This document discusses k-space and how it relates to MRI image formation. It explains that k-space is a mathematical representation of spatial frequencies, not a real space, and that each point in an MR image is reconstructed based on all points in k-space. It also describes how MRI uses magnetic field gradients to spatially encode the nuclear magnetic resonance signal and fill k-space during the image acquisition process.

1. Topics
• spatial encoding - part 1

2. K-space, the path to MRI.K-space, the path to MRI.
ENTER IF YOU DAREENTER IF YOU DARE

3. 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

4. k-space and the MR Image
x
y
f(x,y)
kx
ky
K-spaceK-space
F(kx,ky)
Image-spaceImage-space

5. 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

6. 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
ℑ
K-space
F(kx,ky)
f x y
N
F k k e
xk yk
kk
x y
j
N
x j
N
yNN
yx
( , ) ( , )=
+






=
−
=
−
∑∑
1
2
2 2
0
1
0
1 π π
image-space

7. 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.

8. Low Spatial Frequency

9. Higher Spatial Frequency

10. low spatial
frequencies
high spatial
frequencies
all
frequencies

11. Spatial Frequencies
• low frequency = contrast
• high frequency = detail
• The most abrupt change occurs at an
edge. Images of edges contain the
highest spatial frequencies.

12. Waves and Frequencies
• simplest wave is a cosine wave
• properties
–frequency (f)
–phase (φ)
–amplitude (A)
f x A f x( ) cos ( )= +2π φ

13. Cosine Waves of
different frequencies
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1

14. Cosine Waves of
different amplitudes
-4
-3
-2
-1
0
1
2
3
4

15. Cosine Waves of
different phases
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1

16. k-space Representation of Waves
image space, f=4 k-space
-128 -96 -64 -32 0 32 64 96 128

17. k-space Representation of Waves
image space, f=16 k-space
-128 -96 -64 -32 0 32 64 96 128

18. k-space Representation of Waves
image space, f=64 k-space
-128 -96 -64 -32 0 32 64 96 128

19. Complex Waveform Synthesis
f4 + 1/2 f16 + 1/4 f32
Complex waveforms can be
synthesized by adding simple
waves together.

20. 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

21. k-space Representation of
Complex Waves
“square” wave
image space k-space
-128 -96 -64 -32 0 32 64 96 128

22. 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

23. 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

24. 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

25. 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

26. 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.

27. RF signal
A/D
conversion
image space
FT
k-space

28. 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.

29. 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.

30. 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.

31. 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

32. 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.