There is an excellent guide to MRI at mriquestions.com. Most images in these notes come from that site, courtesy of Allen D. Elster, MRIquestions.com.

The goal is to isolate signals from a 2D slice perpendicular to B0, which is in the $z$ direction.

Recall from the notes on gradients that we can vary the strength of $B_0$ along any axis. The direction of $B_0$ does not change ... just its strength.

$B_0$ can be made to vary linearly along the $z$ axis such that the strength at position $z$ is $$B_0 + z\ G_z$$

where constant $G_z$ is the gradient of the field strength. The image below shows this.

Above, note that the gradient, $z\ G_z$, is zero at at $z=0$. This is the centre of the MRI bore and $z$ coordinates extend in both directions from the centre.

The changing field strength causes a corresponding change in precession frequency along the $z$ direction. Recall also from the notes on precession that the frequency of precession is proportional to the strength of the field. For a simple $B_0$ field, $$\omega_0 = 2 \pi\ \gamma\ B_0$$

But, under the influence of $G_z$, the precision frequency of proton at position $z$ will be $$\omega = 2 \pi\ \gamma\ (B_0 + z\ G_z)$$

For example, one end could precess at 64,000,000 Hz and the other end at 64,009,000 Hz. With 100 slices in that direction, each slice would have protons that precess in a 90 Hz range:

The transverse $B_1$ pulse is used rotate the protons $90^o$ into the transverse plane. But, for this to happen, the pulse frequency must match the proton precession frequency.

So a $B_1$ pulse can be made to contain only those frequencies present in one slice along the $z$ axis. In the frequency domain, such a signal would look like

and the corresponding signal in the time domain would be a truncated $\textrm{sinc}(t) = {\sin(t) \over t}$ waveform, as shown below (this is explainable with a Fourier Transform, but is not discussed here):

So the transverse $B_1$ pulse can be made within one of these ranges, so the only protons within that range of precession frequencies will resonate. That is, only protons within a particular slice will resonate, so protons from other slices will produce no signal.

The $G_z$ gradient is called the "slice gradient".

Note 1: Since the RF pulse has a limited duration (2-4ms), a perfect sinc, which has an infinite width, is not possible and a somewhat different pulse is used in reality.

Note 2: The range of frequencies induced by the gradient in the z direction will cause dephasing of the precession, so a short negative gradient in z must subsequently be added to bring the precessions in phase again.

The MR machine has a read coil which measures the oscillating magnetic field in the transverse ($xy$) plane that has been been selected in Step 1. The coil can determine both the magnitude and phase of the signal at any time.

However, there's only one signal coming from the selected slice. That signal is the sum of all the individual signals of all the individual spinning protons. So the read coil cannot localize a signal within the slice.

However, there's a nice trick that can be done:

After the $B_1$ pulse, a gradient, $G_{xy}$, can be applied in some direction within the slice. This direction is perpendicular to $z$. The gradient is really the sum of a gradient, $G_x$ in the $x$ direction and another gradient, $G_y$, in the $y$ direction. The vector sum gives $$G_{xy} = G_x + G_y$$

$G_{xy}$ is called the read gradient and is applied while the read coil is measuring the signal.

With the read gradient applied, the precession frequency of protons in the transverse plane will vary linearly with distance along $G_{xy}$.

The video below shows precession in the transverse plane. A blue arrow, which appears, indicates the direction and magnitude of the gradient, $G_{xy}$.

Some notes about the video:

• The precession frequency varies linearly with distance from (0,0) at the centre of the image.
• All the precessions of a particular frequency are on a line perpendicular to $G_{xy}$.
• $G_{xy}$ adds to $B_0$ in $G_{xy}$'s positive direction from (0,0) and subtracts from $B_0$ in $G_{xy}$'s negative direction from (0,0), as shown in the graph above.

  In the video above, if $G_{xy}$ is large enough, its is possible for it to overwhelm $B_0$ in the negative direction and cause the spins to reverse. However, this does not happen in an MRI machine because the magnitude of $G_{xy}$ is much, much smaller than that of $B_0$ and only changes the frequencies slightly.

• When the gradient becomes zero, the only remaining field is that of $B_0$, which has the same strength everywhere. So the precession frequency reverts to its original frequency everywhere when the gradient is turned off. But the precessions are now out of phase.

Consider a $G_{xy}$ that points only upward (i.e. in the positive $y$ direction). This gradient will cause higher-frequency precession on the top and lower-frequency precession on the bottom, as seen at the beginning of the video above.

Here's the interesting bit:

The read coil can only read the sum of all the precessions, which appears as a time-varying signal.

However ... the Fourier transform can convert that time-varying signal into a spectrum of amplitudes at different frequences, as we've seen before:

Each frequency corresponds to a particular distance along the direction of the read gradient, $G_{xy}$ (e.g. one row of the image when $G_{xy}$ points straight upward). So the amplitude at a particular frequency is the sum of all precessions at that distance in the direction of $G_{xy}$, which form a line perpendicular to $G_{xy}$ (as seen in the video).

In other words, the Fourier transform gives us, at each distance (corresponding to a frequency) the projection of the signal at that distance.

In a CT sinogram, one line of the sinogram is the projection of the densities in a particular direction.

Similarly, in MRI under a read gradient, the amplitudes at each frequency make one lines of an "MRI sinogram".

Given the projections of reponses along a number of read directions, filtered backprojection can be used to reconstruct the response at individual pixels within the slice.

But filtered backprojection is no longer used in MRI reconstruction. Instead, various "phase encoding" methods are used. These are discussed in the next lectures.