240 spatial encoding
TRANSCRIPT
Topics
• spatial encoding - part 1
K-space, the path to MRI.K-space, the path to MRI.
ENTER IF YOU DAREENTER IF YOU DARE
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
x
y
f(x,y)
kx
ky
K-spaceK-space
F(kx,ky)
Image-spaceImage-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)
x
y
f(x,y)
kx
ky
K-space
F(kx,ky)
f x yN
F k k exk yk
kkx y
jN
x jN
yNN
yx
( , ) ( , )
12
2 2
0
1
0
1
image-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
allfrequencies
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 f x( ) cos ( ) 2
Cosine Waves ofdifferent frequencies
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Cosine Waves ofdifferent amplitudes
-4
-3
-2
-1
0
1
2
3
4
Cosine Waves ofdifferent 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 besynthesized by adding simplewaves together.
k-space Representation of Complex Waves
f4 + 1/2 f16 + 1/4 f32
-128 -96 -64 -32 0 32 64 96 128
image space k-space
k-space Representation of Complex Waves
“square” wave
image space k-space
-128 -96 -64 -32 0 32 64 96 128
Reconstruction of square wave from truncated k-space
truncated space (16)
image space k-space
-128 -96 -64 -32 0 32 64 96 128
reconstructed waveform
Reconstruction of square wave from truncated k-space
truncated space (8)
image space k-space
-128 -96 -64 -32 0 32 64 96 128
reconstructed waveform
Reconstruction of square wave from truncated k-space
truncated space (240)
image space k-space
-128 -96 -64 -32 0 32 64 96 128
reconstructed waveform
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
A/Dconversion
image space
FT
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.