The Fourier Method

According to the Fourier method, the measured line integral p r,4 ) in a sinogram is related to the count density distribution A(x,y) in the object obtained by the Fourier transformation. The projection data obtained in the spatial domain (Fig. 4.2a) can be expressed in terms of a Fourier series in the frequency domain as the sum of a series of sinusoidal waves of different amplitudes, spatial frequencies, and phase shifts running across the image (Fig. 4.2b). This is equivalent to sound waves that are composed of many sound frequencies. The data in each row of an acquisition matrix can be considered to be composed of sinusoidal waves of varying amplitudes and frequencies in the frequency domain. This conversion of data from spatial domain to frequency domain is called the Fourier transformation (Fig. 4.3). Similarly the reverse operation of converting the data from frequency domain to spatial domain is termed the inverse Fourier transformation. 

In the Fourier method of backprojection, the projection data in each profile are subjected to the Fourier transformation from spatial domain to frequency domain, which is symbolically expressed as 

A filter H(v), in the frequency domain is applied to each projection, i.e., 

Finally, the inverse Fourier transformation is performed to obtain filtered projection data in the spatial domain, which are then backprojected in the same manner as in the simple backprojection. With the use of faster computers, the Fourier technique of filtered backprojection has gained wide acceptance in reconstruction of images in nuclear medicine. 

Several factors affect the filtered backprojection. Adequate sampling of projections (both linear and angular projections related to r and of the sinogram) is needed for accurate backprojection. Data noise, positron range, noncolinearity, scattering, and random events are not taken into consideration in the method. Also in this model, detectors are assumed to be point 

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The Fourier method is not a requirement, and direct sinusoidal fitting procedures are also used to fit the data from a set of images. A number of specialized procedures have been described over the years and it is worth noting that extracting the amplitude and phase may be done as a simple extension to conventional linear regression. 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

A fundamental property of the Fourier transform is that of superposition. The usefulness of the Fourier method lies in the fact that one can separate a function into additive components, treat each one separately, and then build up the full result by summing the individual results. It is a beautiful and explicit example of the stepwise refinement of complex problems. In stepwise refinement, one successfully tackles the most difficult tasks and solves problems far beyond the mind s momentary grasp by dividing the problem into its ultimately simple pieces. The full solution is then obtained by reassembling the solved pieces. 

In spite of these difficulties great progress has been made by Pinsker (1953) and Weinstein (1956) and by Cowley and Rees (1958). In this work the Fourier method has been used to delineate potential distribution in molecules with considerable clarity. The results are of interest in the present context because it is already clear that electron diffraction is much more powerful than X-ray diffraction for the detection of hydrogen atoms in organic molecules. Future results may therefore be important in the study of overcrowded aromatic molecules. 

The Fourier method is best suited to cartesian coordinates because the expansion functions QtkR/LR etlr Lr are just the eigenfunctions of the kinetic energy operator. For problems including the rotational degree of freedom other propagation methods have been developed (Mowrey, Sun, and Kouri 1989 Le Quere and Leforestier 1990 Dateo, Engel, Almeida, and Metiu 1991 Dateo and Metiu 1991). 

FOURIER SERIES AND ORTHOGONAL FUNCTIONS, Harry F. Davis. An incisive text combining theory and practical example to introduce Fourier series, orthogonal functions and applications of the Fourier method to boundary-value problems. 570 exercises. Answers and notes. 4I6pp. 5H x 8b. 65973-9 Pa. 8.95... 

G. Katz, et at. The Fourier method for triatomic systems in the search for the optimal coordinate system, /. Chem. Phys. 116 (11) (2002) 4403-4414. 

R. Kosloff, The fourier method. In C. Ceijan, editor. Numerical Grid Methods and Their Application to Schr6dinger s Equation, volume NATO ASI Ser. C 412, 175-194, Kluwer Academic Publishers, The Netherlands, (1993). 

Waser, J. Pictorial representation of the Fourier method of X-ray crystallography, J. Chem. Educ. 45, 446-451 (1968). 

Beevers, C. A., and Lipson, H. The crystal structure of copper sulphate pen-tahydrate, CuS04-5H20. Proc. Roy. Soc. (London) A146, 570-582 (1934). Lonsdale, K. An X-ray analysis of the structure of hexachlorobenzene using the Fourier method. Proc. Roy. Soc. (London) A133, 536-552 (1931). 

In the Fourier method each path contributing to Eq. (4.13) is expanded in a Fourier series and the sum over all contributing paths is replaced by an equivalent Riemann integration over all Fourier coefficients. This method was first introduced by Feynman and Hibbs to determine analytic expressions for the harmonic oscillator propagator and has been used by Miller in the context of chemical reaction dynamics. We have further developed the approach for use in finite-temjjerature Monte Carlo studies of quantum sys-tems, and we have found the method to be very useful in the cluster studies discussed in this chapter. 

To develop the Fourier method we let x,- be one of the 3u coordinates for our M-particle system. For a given path the coordinate is expressed parametrically as... 

There have been two principal methods developed to evaluate the kinetic energy using path integral methods. One method, based on Eq. (3.5), has been termed the T-method and the other, based on Eq. (4.1), has bwn termed the //-method. In discretized path integral calculations the T-method and the //-method have similar properties, but in the Fourier method the expressions and the behavior of the kinetic energy evaluated by Monte Carlo techniques are different. 

In contrast the //-method expression for the kinetic energy in the Fourier method is obtained by utilizing Eqs. (4.1) and (4.15). The required derivatives from ojjeration of the Hamiltonian ojjerator on the density matrix can be evaluated analytically, and the resulting expression is given by... 

We close this section by assessing the Fourier method with model studies on two one-dimensional systems. The first is the harmonic oscillator whose Fourier kinetic (in the H method) and potential energies as a function of fc , can be evaluated analytically and are given by... 

This equation results from the assumption that the diffraction line has the shape of an error curve [9.1].) Once B has been obtained from Eq. (9-1), it can be inserted into Eq. (3-13) to yield the particle size t. There are several other methods of finding B from Bf compared with Warren s method, they are somewhat more accurate and considerably more intricate. These other methods involve Fourier analysis of the diffraction lines from the unknown and from the standard, and considerable computation [G.30, G.39]. Even the approach involving Eq. (9-1) can be difficult if the line from the standard is a resolved Ka. doublet it is then simpler to use a Kfi line. The Fourier methods automatically take care of the existence of a doublet. 

This expression was used for the P ri) extraction directly from f x), but as this did not remove spurious oscillations, a smoothing procedure was applied, and the method was tested on composite specimens prepared by mixing known quantities of samples of nickel hydroxide, whose crystallite size distributions were previously determined. The Fourier method is in principle an exact... 

In the historical survey of the spectral methods given by Canute et al [22], it was assumed that Lanczos [101] was the first to reveal that a proper choice of trial functions and distribution of collocation points is crucial to the accuracy of the solution of ordinary differential equations. Villadsen and Stewart [203] developed this method for boundary value problems. The earliest applications of the spectral collocation method to partial differential equations were made for spatially periodic problems by Kreiss and Oliger [94] and Orszag [139]. However, at that time Kreiss and Oliger [94] termed the novel spectral method for the Fourier method while Orszag [139] termed it a pseudospectral method. 

Fig. 6.1.6 Principle of spectroscopic resolution by the Fourier method in combination with reconstruction of the spin density from projections, (a) Arrangement of measured signals Vi (I, G) in IG space before Fourier transformation over G. (b) Interpolation scheme in tg space after Fourier transformation over G for separation of space and frequency coordinates.

The Random Walk. The most compelling discrete effective theory of diffusion is that provided by the random walk model. This picture of diffusion is built around nothing more than the idea that the diffusing entities of interest exercise a series of uncorrelated hops. The key analytic properties of this process can be exposed without too much difficulty and will serve as the basis of an interesting comparison with the Fourier methods we will undertake in the context of the diffusion equation. 

N is set to the nearest integer to XN. For ease of computation in the Fourier method, N should be even and so if N is odd, the next highest even integer is used. The result of this calculation of N ensures that the mobile molecule can always be placed to touch the static molecule and there will remain grid cells at the boundary unoccupied by the protein. 

The filtered backprojection can be applied to 3D image reconstruction with some manipulations. The 3D data sinograms are considered to consist of a set of 2D parallel projections, and the FBP is applied to these projections by the Fourier method. The iteration methods also can be generally applied to the 3D data. However, the complexity, large volume, and incomplete sampling of the data due to the finite axial length of the scanner are some of the factors that limit the use of the FBP and iterative methods directly in 3D reconstruction. To circumvent these difficulties, a modified method of handling 3D data is commonly used, which is described below. 

Choosing the generating function UNg as the polynomial, UNf (q) = (q - qx)(q - 2) " (q qi) (q q ) leads to the well-known Lagrange interpolation formula. Figure 4 shows the expansion function g (q) which is based on the zeros of the Cheby-chev orthogonal polynomial of order Ng. Another choice appropriate for evenly distributed sampling points is based on the global function NNg(q) = sin(2 Tr /A ). It is closely related to the Fourier method described in the next section. 

An examination of the Fourier method, which is a special case of an orthogonal collocation representation, elucidates the main considerations of representation theory. It will be shown that by optimizing the representation the quantum limit of one point per unit phase space volume of h can be obtained. Moreover, the Fourier method has great numerical advantages because of the fast nature of the algorithm (22-26). This means that the numerical effort scales semilinearly with the represented volume of phase space (27). 

The phase space representation of the Fourier method is of a rectangular shape. The volume in phase space covered by the Fourier representation is calculated as follows The length of the spatial dimension in phase is L, and the maximum momentum is pmax. Therefore, the represented volume becomes Y = 2L-pmax, where the factor of two appears because the momentum range is from -pmax to + pmax. Using the fact that p = ftk, the phase space volume can be expressed as... 

The computational scaling properties of the Fourier method are a result of the scaling properties of the FFT algorithm which scales as 0(Nf, log Ng). As a result the phase space volume determines the scaling of the computational effort 0(Y log T). 

Figure 7 The four steps in the application of the kinetic energy operator by the Fourier method (A - B - C - D) (lower panel) coordinate space (upper panel) momentum space. A is the original wave function i i(g). B represents the wave function in momentum space 4>(p) obtained by Fourier transform of ty(q). C is the application of the kinetic energy operator in momentum space. (p) = p2/2M ij>(p) where T = p2/2M is also shown. D is (<7) = h is the final...

The operation scaling law of the Fourier method is determined by the forward and reverse unitary transformations from coordinate to momentum space. In general, they scale as 0(N2R) but with the use of the fast Fourier transform (FFT) algorithm this scaling is reduced to 0(NH log NR). 

Figure 8 Comparison of the kinetic energy operator spectrum for the Fourier method (solid) with the fourth- (FD-4) and sixth-order (FD-6) finite difference method.

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