Fourier space

Within the cavity v 0, which in Fourier space is 0. Thus, only two of the tluee components of are... 

From the Omstein-Zemike equation in Fourier space one finds that... 

In a diffraction experiment a quantity F(S) can be measured which follows from equation (B 1.17.8) and equation (B 1.17.9) in Fourier space as... 

One of the most efficient algorithms known for evaluating the Ewald sum is the Particle-mesh Ewald (PME) method of Darden et al. [8, 9]. The use of Ewald s trick of splitting the Coulomb sum into real space and Fourier space parts yields two distinct computational problems. The relative amount of work performed in real space vs Fourier space can be adjusted within certain limits via a free parameter in the method, but one is still left with two distinct calculations. PME performs the real-space calculation in the conventional manner, evaluating the complementary error function within a cutoff... 

The integrals in Eqs. (17) and (18) are called convolution integrals. In Fourier space they are products of the Fourier transforms of c r). Thus, Eq. (18) is a geometric series in Fourier space, which can be summed. Performing this summation and returning to direct space, we have the OZ equation... 

In the Fourier space, the order parameter 0(r) for periodic structures is approximated by the Fourier series... 

The calculation of the free energy by minimization in the Fourier space gives almost the same values of the free energy, but a significant number of shells is necessary [26]. 

It is often useful to deal with the statistics in Fourier space. The Fourier transform of the correlation is called the power spectrum... 

Equivalently, in Fourier space, where tilde denotes the Fourier transform,... 

Considering the diagonalized form (5) of the image formation equation, a very tempting solution is to perform straightforward direct inversion in the Fourier space and then Fourier transform back to get the deconvolved image. 

The regularized solution is easy to obtain in the case of Gaussian white noise if we choose a smoothness prior measured in the Fourier space. In this case, the MAP penalty writes ... 

Shifting the origin in the Fourier space by uci, we obtain the wave-function FT[0(r)]e > , from which the lens aberration term can be eliminated in principle by multiplication with the inverse of the aberration phase factor e . The inverse Fourier transform gives finally the amplitude and phase of the true object wave 0 (f). 

Fig. 1.7 Two-dimensional objects in position three individual objects. Likewise, the Fourier space (top row) and their Fourier transform, transform is additive and the signal functions corresponding to the shape of the signal func- corresponding to each of the objects are shown tion S(kXr ky) (bottom row). The actual object, a for comparison, letter i inside a circle, is shown as the sum of...

The diffusion equation describes the evolution of the mean test particle density h(r,t) at point r in the fluid at time t. Denoting the Fourier transform of the local density field by hk(t), in Fourier space the diffusion equation takes the form... 

More general access to the coherent dynamic structure factor was provided by Akcasu and coworkers [93-95], starting from the assumption that the temporal evolution of the densities in Fourier space p(Q,t)... 

Submatrix ranking operators are belonging to the class of image-space operators (Haberacker [42]) in contrast to Fourier-space operators. 

From the computational point of view the Fourier space approach requires less variables to minimize for, but the speed of calculations is significantly decreased by the evaluation of trigonometric function, which is computationally expensive. However, the minimization in the Fourier space does not lead to the structures shown in Figs. 10-12. They have been obtained only in the real-space minimization. Most probably the landscape of the local minima of F as a function of the Fourier amplitudes A,- is completely different from the landscape of F as a function of the field real space. In other words, the basin of attraction of the local minima representing surfaces of complex topology is much larger in the latter case. As far as the minima corresponding to the simple surfaces are concerned (P, D, G etc.), both methods lead to the same results [21-23,119]. 

For low-Reynolds-number fluids the second term in the right-hand side of the Navier-Stokes equation can be neglected. Additionally, assuming that the viscous relaxation occurs more rapidly than the change of the order parameter, the acceleration term in Eq. (65) can be also omitted. Such approximations are validated in the case of polymer blends, for which they become exact in the limit of infinite polymer length, N —> oo. After these approximations, the NS equation can be easily solved in the Fourier space [160]. 

The other useful operation available within the Fourier space analysis is a cross-sectioning of images, which allows us to obtain quantitative information about their similarity or differences [218]. The correlation function C(r) is calculated as... 

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