Fourier transform imaging

MR imaging technique using the Fourier transformation FT method to reconstruct the images. The most frequently used techniques are two dimensional Fourier transform 2DFT imaging and three dimensional Fourier transform 3DFT imaging: reconstruction of image data involves two or three Fourier transforms. In MR imaging, data for one spatial dimension is obtained by frequency encoding while additional dimensions are obtained using phase encoding. In both processes data are acquired in the Fourier transform space of the image space, hence Fourier transformation is needed to obtain the standard image data

The first dimension of an image plane is obtained with frequency encoding (see frequency encoding (I), Fig. 1). As can be seen in the figure, each pixel column in the schematic drawing is represented by a different frequency according to the Larmor equation. The signal coming from the patient during the readout process, a time dependent radiofrequency burst f(t) is measured and then Fourier transformed to yield a frequency spectrum A(w ). The signal intensities A at the various frequencies originate from different image columns.

The second dimension in a plane can be obtained in several ways. Infrequently, one uses Fourier Zeugmatography. The standard method to measure data in the second dimension in MR imaging is by phase encoding. In this method, a gradient G_n perpendicular to the frequency encoding direction (see phase-encoding (I), Fig. 1) is increased after the repetition time TR in standard sequences and after each gradient echo GRE in echo planar imaging EPI . Hence, the phase angle Df of the spins in each column is systematically increased after each repetition and dependent on their position. Relative to the centre columns, spins in columns towards the left lag behind, while the ones towards the right precess faster. If the projection of the spin vectors in each column is plotted as a function of repetition time and therefore as a function of the systematical increase of the gradient, this projection traces a sine wave (trigonometric functions). The farther away from the centre the spins are located, the higher is the frequency of this sine wave. Hence, the second spatial dimension has been encoded by assigning to each column a different frequency at which the phase is turning around. This process can be applied in an additional dimension, resulting in one frequency encoded and two phase encoded dimensions to yield 3DFT.

Hence, the spatial localization in Fourier imaging is achieved as follows. During the readout process, the signals are frequency encoded in one spatial dimension to yield a signal f_I(t) , which upon a first Fourier transformation yields A_I(ð). A_I(ð) differs e.g. for each pixel column. By incrementing a phase encoding gradient systematically in the other spatial dimension, the readout process is repeated as many times as necessary to acquire enough data to assign an unambiguous number to each pixel in the image matrix. For each column we therefore have n (e.g. 256) values of A_i(ð) . These data are Fourier transformed a second time, yielding amplitudes D(ð,i) at the various phase angle frequencies, which were encoded for each pixel row. The amplitudes D correspond to the signal intensities for each pixel in the image and spatial localization of the signal in two dimensions has been accomplished. Obviously, frequency and phase encoding directions can be interchanged and are preferrably chosen to minimize backfolding artefacts and ghost artefacts. In 3DFT the process described is repeated with additional phase encoding by applying a systematically varying gradient field in the third spatial dimension. The number of repetitions needed depends on the in-plane or volume dimensions and the desired spatial resolution.

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