M R I:The

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M R IThe Meaning of k-space

• Nathan Yanasak Ph.D. (and also Jerry Allison
  Ph.D., Chris Wright B.S.,
• Tom Lavin B.S.),
• Department of Radiology
• Medical College of Georgia

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3 Big Questions for todayHow can we
represent data using the frequency-domain?How
does this apply to an image? How do we use
magnetic gradients to encode positions in
space, and to acquire an image?
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How can we represent data using the
frequency-domain?
K-space is a representation of a function
(i.e., an image, which is a 2D spatial function
of intensities) in the frequency domain.
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Time-domain vs. Frequency-domaintime
secondsfrequency 1/seconds HzIf we can
measure some data during a time interval and draw
a curve to describe the data content, we could
equally describe the curve in units of time or
frequency.
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Time-domain vs. Frequency-domain
A
t1/f
t1(1/f1)
A
f 1/t
f1
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Time-domain vs. Frequency-domainWhat do
these curves represent? Time-domain (more
intuitive) the function takes different values
at different points in time.Frequency-domain
(not intuitive) the function has varying amounts
of spectral components at different
frequencies.Lets try to understand that last
point.
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Frequency-Domain Basics
In 1-D, we can create a complicated wave by
adding simple, periodic waves of different
frequency together.
A
AB
B
In this example, we could keep going to create a
square wave, if we wanted.
C
ABCD
D
ABCDEF
E
F
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Frequency-Domain Basics
This process works in reverse as well we can
decompose a complicated wave into a combination
of simple component waves.
The mathematical process for doing this is known
as a Fourier Decomposition.
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Fourier Transform Basics
Each component is known as a harmonic of the
waveform.
Fundamental
1st Harmonic
These are the spectral components to which I was
referring previously.
2nd Harmonic
3rd Harmonic
4th Harmonic
5th Harmonic
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Fourier Transform Basics
For each different frequency component, we need
to know the amplitude
and the phase, to construct a unique wave.
f
f
f
Amplitude change
Phase change
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Fourier Transform Basics
Heres the representation of this waveform as a
plot of amplitudes and phases
Time-domain representation
f
Frequency-domain representation
t
phase
Fourier transform moves us from one
representation to another and back.
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X-domain vs. 1/X-domainWe arent
just limited in using the Fourier transform to
move between time-domain and frequency-domain.
We could equally use another variable (e.g.,
spatial position x). position
mmwavelength 1/mm
The k in k-space is related to
wavelength.Diagnostic question in the
Time-domain how much distance lies between to
dark spots in an image?Rephrased in the
Frequency-domain what is the wavelength of
primary spectral component (i.e., with a length
equal to the distance between two dark spots on
an image)?
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Fourier Transform Basics
Time-domain representation
Frequency-domain representation
Looks like a measurement over time, but it might
as well be a measurement of intensity as a
function of space.
Looks like a measurement in frequency units, but
it might as well be in wavelengths.
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How does this apply to an MRI image?
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K-space
For 2-D MR image, k-space stores amplitude and
phase information (as a complex number), for each
simple component. This can be used to
reconstruct a very complicated 2-D waveform
(i.e., the image) via Fourier transform ? k is
transformed to x.
Harmonics with long wavelengths? stored near the
middle of k-space
Harmonics with short wavelengths ? stored near
the periphery of k-space
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If k-space is only partially filled, the image
may not be a complete representation of the
anatomy (just like our 1-D example, where we sum
just a few simple waves).
Key Whitefilled Blackunfilled
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How do we acquire data in the frequency-domain
and fill up k-space?
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Imaging using spatial position (not!)

• Even though an MRI image displays spatial
  structure, the MRI technique does NOT image each
  individual voxel (3D pixel) sequentially.

t1
t3
t2
This would actually take a great deal of time, as
we shall see.
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Imaging Using K-space

• Heres an analogy you are a director of a large
  choir, and you are trying to determine how well
  each member sings (or what their best range is).
  But, you dont wish to humble any one singer by
  asking them to sing solo (not like real life).

Unknown range of singers Soprano Alto Tenor Bari
tone Bass
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K-space

• So, instead, you first group the choir into two
  different ensembles and ask them to sing a round
  Row, Row, Row Your Boat. Each member of an
  ensemble sings the same notes, but different
  ensembles come in at different times.

Ensemble 1 Ensemble 2
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K-space

• These two groupings produce a total sound (it may
  clash, it may be beautiful, but you hear one
  sound).

Ensemble 1 Ensemble 2
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K-space

• then, you regroup the choir into different
  ensembles and ask them to sing together
• and again
• and so on

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K-space

• By hearing the ensembular sound from various
  different groupings, you can pick out the
  individuals. Ensembles containing a particularly
  bad or good singer will persistently sound bad or
  good.
• Take all of the heard notes from different
  ensembles ? reduce them to a table to determine
  who is bad and good.
• Choose particular ensembles to help.

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K-space

• This gig is uplets map the analogy to MRI.
• Your receiving coil is ALWAYS listening to ALL
  excited spins during readout (just like the
  choir).

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K-space

• but you prepare the sample (the ensembles)
  differently each time you get ready to listen.
  Spins in different places are grouped up to sing
  either in unison (in-phase) or in dissonance
  (out-of-phase), as per our ensembles. With MRI,
  were listening to the total sound at a given
  time.

t3
t1
t2
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K-space

• The sound adds together to yield a net amplitude
  and net phase. We store the amplitude (like
  decibels) and phase (less intuitive) in a
  table. In 2D imaging, this table is arranged
  in a two-dimensional fashion. We say that the
  data in this table resides in a mathematical
  space called k-space.

So, each value in the table corresponds to a
signal strength and phase generated by a
particular ensembular grouping of spins.
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K-space

• Remember our example ensembles that are singing
  in- or out-of-phase? If we are smart in our
  method for picking the ensembles, their spatial
  distributions form harmonics (albeit spatial
  harmonics). But in 2D, they can extend in any
  direction along a plane.

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K-space
If spins in an excited slice were prepared such
that they precess in these particular ensembular
groupings, the resulting signal could only be
constructed with a unique set of magnetization
amplitudes from each position. Thus, we could
apply a Fourier transform to our k-space table to
determine the transverse magnetization at every
position.
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Phase and Frequency Encoding

• Consider an MRI image composed of 169 voxels
• (13 13 matrix)
• All voxels have the same precessional frequency
  and are all in phase after the slice select
  gradient and RF pulse.

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Phase and Frequency Encoding(continued)
Direction of increasing gradient

• When the X phase encode gradient is on, spins
  in the right column have relatively higher
  precessional frequency and advanced phase. Spins
  in the left column have reduced precessional
  frequency and retarded phase.

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Phase and Frequency Encoding(continued)
Direction of increasing gradient

• Frequency-encoding performs the same trick along
  the other axis.

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