Transcript Slides on spatial encoding, part 1

Topics
• spatial encoding - part 1
K-space, the path to MRI.
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What is k-space?
• a mathematical device
• not a real “space” in the patient nor in
the MR scanner
• key to understanding spatial encoding
of MR images
k-space and the MR Image
ky
y
kx
x
f(x,y)
Image-space
F(kx,ky)
K-space
k-space and the MR Image
• each individual point in the MR image
is reconstructed from every point in
the k-space representation of the
image
– like a card shuffling trick: you must have all of
the cards (k-space) to pick the single correct
card from the deck
• all points of k-space must be collected
for a faithful reconstruction of the
image
Discrete Fourier Transform
F(kx,ky) is the 2D discrete Fourier transform of the
image f(x,y)
N 1
N 1
1
f ( x, y )  2   F ( kx, ky )e
N k x 0 k y 0
 2
j
 N
xkx  j 2N yky 
y
x
f(x,y)
image-space
ky

kx
F(kx,ky)
K-space
k-space and the MR Image
• If the image is a 256 x 256 matrix size,
then k-space is also 256 x 256 points.
• The individual points in k-space
represent spatial frequencies in the
image.
• Contrast is represented by low spatial
frequencies; detail is represented by
high spatial frequencies.
Low Spatial Frequency
Higher Spatial Frequency
low spatial
frequencies
high spatial
frequencies
all
frequencies
Spatial Frequencies
• low frequency = contrast
• high frequency = detail
• The most abrupt change occurs at an
edge. Images of edges contain the
highest spatial frequencies.
Waves and Frequencies
• simplest wave is a cosine wave
• properties
– frequency (f)
– phase ()
– amplitude (A)
f ( x )  A cos (2 f x  )
Cosine Waves of
different frequencies
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
Cosine Waves of
different amplitudes
4
3
2
1
0
-1
-2
-3
-4
Cosine Waves of
different phases
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
k-space Representation of Waves
image space, f=4
k-space
-128
-96
-64
-32
0
32
64
96
128
k-space Representation of Waves
image space, f=16
k-space
-128
-96
-64
-32
0
32
64
96
128
k-space Representation of Waves
image space, f=64
k-space
-128
-96
-64
-32
0
32
64
96
128
Complex Waveform Synthesis
f4 + 1/2 f16 + 1/4 f32
Complex waveforms can be
synthesized by adding simple
waves together.
k-space Representation of
Complex Waves
image space
k-space
-128
f4 + 1/2 f16 + 1/4 f32
-96
-64
-32
0
32
64
96
128
k-space Representation of
Complex Waves
image space
k-space
-128
“square” wave
-96
-64
-32
0
32
64
96
128
Reconstruction of square wave
from truncated k-space
image space
k-space
-128
reconstructed waveform
-96
-64
-32
0
32
64
96
128
truncated space (16)
Reconstruction of square wave
from truncated k-space
image space
k-space
-128
reconstructed waveform
-96
-64
-32
0
32
64
96
truncated space (8)
128
Reconstruction of square wave
from truncated k-space
image space
k-space
-128
reconstructed waveform
-96
-64
-32
0
32
64
96
128
truncated space (240)
Properties of k-space
• k-space is symmetrical
• all of the points in k-space must be known
to reconstruct the waveform faithfully
• truncation of k-space results in loss of
detail, particularly for edges
• most important information centered
around the middle of k-space
• k-space is the Fourier representation of the
waveform
MRI and k-space
• The nuclei in an MR experiment
produce a radio signal (wave) that
depends on the strength of the main
magnet and the specific nucleus being
studied (usually H+).
• To reconstruct an MR image we need
to determine the k-space values from
the MR signal.
RF signal
FT
A/D
conversion
image space
k-space
MRI
• Spatial encoding is accomplished by
superimposing gradient fields.
• There are three gradient fields in the
x, y, and z directions.
• Gradients alter the magnetic field
resulting in a change in resonance
frequency or a change in phase.
MRI
• For most clinical MR imagers using
superconducting main magnets, the main
magnetic field is oriented in the z direction.
• Gradient fields are located in the x, y, and
z directions.
MRI
• The three magnetic gradients work together
to encode the NMR signal with spatial
information.
• Remember: the resonance frequency
depends on the magnetic field strength.
Small alterations in the magnetic field by the
gradient coils will change the resonance
frequency.
Gradients
• Consider the example of MR imaging in the
transverse (axial) plane.
Z gradient: slice select
X gradient: frequency encode (readout)
Y gradient: phase encode
Slice Selection
• For axial imaging, slice selection occurs
along the long axis of the magnet.
• Superposition of the slice selection gradient
causes non-resonance of tissues that are
located above and below the plane of
interest.