detector pixels on which the slices are projected in the same way as required for direct imaging. 

The key slice/projection theorem was first formulated in a radio astronomy context by Bracewell [5] and later exploited in NMR by Nagayama et al. [6] and Bodenhausen and Ernst [7, 8]. Consider the case of a typical plane 5(Fi 2) from a three-dimensional NMR spectrum S Fi,F2,F ). In order to obtain a projection at some angle a, the theorem postulates that the time domain response should be sampled along a slice through the origin at this same angle a. This requires that the evolution parameters C and t2 be varied jointly [7-13] ... 

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)...

Balta and Vonk [22], p. 15) and on several modern technologies17. The theorem deals with projections and slices. It explains the weird information on structure that we retrieve if we study the scattering intensity cut from a pattern along a line that is extending outward from the center of the pattern. In fact, the respective intensity curve is called a slice (or a section). Last but not least, the theorem reveals an elegant way to overcome the recognized problem. 

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