1. MRI Image Formation
Karla Miller
FMRIB Physics Group

2. Image Formation Gradients and spatial encoding Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts

3. MR imaging is based on precessionz y x
[courtesy William Overall]
Spins precess at the Larmor rate:
= g (B0 + DB)
field strength
field offset

4. 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 B0
High field
Low field
Make color gradient on arrows & fast->slow precession?

5. 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 B0
Fast
precession
High field
Low field
Make color gradient on arrows & fast->slow precession?
Slow
precession

6. Simple “imaging” experiment (1D)
increasing
field

7. 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

8. 2D Imaging via 2D Fourier Transform
1D Signal
1D “Image”
1DFT
2D Signal kx ky
2D Image x y
2DFT

9. Analogy: Weather Mapping

10. (frequency-, or k-space)
2D Fourier Transform
2DFT kx ky x y
Measured signal
(frequency-, or k-space)
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

11. 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

12. Image Formation Gradients and spatial encoding Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts

13. Sampling k-space
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)

14. Sampling k-space kx ky
kx
2 kxmax
What is the highest frequency we need to sample in k-space (kmax)?
How close should the samples be in k-space (Dk)?

15. Frequency spectrum What is the maximum frequency we need to measure?
Or, what is the maximum k-space value we must sample (kmax)? FT
-kmax
kmax

16. Frequency spectrum

17. Frequency spectrum

18. Frequency spectrum

19. Frequency spectrum

20. Frequency spectrum

21. Frequency spectrum
Higher frequencies make the reconstruction look more like the original object!
Large kmax increases resolution (allows us to distinguish smaller features)

22. 2D Extension increasing kmax kymax kxmax kxmax kymax 2 kxmax
kmax determines image resolution
Large kmax means high resolution !

23. Sampling k-space kx ky
kx
2 kxmax
What is the highest frequency we need to sample in k-space (kmax)?
How close should the samples be in k-space (Dk)?

24. Nyquist Sampling Theorem
A given frequency must be sampled at least twice per cycle in order to reproduce it accurately
1 samp/cyc
2 samp/cyc
Cannot distinguish
between waveforms
Upper waveform is
resolved!

25. Nyquist Sampling Theorem
increasing field
Insufficient sampling
forces us to interpret
that both samples are
at the same location:
aliasing

26. Aliasing (ghosting): inability to differentiate between 2 frequencies makes them appear to be at same location x x
Aliased image
Applied FOV
max ive
frequency
max ive
frequency

27. k-space relations: FOV and Resolutionkx ky
kx
2 kxmax
x = 1/(2*kxmax)
FOV = 1/kx

28. k-space relations: FOV and Resolutionkx ky
kx
2 kxmax
2 kxmax = 1/x
xmax = 1/kx
k-space and image-space are inversely related: resolution in one domain determines extent in other

29. k-space Image Full-FOV, Full sampling high-res 2DFT Full-FOV, low-res:
blurred
Reduce kmax
Low-FOV,
high-res:
may be aliased
Increase k

30. Image Formation Gradients and spatial encoding Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts

31. Visualizing k-space trajectories
kx(t) =  Gx(t) dt
ky(t) =  Gy(t) dt
k-space location is proportional to accumulated area under gradient waveforms
Gradients move us along a trajectory through k-space !

32. 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

33. Echo-planar Imaging (EPI) Acquisition
Single-shot (snap-shot): acquire all data at once

34. Many possible trajectories through k-space…

35. 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

36. Image Formation Gradients and spatial encoding Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts

37. Partial k-space
If object is entirely real, quadrants of k-space contain redundant information 2 1
c+id
a+ib
aib
cid ky 3 4 kx

38. Partial k-space c+id a+ib aib cid
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
a+ib
measured data
aib
cid ky
missing data kx

39. Multiple approaches Acquire half of each frequency encodekx ky kx ky
Acquire half of each
frequency encode
Reduced phase
encode steps

40. 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

41. 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

42. Image Formation Gradients and spatial encoding Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts

43. Slice Selection RF
frequency 0
gradient Gz
excited slice

44. 2D Multi-slice Imaging t1 t2 t3 t4 t5 t6
excited slice
All slices excited and acquired sequentially (separately)
Most scans acquired this way (including FMRI, DTI)

45. “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

46. Image Formation Gradients and spatial encoding Sampling k-space
Trajectories and acquisition strategies
Fast imaging
Acquiring multiple slices
Image reconstruction and artifacts

47. Motion Artifacts PE
Motion causes inconsistencies between readouts in multi-shot data (structurals)
Usually looks like replication of object edges along phase encode direction

48. Gibbs Ringing (Truncation)
Abruptly truncating signal in k-space introduces “ringing” to the image

49. EPI distortion (warping)
field offset
image distortion
Field map
EPI image
(uncorrected)
Magnetization precesses at a different rate than expected
Reconstruction places the signal at the wrong location

50. EPI unwarping (FUGUE) Field map tells us where there are problems
uncorrected
corrected
Field map tells us where there are problems
Estimate distortion from field map and remove it

51. EPI Trajectory Errors
Left-to-right lines offset from right-to-left lines
Many causes: timing errors, eddy currents…

52. 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

53. Image Formation Tutorial
Matlab exercises (self-contained, simple!)
k-space sampling (FOV, resolution)
k-space trajectories
Get file from FMRIB network:
Instructions in PDF
Go through on your own (or in pairs), we’ll discuss on Thursday