Fourier imaging

Figure 6. A pulse sequence for a two-dimensional Fourier imaging using a spin echo. A Gy gradient for a fixed period is imposed for a "phase" modulation to the signal, encoding position dependent information along the y-axis. The magnitude of Gy gradient is varied with a fixed increment for each scan of the sequence.

This process is carried out using a computer, subsequent to obtaining the two-dimensional signal, S(kx, ky). A pulse sequence for two-dimensional Fourier imaging is shown in Figure 6. An example of the two-dimensional Fourier transform process is shown for two circular objects in Figure 7. 

These are non-linear equation for the Fourier images of functions cp and i which like non-linear Schroedinger equation and require a special solution. 

From the absolute values and internal consistency of ADP, visualized as thermal (or displacement) ellipsoids . These parameters, especially in anisotropic approximation, tend to act as sinks for all kinds of random and (neglected) systematic errors. Thus, for a strongly absorbing crystal (in the absence of intensity correction) the thermal ellipsoids of all atoms will approximate the Fourier image of the crystal s outer shape. An unreasonably small or large... 

The next milestone, in the history of NMR [Frel], was the extension of the NMR spectrum to more than one frequency coordinate. It is called multi-dimensional spectroscopy and is a form of nonlinear spectroscopy. The technique was introduced by Jean Jeener in 1971 [Jeel] with two-dimensional (2D) NMR. It was subsequently explored systematically by the research group of Richard Ernst [Em 1 ] who also introduced Fourier imaging [Kuml]. Today such techniques are valuable tools, for instance, in the structure elucidation of biological macromolecules in solution in competition with X-ray analysis of crystallized molecules as well as in solid state NMR of polymers (cf. Fig. 3.2.7) [Sch2]. 

Fig. 5.4.2 [Ljul] Trajectories in k space, (a) Line-scan method, (b) Back-projection method, (c) Modified back-projection method, (d) Fourier imaging, (e) Echo-planar imaging.

Spin-warp imaging denotes the most common form of Fourier imaging with data acquisition in Cartesian k-space coordinates [Edel]. Instead of a variable evolution time t for phase encoding, the gradient amplitude is stepped during t in order to halt phase evolution from spin interactions other than with the applied space-encoding gradient. 

Fig. 6.2.3 [Wehl] Principle of spectroscopic Fourier imaging with slice selection. To separate spatial and spectroscopic responses, the spectroscopic evolution must be constant during the space-encoding period fj. The spectroscopic signal is acquired during the detection time tz in the absence of a gradient.

Fit , 6.3.2 jHoulj Pulse sequence for 2D Fourier imaging in the rotating frame, in these directions can be written as... 

The gradient is static in the sample ftame so that the rotating spin density leads to a time-independent signal. The phase tp is varied in small steps for imaging with reconstruction from projections, or it is switched by multiples of n/2 for Fourier imaging methods. 

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