B Lineintegral projection reconstruction

i. Basic principles. Projection reconstruction using 2-D and 3-D image reconstruction algorithms is well known, especially in the areas of X-ray CT and radionu-clide emission tomography, as previously discussed. Although the image can be reconstructed in several different ways, the basic forms of data collection are similar. Lineintegral projection data are obtained in angular steps, by rotating the object a total of either i80° or 360°. The most familiar and convenient way to reconstruct 2-D or 3-D images is through the Fourier convolution method, which can y be summarized as follows. The reconstructed 2-D image f (x, y) is given by f (x, y) =

where p$(x') is projection data, h(x') is the filter kernel that corrects 1 /r blurring caused by circular symmetric linear superposition, and (xy') is the coordinate system rotated by an angle $ from the original coordinates (x, y).

In Fourier transform NMR, the nuclear signal can be considered to be the inverse Fourier transform of the spatial domain spin density function. If a plane at z = z0 is selected, then the FID at an angular view $ can be expressed as

s$ (t ) = Mo jj f (x y '; zo) exp[i y x ' Gx < t ] dy' dx'.

Although the FID signal s$ (t ) appears in the time domain, it represents the Fourier domain projection data. Therefore, the projection data p$ (x') are obtained through the Fourier transform of the FID signal as

The basic form of projection data obtainable in Fourier transform NMR is similar to the data obtained in X-ray CT. In Fig. 21, spin-echo signals or FIDs are obtained at different angular views through the application of the field gradient and RF excitation sequences. As a first step, all

FIGURE 21 FID, RF pulse timing, and field gradient sequences for the basic single-slice line-integral projection reconstruction. Gradient pulse sequence: Gx(0) = Gcos0; Gy(0) = Gsin0;0° < 0 < 180° or 0° < 0 < 360°, where G is the maximum value of the reading gradient.

the spins in the sample are excited with a 90° RF pulse, and a subsequent 180° RF selects the slice. After spins are refocused they generate the spin-echo or FID signal (Fig. 21). After 180° or 360° rotation of projection with an appropriate step through the adjustment of the field gradients Gx and G y, a complete projection data set sufficient for reconstruction of a slice at a given plane zo is obtained. At this point, 2-D image reconstruction can proceed according to Eq. (54); that is, each echo or FID signal s0 (t) is Fourier-transformed, convolved with a filter kernel, and backprojected.

ii. Slice (plane)-encoded multislice LPR. The single-slice line-integral projection technique explained earlier can be extended to achieve multislice imaging through several encoding techniques, for example, the plane-encoding technique explained in the following paragraphs.

Let us assume that the number of planes is n. For the data set at a view 0\, the same Gxy and Gz are applied n times, each with a different frequency composition of RF pulses. The RF pulses are specially tailored to assign desired phases to the designated slices. To obtain a complete set of view data corresponding to the n planes, the acquisition of data is repeated n times with differently composed RF pulses.

The key to this method lies in the encoding of signals according to the RF pulse sequence. A simple illustration of the encoding procedure using a coding matrix is as follows. Let the FIDs obtained at each 180° composite RF pulse sequence be S0o(t), S0o(t),..., S0o (t). Each FID is a composite of the line-integral projection sets, which include data from several planes at an angular view 00, that is, s00z0(t), s00z1(t), and so on. Therefore, composite FIDs, Si(t), S?0(t),..., SUt) can be given as

From Eq. (57), the desired FID signal s$iZi, which corresponds to the FID of slice zi , can be obtained through matrix inversion.

Examples of coding matrices include the Hadamard matrix and the Fourier matrix. The advantage of this method is the statistical improvement gained as a result of the increase in total scanning time.

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