2D Fourier transformation

It is complete because of fiber symmetry. The 2D Fourier transform of this image is not related to the searched slice, but to a projection of the correlation function. In contrast, the sought-after slice in real space... 

By means of this procedure our problem is not only reduced from three to two dimensions, but also is the statistical noise in the scattering data considerably reduced. Multiplication by —4ns2 is equivalent to the 2D Laplacian89 in physical space. It is applied for the purpose of edge enhancement. Thereafter the 2D background is eliminated by spatial frequency filtering, and an interference function G(s 2,s ) is finally received. The process is demonstrated in Fig. 8.27. 2D Fourier transform of the interference function... 

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. 

D reconstruction can be performed by restoring the 3D Fourier space of the object from a series of 2D Fourier transforms of the projections. Then the 3D object can be reconstructed by inverse Fourier transformation of the 3D Fourier space. For crystalline objects, the Fourier transforms are discrete spots, i.e. reflections. In electron microscopy, the Fourier transform of the projection of the 3D electrostatic potential distribution inside a crystal, or crystal structure factors, can be obtained from HREM images of thin crystals. So one can obtain the 3D electrostatic potential distribution (p(r) inside a crystal from a series of projections by... 

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

With 2D WIN-NMR, 2D Fourier transformation may be accomplished with the commands xfb, xtrf, xf2 and xfl accessible via the Process pull-down menu (Fig. 5.5). 

Load the raw data obtained for glucose with the inverse 2D CH-COSY experiment D NMRDATA GLUCOSE 2D CH GCHICOMQ 001001. SER and perform a 2D Fourier transformation following the guidelines given above. Enter the Manual phase correction option and perform a phase corrections in F2 and Ft according to the procedure outlined above. Try to phase all peaks for positive absorption and store the spectrum (... 001001. RR). 

After 2D Fourier transformation J-Resolved spectra usually contain a distortion along the horizontal line leading through the centre of the matrix. In order to get rid of this distortion and to separate chemical shifts from homonuclear J-couplings, the whole matrix is tilted. With 2D WIN-NMR a Tilt command is available which automatically adjusts the corresponding parameters (Tilt factor) and performs a tilt operation. 

The microdomain orientation as a function of the electric field strength was monitored by a series of scanning force microscopy (SFM) images taken in the center between the electrodes. The entire electrode length of 6 mm was screened in steps of a few tens of microns. From the azimuthal intensity distribution of the 2D Fourier transformations of the SFM images, the orientational order parameter P2 was calculated according to ... 

The angle (p quantifies the in-plane direction, with cp = 0° corresponding to the direction along the stripe-like electrodes. For an alignment of the lamellae along the field direction (maximum azimuthal intensity distribution of the 2D Fourier transform intensity at (p = 90°), P) ranges from 0 to -0.5 with P2 = -0.5 corresponding to the fully oriented case. 

Fig. 7. Unfiltered 2D Fourier transform of the image in Fig. 6. The rings detected around the central spot (see inset) and the other lattice spots have twice the radius of the 2D Fermi contour of the surface state. The lines in the power spectrum reflect the residual mechanical vibrations of the experimental setup.

Applying a 2D Fourier transform to Eqs. (47) and (48) with respect to the variable, s, leads to the set of ordinary differential equations for the (Fourier) transform function, (pk(z)... 

The 2D PE signal is computed using the response function given by Equations (21)-(23). Since we consider the 2D response on the time scale smaller than the dephasing times, only the coherent component of the response function [Equation (21)] contributes to the signal. The 2D Fourier transform PE signal determined by Equations (22) and (30) has the following form (17) ... 

The first two terms in the bracket give—after a 2D Fourier transform—the diagonal peaks in the spectrum, where as the last two terms give the cross-peaks. 

In two-dimensional techniques, prior to the observation pulse with the detection period t2, an rf pulse is applied with the evolution period y between the two pulses. A second time dimension (COSY) is created by repeating the same experiment with the incrementation of H. For each value of ti a free induction decay (FID) is recorded and, after 2D Fourier transformation, the desired 2D frequency spectrum S(wi,w2) is obtained. In the NOESY spectroscopy, the mixing period consisting of two 90° pulses separated by the mixing time Tm is used. The general experimental scheme for... 

The samples of the echo readout correspond to the t2 interval, and the phase encode steps to the t interval (not a true time interval of course), of a 2D spectroscopic acquisition. A 2D Fourier transform yields an image in which the two frequency axes correspond to the x and y spatial axes. This method is a distinct improvement over the projection method... 

Figure 3.38 In situ STM image (a) and the corresponding 2D Fourier transform (b) of a compressed 2D hep Pbads overlayer showing a higher order superstructure with moir6 pattern (cf. Fig. 3.18) in the system Au(lll)/5 x 10 M Pb(C104)2 + 10 M HCIO4 at AB = 50 mV and T = 298 K with = 40 nA and Ft -Ir tip 13.300].

Here, q is the 2D Fourier transform wavevector conjugate to r = (x,y) and referenee to it and to a -frequency will be suppressed when possible. The assoeiated semi-infinite screening function, K (zi,Z2), which satisfies the inversion relation... 

A 2D spectrum is obtained by 2D Fourier transformation over the spinner phase and the acquisition time t2. It exhibits spinning sideband signals in both dimensions. Those in the additional dimension o)i are characteristic of the molecular order. No sidebands appear in this direction if the sample is isotropic. The 2D sideband signals can be analysed to obtain the orientational distribution function. 

For 2D systems with circular symmetry, the relevant information is in the radial direction. Therefore, instead of calculating its 2D Fourier transform by (4.1.10), evaluation of a radial ID Fourier transformation is sufficient. In magnetic resonance imaging [Majl]... 

Fig. 5.4..3 [Blu4] Imaging and diffraction by example of. W similar glass capillaries immersed in water. The image 5(jc, > ) of the spin density derives from the acquired signal s(ky. A, ) by 2D Fourier transformation. Formation of the magnitude square l.v(A , ) discards the signal phase...

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