storing data in k-space

what the Fourier

transform does

spatial encoding

k-space examples

we will review:

How K-Space Works

This is covered in the What is K-Space? tutorial. Knowledge of what kind of data we are collecting is essential.

Don’t worry, no maths!

We’ll then draw on these elements of understanding to explain spatial encoding.

© D M Higgins

The Fourier transform allows us to determine the constituent waves of any signal

FT

The signals that we measure in MRI are a combination of signals from all over the object being imaged.

It so happens that any signal (even if you simply make one up and draw a squiggle) is composed of a series of sine waves, each with an individual frequency and amplitude. The Fourier transform takes the signal (or squiggle!) and works out what those constituent frequencies and amplitudes are. (That is to say, it converts the signal from the time domain into the frequency domain.) This signal

is made up of this lot

The Fourier transform can tell us what these

signals are (frequency and

amplitude).

In MRI, we spatially encode the signal with magnetic field gradients which make frequency and (rate of change of) phase relate to position. Since the Fourier transform can separate out the frequencies in an MR signal, we can work out where to plot the correct signal intensities, to create the image. You’ll see this in action later on.

© D M Higgins

choir analogycombined:

image / object

radiofrequency signals

ph

ase enco

din

g

frequency encoding

frequency encodinga phase encoding step

Spatial Encoding

Remember that each signal we measure in MRI is a mixture of the signals from every pixel / voxel in the sample. The challenge we face is to separate out the signal from these voxels, so that the magnitude of the signal emitted from that point may be used to plot a grey-scale pixel value into the image. We will see how frequency encoding allows us, by the Fourier transform, to determine which column the signals came from. (But what about the rows? That is where phase encoding comes in.)

Here we have a 6-pixel image. Note that the most signal is coming from the upper middle pixel.

In the choir analogy, the upper middle choir member sings the loudest. (Click the speaker icons.)

When we record an MR signal, we can only record the whole choir. We have no idea who was singing at what volume, without spatial encoding.

Here the signals coming from each pixel are illustrated (or notes from each choir member). Notice the higher amplitude of the upper middle pixel. Also notice that without any encoding, the only difference between the signals is their amplitude.

If we frequency encode the signals, the only change is that each column now has a unique frequency. The amplitudes of the signals have not changed. The upper middle pixel is still the loudest. Click the choir members speaker icons now. Click also the combined sound.

It’s as if the choir members sing the same thing, for the same duration and at the same volume, but some now have changed their pitch. We can tell something has changed when we hear their combined sound. Although we may not be able to deduce which column of choir members are singing which pitch, the Fourier transform can. So we use it to divide up the total sound of the choir into the component parts which came from the separate columns. [Note that we still need to determine the rows, so we can tell that the middle upper pixel, for example, is emitting a stronger signal than the rest, and plot a bright pixel at that point.]

Then, one phase encoding step is shown. Because there are only two rows of pixels in this simple image, we need only two phase encoding steps. One, with no phase encoding at all, and a second with a maximum phase difference (180°). If there were more pixels in the phase encoding direction there would be more phase encoding steps in between. Let’s see what this achieves.

© D M Higgins

radiofrequency signals

image / object

ph

ase enco

din

g

frequency encoding

FT

FT

frequency encodinga phase encoding step

A C E

B D F

A+B

C+DE+F

E-F

C-D

A-B

We could work out pixel C like this:

(C+D) + (C-D) = 2C

[…but we don’t…]

No phaseencoding

180° phaseshift

A+B+C+D+E+F

A+B+C+D+E+F

freq.

freq.

With no phase encoding we get a signal which is the sum of all the signals

in the object.

The FT separates out the total signal at each frequency that was

introduced by frequency encoding.

…the totals for each frequency are different.

Since a 180° phase shift is introduced, when we measure another MR

signal…

Here, for two rows of pixels and one phase encoding step, the simultaneous equations used to work out the amplitude for pixels C and D seem fairly straightforward. But what about when there are, say, 256 phase encoding steps, for 256 pixels in the phase encoding direction? A second Fourier transform is used. Hopefully this slide should convince you that it is at least possible to work out individual pixel values using phase encoding. [Remember, there are many phase encoding steps in a real image with many more pixels.] We will see why a second Fourier transform is the ideal tool when multiple phase encoding steps have been used to encode the image.

© D M Higgins

Q. Why not just frequency encode in both directions?

A. It doesn’t work!

0 0 0

0 0 0

0 0 0

- frequency encode +

- fr

equ

ency

en

cod

e +

there are always two or more (or many!)

pixels with the same frequency

0 +1 +2

-1 0 +1

-2 -1 0

-1 0 +1

-1 0 +1

-1 0 +1

Instead, we introduce different amounts of

phase change over many steps

Frequency encoding has been performed in one

direction. A nominal change in frequency has

been indicated.

Now we have frequency encoded in the other

direction. [+1 top row, -1 bottom row.] What do you

notice?

In other words, since the Fourier transform can only distinguish the total signal of any particular frequency, we

won’t be able to assign the correct signal intensities to unique pixels!

© D M Higgins

Recall: phase change

ph

ase enco

din

g

frequency encoding

phase record

rate of change of phase

= frequency!We create the MR image with two Fourier transforms, one in each direction.

phase change

each signal is from the

whole knee

For the next few moments, we are going to consider only the phase changes that we introduce at the top of the knee. There are

different phase changes going on in the other parts of the knee, but we shall ignore them for

the moment. However, do remember that whenever we acquire an MR signal it has

come as a mixture of all the signals from the whole knee, not just the top of the knee that

we are currently considering.

We shall start with a large yellow arrow in this direction, indicating a large positive phase

change compared with no phase encoding. [Phase changes are introduced by changing the frequencies in

this direction for a short period of time, so that after this has occurred the signal has either got ahead or is lagging behind.]

We will record the phase change we used at the top of the knee for each

signal, here.

We reduce the phase change slightly at the top of the knee, record the MR signal from the whole knee, and store

this data in the next k-space line.

Again we reduce the phase change slightly at the top of the knee, record the MR signal from the whole knee,

and store this data in the next k-space line.

and againand so onWe started with a large positive phase

change, and ended up with a large negative phase change. Over all the

phase encoding steps, we have a rate of change of phase.

We all know intuitively that velocity is the same as the rate-of-change of distance [v = dx/dt]. Your car may travel at 70 miles-per-hour, for example. Acceleration is the same as the rate-of-change of velocity [a = dv/dt]. Similarly, frequency is simply the rate-of-

change of phase!

We made a phase record of what happened at the top of the knee only. In fact, a different rate-of-change of phase has been introduced

for every level in the knee (rows in the image).

Large +ve to large -ve at the top

Medium +ve to medium -ve here

No change here

Medium -ve to medium +ve here

Large -ve to large +ve at the bottomBecause the Fourier transform can separate out different frequencies (even ones that are made from a rate-of-change of phase over many phase encoding steps), we now have a way of determining the total signal from each of the

rows, just like we did with the Fourier transform for the columns, earlier.

© D M Higgins

1DFT

final image

1DFT1DFT

1DFT

This is it’s k-space.

Performing the Fourier transform in one direction sorts out the columns. But

the data in the other direction is still encoded.

The order of Fourier transforms does not matter

© D M Higgins

final image

1DFT1DFT1DFT

1DFT

© D M Higgins

Notice how the Fourier transform has already

deduced that these columns do not contain much signal.

However, the row information is still encoded.

You might ask: Why use Fourier transforms? I thought the images were made from the superposition of many spatial frequencies as shown in the first k-space tutorial! We could superimpose all the spatial frequencies from k-space, but for a 256*256 image, this would mean 65536 arrays of 256*256 pixel values to cope with! That’s a lot of calculation. The Fourier transform is a short cut to the correct image data.