Unaliasing by Fourier Encoding the Overlaps Using the Temporal Dimension UNFOLD for Parallel Imaging

1
Unaliasing by Fourier Encoding the Overlaps Using
the Temporal Dimension (UNFOLD) for Parallel
Imaging

• Esin Ozturk
• Parallel Imaging Seminar
• 07/05/2005

2
Principles of UNFOLD

• UNaliasing by Fourier-encoding the Overlaps using
  the temporaL Dimension (UNFOLD) is a flexible
  way of encoding spatiotemporal information with
  MRI.
• UNFOLD reduces the dynamic (k,t) FOV
• Reduces the total amount of spatial information
  acquired along k axes
• Reduced FOV ? Aliasing ? Overlap of spatially
  distinct points
• UNFOLD uses time to label the overlapped
  components such that a Fourier transform through
  time can resolve them.

3
Aliasing
Regular sampling
Reduced sampling
4
Discrete Fourier Transform Principles

• DFT
• DFT Shift Theorem

5
How is time used to label aliased spatial
components?

• Given the object O(x) and the sampling function
  S(k), its image I(x) is equal to,
• I(x) FT (S(k)) O(x) PSF(x) O(x) 1
• where PSF(x) FT(S(k)).
• UNFOLD involves shifting the sampling function in
  the phase encode direction.
• A shift of S(k) by a fraction f of a line results
  in a linear phase shift applied to PSF(r)
  altering the phase of all but center peak.
• Through the convolution (Eq 1), the phase of
  the PSF peak is passed to the corresponding
  replica of the object.
• P0 unchanged, P1 phase shifted by an angle 2pf
• ? Aliased P0 P0P1ei2pf

6
Aliasing with a phase shift
Shifted sampling function
7
Time labeling (cont)

• In a dynamic study, the shift in the sampling
  function can be varied from image to image.
• This time varying shift can be used to label
  and then resolve the various components that are
  overlapped, by modulating their phase as a
  function of time in a controlled way.

8
Example

• Fig 4 of UNFOLD Paper

• Assume a time series of images are acquired.
• If the phase of the odd images are shifted
• by 0.5 lines,
• ei2pf eip -1
• The value of an image point oscillates between,
• P0P1 for even images
• P0-P1 for odd images
• A Fourier transform through time then shows
• two peaks.
• Since P0 and P1 are seperated, the time
  dependence of one point can be obtained by
  filtering out the spectrum associated with the
  other point applying a FT to the result.

9
Dynamic object
If the object is dynamic, there will be a range
of frequencies Instead of delta functions in the
temporal frequency response.
10
FOV requirements

• More dynamic points have wider range of spectrum.
  
• If the FOV is kept larger than the size of the
  dynamic region, it is possible to make sure that
  two dynamic regions wont overlap.
• This overlap would cause difficulty in separating
  the two points.

11
Generalized UNFOLD

• If there are n overlapped points due to aliasing,
  UNFOLD can resolve them using n differently
  shifted k-space patterns.
• If the k-space sampling is not Cartesian, still
  k-space is partially covered so that a full
  k-space matrix is covered in n time frames. The
  sampling functions of n frames are used to
  generate aliased n images to from the time
  series.
• Temporal frequency of a location is generated,
  and the peak at the DC is filtered to generate
  the original image.

12
Example- Cardiac Imaging
a) Original image, regions A and B are shown b)
One aliased image from the time series, by
acquiring odd k lines for odd images, and even k
lines for even images c) UNFOLDed image d)
Difference image (c-a)
13
Example of spiral UNFOLD

• Original image -
• 6 spiral interleaves
• b) Interleaves reduced by 2
• c) Interleaves reduced by 3
• d) Interleaves reduced by 6
• Original image
• Temporal frequency
• UNFOLDed image

Aliasing does not come from a single point, but
it is guaranteed to be zero at DC with proper
choice of reduction factor and k-space shift.
14
Discussion

• UNFOLD reduces the total scan time necessary for
  a given temporal frame.
• Decrease in total scan time, better spatial or
  temporal resolution, or larger spatial coverage
  or double TR
• UNFOLD assumes the signal in a given location
  varies slowly with time.
• UNFOLD assumes more than one spatial location can
  share the same temporal bandwidth without
  overlap.
• UNFOLD assumes enough is known about the shape of
  the temporal spectra to separate the contributing
  points.

15
References

• Original UNFOLD paper
• Madore et al. Magn Reson Med. 1999
  Nov42(5)813-28
• UNFOLD for Parallel and Partial Fourier Imaging
• Madore B. Magn Reson Med. 2002 Sep48(3)493-501
  
• Adaptive sensitivity encoding incorporating
  temporal filtering (TSENSE).
• Kellman et al. Magn Reson Med. 2001
  May45(5)846-52.
• UNFOLD-SENSE a parallel MRI method with
  self-calibration and artifact suppression.Madore
  B. Magn Reson Med. 2004 Aug52(2)310-20.