MRI Image Formation
Karla Miller
FMRIB Physics Group
Image Formation
•
•
•
•
•
•
Gradients and spatial encoding
Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts
MR imaging is based on precession
z
y
x
[courtesy William Overall]
Spins precess at the Larmor rate:
 = g (B0 + DB)
field strength
field offset
Magnetic Gradients
Gradient: Additional magnetic field which varies
over space
– Gradient adds to B0, so field depends on position
– Precessional frequency varies with position!
– “Pulse sequence” modulates size of gradient
High field
B0
Low field
Magnetic Gradients
• Spins at each position sing at different frequency
• RF coil hears all of the spins at once
• Differentiate material at a given position by selectively
listening to that frequency
High field
Fast
precession
Low field
Slow
precession
B0
Simple “imaging” experiment (1D)
increasing
field
Simple “imaging” experiment (1D)
Signal
Fourier transform
“Image”
position
Fourier Transform: determines amount of material at a
given location by selectively “listening” to the
corresponding frequency
2D Imaging via 2D Fourier Transform
1DFT
1D “Image”
1D Signal
2DFT
ky
y
kx
2D Signal
x
2D Image
Analogy: Weather Mapping
2D Fourier Transform
2DFT
ky
kx
Measured signal
(frequency-, or k-space)
y
x
Reconstructed
image
FT can be applied in any number of dimensions
MRI: signal acquired in 2D frequency space (k-space)
(Usually) reconstruct image with 2DFT
Gradients and image acquisition
• Magnetic field gradients encode spatial position in
precession frequency
• Signal is acquired in the frequency domain (k-space)
• To get an image, acquire spatial frequencies along
both x and y
• Image is recovered from k-space data using a Fourier
transform
Image Formation
•
•
•
•
•
•
Gradients and spatial encoding
Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts
Sampling k-space
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
FT
Perfect reconstruction of an object would require
measurement of all locations in k-space
(infinite!)
Data is acquired point-by-point in k-space
(sampling)
Sampling k-space
Dkx
ky
kx
2 kxmax
1. What is the highest frequency we need to
sample in k-space (kmax)?
2. How close should the samples be in k-space
(Dk)?
Frequency spectrum
What is the
maximum
frequency we
need to measure?
FT
Or, what is the
maximum kspace value we
must sample
(kmax)?
-kmax
kmax
Frequency spectrum
Frequency spectrum
Frequency spectrum
Frequency spectrum
Frequency spectrum
Frequency spectrum
Higher frequencies
make the
reconstruction look
more like the original
object!
Large kmax increases
resolution (allows us
to distinguish smaller
features)
2D Extension
increasing
kmax
kymax
kxmax
kxmax
kymax
2 kxmax
kmax determines image resolution
Large kmax means high resolution !
Sampling k-space
Dkx
ky
kx
2 kxmax
1. What is the highest frequency we need to
sample in k-space (kmax)?
2. How close should the samples be in k-space
(Dk)?
Nyquist Sampling Theorem
A given frequency must be sampled at least twice per
cycle in order to reproduce it accurately
1 samp/cyc
Cannot distinguish
between waveforms
2 samp/cyc
Upper waveform is
resolved!
Nyquist Sampling Theorem
increasing field
Insufficient sampling
forces us to interpret
that both samples are
at the same location:
aliasing
Aliasing (ghosting): inability to differentiate between 2 frequencies
makes them appear to be at same location
x
max ive
frequency
Applied FOV
x
max ive
frequency
Aliased image
k-space relations:
FOV and Resolution
Dkx
ky
FOV = 1/Dkx
Dx = 1/(2*kx
kx
max)
2 kxmax
k-space relations:
FOV and Resolution
Dkx
ky
xmax = 1/Dkx
2 kx
max
kx
= 1/Dx
2 kxmax
k-space and image-space are inversely related:
resolution in one domain determines extent in other
k-space
Image
Full-FOV,
high-res
Full sampling
Reduce
kmax
Increase Dk
2DFT
Full-FOV,
low-res:
blurred
Low-FOV,
high-res:
may be
aliased
Image Formation
•
•
•
•
•
•
Gradients and spatial encoding
Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts
Visualizing k-space trajectories
kx(t) = g Gx(t) dt
ky(t) = g Gy(t) dt
k-space location is proportional to accumulated
area under gradient waveforms
Gradients move us along a trajectory through kspace !
Raster-scan (2DFT) Acquisition
Acquire k-space line-by-line (usually called “2DFT”)
Gx causes frequency shift along x: “frequency encode” axis
Gy causes phase shift along y: “phase ecode” axis
Echo-planar Imaging (EPI) Acquisition
Single-shot (snap-shot): acquire all data at once
Many possible trajectories through k-space…
Trajectory considerations
• Longer readout = more image artifacts
– Single-shot (EPI & spiral) warping or blurring
– PR & 2DFT have very short readouts and few artifacts
• Cartesian (2DFT, EPI) vs radial (PR, spiral)
– 2DFT & EPI = ghosting & warping artifacts
– PR & spiral = blurring artifacts
• SNR for N shots with time per shot Tread :
SNR   Ttotal =  N  Tread
Image Formation
•
•
•
•
•
•
Gradients and spatial encoding
Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts
Partial k-space
If object is entirely real, quadrants of k-space
contain redundant information
2
ky
1
c+id
a+ib
aib
cid
3
4
kx
Partial k-space
Idea: just acquire half of k-space and “fill in” missing data
Symmetry isn’t perfect, so must get slightly more than half
1
c+id
ky
a ib
kx
a+ib
c id
measured data
missing data
Multiple approaches
ky
ky
kx
Acquire half of each
frequency encode
kx
Reduced phase
encode steps
Parallel imaging
(SENSE, SMASH, GRAPPA, iPAT, etc)
Surface
coils
Object in
8-channel array
Single coil
sensitivity
Multi-channel coils: Array of RF receive coils
Each coil is sensitive to a subset of the object
Parallel imaging
(SENSE, SMASH, GRAPPA, iPAT, etc)
Surface
coils
Object in
8-channel array
Single coil
sensitivity
Coil sensitivity to encode additional information
Can “leave out” large parts of k-space (more than 1/2!)
Similar uses to partial k-space (faster imaging,
reduced distortion, etc), but can go farther
Image Formation
•
•
•
•
•
•
Gradients and spatial encoding
Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts
Slice Selection
RF
0
frequency
gradient
Gz
excited slice
2D Multi-slice Imaging
excited slice
t1
t2
t3
t4
t5
t6
All slices excited and acquired sequentially (separately)
Most scans acquired this way (including FMRI, DTI)
“True” 3D imaging
excited volume
excited volume
Repeatedly excite all slices simultaneously, k-space
acquisition extended from 2D to 3D
Higher SNR than multi-slice, but may take longer
Typically used in structural scans
Image Formation
•
•
•
•
•
•
Gradients and spatial encoding
Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts
Motion Artifacts
PE
Motion causes inconsistencies between readouts in
multi-shot data (structurals)
Usually looks like replication of object edges along
phase encode direction
Gibbs Ringing (Truncation)
Abruptly truncating signal in k-space introduces
“ringing” to the image
EPI distortion (warping)
field offset
Field map
image distortion
EPI image
(uncorrected)
Magnetization precesses at a different rate than expected
Reconstruction places the signal at the wrong location
EPI unwarping (FUGUE)
field map
uncorrected
corrected
Field map tells us where there are problems
Estimate distortion from field map and remove it
EPI Trajectory Errors
Left-to-right lines offset from right-to-left lines
Many causes: timing errors, eddy currents…
EPI Ghosting
Shifted trajectory is sum of 2 shifted
undersampled trajectories
Causes aliasing (“ghosting”)
To fix: measure shifts with reference
scan, shift back in reconstruction
=
+
undersampled
Image Formation Tutorial
Matlab exercises (self-contained, simple!)
k-space sampling (FOV, resolution)
k-space trajectories
Get file from FMRIB network:
http://www.fmrib.ox.ac.uk/karla/misc/imageform.tar
Instructions in PDF
Go through on your own (or in pairs), we’ll
discuss on Thursday