The Fourier transform and the phase-correction algorithm s

Although the Bruker software has of course routines to perform the image reconstruction and phase correction, we decided to write our own programs to execute these tasks on the workstations. The reasons for this were partly because of the poor explanation in the manuals about how the Bruker routines executed the phase correction, but also because of an apparent intermittent error in the phase correction. Sometimes in one experiment the machine would produce nice flat phase images and sometimes a completely undulating phase stripe. In addition, the Bruker System is very slow and the procedure used to estimate the phase parameters takes quite a long time.

The first critical step for image reconstruction when the phase information is required is to find exactly the time domain centre point, as indicated by the arrow in figure 2.24(a). This point corresponds to the time zero and is the first point to be passed to the DFT algorithm. If the centre point is too far away from the first point, the phase image will consist of black and white stripes and it will be very difficult to correct this in the frequency domain. The time domain centre point corresponds to the point where the signal is maximum (the isocentre of the phase encode and readout gradients). Once this is found the next step is to match it with the centre of the image matrix (for example for a 256 by 256 matrix the centre of the time data should be at the coordinates 128, 128), and perform the DFT. A software routine was written to implement this procedure to simplify the FT algorithms.

To perform the DFT the Numerical Recipes (Press et al. 1992) routines

C h apter 2 P rin ciples o f N u clear M agn etic Resonance

Transform (FFT). The algorithm for the image reconstruction is divided into 11 steps:

1- First, the time domain Broker file is converted to the workstation format. Briefly,

the Broker hardware oses a processor word of 3 bytes (24 bits) where the first byte on the right is the most significant and the last on the left the least significant. These data are read byte by byte and conveniently converted to a

word of 4 bytes in the workstations as illustrated in figure 2.33;

(a) least significant byte m ost significant byte A, A, A * B, B, C, Broker 2 4 bits word (3 bytes)

{

0 0 0 0 0 C, C, B, Bz B. B, A Az A, A , 3 2 bits positive _{number (4 bytes)} (b)

1 1 1 1 1 C^ C. Q B, Bz B, B. A, Az A, A , 3 2 bits negative number (4 bytes)

Figure 2.33- a) The 3 bytes w o rd structure used by the NMR Bruker system , b) The 4 bytes w o rd o f the w orkstations structure f o r a p o sitive num ber conversion an d f o r a n egative num ber conversion.

2- After the conversion, two time domain matrixes are stored, (real and imaginary).

The time domain centre point for the real image is found, and by moving the data set, matched with the matrix centre. Both, real and imaginary images must

be moved equally;

3- Split each row of the real and imaginary matrix in the middle, and swap the right

and left blocks (this procedure is required by the Numerical Recipes routines);

4- for each matrix row, combine the real and imaginary samples into one array

producing a matrix as in equation (2.87);

910,0) 3(0,0)

9^1,0) 3(1,0)

9%0,1) 3(0,1)

9^1,1) 3(1,1)

9(0,A1-1) 3(0,A1-1)

9%ljV-l) 3(0jV-l)

9CAT-1,0) 3(AT-1,0) 9%1V-1,1) 3(A^-1,0) ...

5- for each row in equation (2.87) apply the Redfield and Kunz treatment followed

C h apter 2 P rin ciples o f N u clear M agnetic R esonance

6- repeat step 3;

7- the result of the real DFT is a complex number, apply the phase correction,

equation (2.86), as described later in this section;

8- split each column in the middle and swap the top and bottom blocks;

9- apply a normal complex DFT to each column;

10- repeat step 8;

11- apply again the phase correction algorithm;

The phase correction is divided into a zero and a first order phase correction, and both are applied after the row DFT and after the column DFT. The zero and the first

order phase correction values are determined for the centre row (step 7) or column (step

11) of the real and imaginary spectrum images, and then repeated to the rest of the

image.

The zero order phase correction consist of finding the right angle to apply in equation (2.86) to all pixels along the image. The technique implemented is that suggested by Ernst (1969) and uses the integral intensity criterion originally applied to NMR spectroscopy. When correctly phased, the dispersion mode signal has a vanishingly small integral, whereas the absorption mode signal has a non-zero value. The ratio between these intensities integrals gives the zero order phase correction,

0Q = tan"'

^ C ^

I /(C O ) Jco

J R m

do

(2 .88)

The baseline offset can be a problem when computing the integral values. To

avoid this, the average value over the first and last few points of the profile are subtracted before the integration. Also to differentiate the noise from the signal during

integration a threshold value is set equal to three times the noise standard deviation of the same points averaged to the baseline correction.

Once the zero order phase correction has been applied to all pixels according to

equation (2.86), the next step is to fit a linear equation to the phase centre row or column profile. This equation is used to determine the phase correction as a function of pixel position and is applied to all rows or columns along the image.

Chapter 2 Principles o f N uclear M agnetic Resonance

column DFT are determined for an image acquired with no electrical current. These values are then stored in a file and used to correct the phase images acquired with the electric current on. These procedures produce phase images where the phase changes are those produced by the electric current magnetic field.

2.4. The B ru k er spin-echo sequence

In the previous section the spin-echo pulse sequence was described. It is slightly different from the actual spin-echo sequence used in the electrical current measurement

experiment, which is presented here. The Bruker spin-echo sequence is called multi-slice

multi-echo or MSVE. It allows us to acquire a sequence of slices with different echos as its name implies, although for the current experiment just one slice is acquired at a time, because it is necessary to synchronise the electrical current pulses with the NMR pulse sequence (see section 3.1). The time diagram of the Bruker MSVE pulse sequence is given in figure 2.34. 1 1 in m m i n g E AD3 n iS to 5 A D 3 <N 5 1 0 . 2 5 m e Receive ■ | G a t d dl^aee gr;adient ISO; rCTOcusing ; gradient d a ta acc^uisition read gradient

Figure 2.34- The M ulti-slice Multi-echo pulse sequence used in the experiments.

The major difference between the MSVE and the spin-echo sequence shown in

figure 2.19 is the position of the phase encoding gradient. The Bruker system applies the phase encode gradient just after the 90° RF pulse. Because of the phase encoding position, a straight line used to appear in the centre of the NMR image. At the beginning of this project this line was eliminated by averaging two images, but this procedure required a longer period of scanning.

C h apter 2 P rin ciples o f N u clear M agnetic Resonance

imperfections. This is not a perfectly pulse and therefore it will not rotate the selected spins by exactly 90° as would be expected. Instead some spins are left behind and may have their rotation equalised to 90° by the equally imperfect 180° pulse. It may also

happen that some spins may be rotated by exactly 90° by the edges of the 180° RF pulse. As shown in figure 2.34, after of the 90° RF pulse all the spins within the

selected slice, that will contribute to the NMR signal are spatially encoded by the phase

encoding gradient. But after the refocusing RF pulse some spins which were not encoded because they were out of the tomographic plane are brought to the selected slice and start to contribute to the NMR signal. Since these spins are not phase encoded they will have a constant phase and their signal after the first FT will not be modulated. In other

words their spatial position in the phase encoding direction have not been codified. Therefore after the second FT the signal corresponding to these spins appears as a line in the centre of the image where the frequency is equal to zero.

This problem was solved by modifying the spin-echo pulse sequence program. The sequence was modified to acquire every other spin-echo signal with a +90° pulse interspersed with a spin-echo signal acquired with a -90° pulse. To compensate from the signal inversion the receiver gain multiplied every echo from the -90° pulse by -1 and the echo from the +90° pulse by +1. Therefore, at the end all spin-echo signals are the same as that of a normal spin-echo sequence. The pulse and gain changing affects only the signal from the 180° RF pulse. Since the 180° pulse is always the same, the signal

from the spins that were not encoded are also the same, but now they are modulated by the receiver gain. The modulation frequency produced by the receiver gain in the signal from the spins that are not phase encoded occurs at the Nyquist frequency for the second

FT. The resulting effect of this modulation is to bring the artifact line from the centre of the image to the border of the field of view.