A generalized approach to parallel magnetic resonance imaging

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A generalized approach to parallel magnetic
resonance imaging

• Daniel Sodickson, Charles McKenzie
• Parallel Imaging Meeting
• 7/13/05

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Outline

• Formation of a generalized theory that allows
  comparison of SENSE and SMASH like methods and
  generation of hybrid techniques that combine the
  advantages of each approach
• Direct inversion
• Image-domain reconstruction
• K-space reconstruction
• Hybrid reconstructions
• Numerical conditioning to practically encode
  matrix inversions
• Coil sensitivity calibrations

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I. Generalized Theory
Sl(kx,ky) ? ? Cl(x,y) exp(- ikxx- ikyy) ?(x,y)
dx dy
Bl(x,y,kx,ky) encoding function
?(x,y) spin density cl(x,y) coil sensitivity of
lth coil (including relaxation induced
variations)
formed by superimposing coil sensitivity
modulation on a gradient modulation
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I. Generalized Theory
Figure 1 -real components -not accounting for
relaxation -each represents different views of
the image, similar to x-ray projections
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I. Generalized Theory
Sl(kx,ky) ? ? Cl(x,y) exp(- ikxx- ikyy) ?(x,y)
dx dy
discretely sampled w/ j pixel index
inverse DFT to get an image from each coil
Or you can group the signals from all coils into
a single index p
So if you know the inverse of the encoding matrix
B, you can calculate your spin density
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I. Generalized Theory
For all pixels
(LNxNy/M x 1)
(NxNy x LNxNy/M)
(NxNy x 1)

• B is overdetermined by a factor of L/M
• inversion of B is time consuming and memory
  intensive
• so for cartesian sampling, an FT can be done
  along the non-coil-encoded directions
• this results in the multiplication of the
  encoding function by a shifted delta function,
  yielding a block diagonal structure which can
  simplify the processing

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I. Generalized Theory
After FT along x
(LNy/M x 1)
(Ny x 1)
(Ny x LNy/M)
Figure 3, top -solidreal components -dashedimagi
ngary -circles indicate one frequency encode(x)
position for which the encoding function applies
Ny
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I. Generalized Theory
Figure 3, bottom -solidreal components -dashedim
agingary -circles indicate one frequency
encode(x) position for which the encoding
function applies
The inversion of B can then be done either
directly, in the image domain, or in k-space
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II. Direct inversion
B
M2
Figure 4a

• Using a generalized inverse procedure such as the
  Moore-Penrose Pseudoinverse where FT is included
  in the inverse

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III. Image based reconstruction
B
M2
Figure 4b

• for the fully encoded case, the additional FT
  would yield a diagonal matrix of shifted delta
  functions (for each component coil) corresponding
  to the coil sensitivities at various locations
• However, undersampling results in multiple
  non-zero values in each row after the FT in the
  phase-encode direction, which causes Nyquist
  aliasing
• Pixel by pixel inversion by an unfolding method
  like SENSE is then required

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IV. K-space reconstruction
B
M2
Figure 4c
Figure 5 -only need to invert/fit a sub-block of
the full matrix and then apply the transform to
the other blocks -reduces dimensionality -major
divergence from SENSE
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V. Hybrid techniques
Figure 6

• Expanding sub-blocks
• Full matrix inversion of max subblock is more
  sensitive to noise and instabilities at higher
  acceleration factors

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V. Hybrid techniques
Figure 8
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VI. Numerical Conditioning

• The generalized inverse of a nonsquare matrix is
  not uniquely defined, so one can choose based on
  desired properties.
• If inverse of a square matrix (BHB) exists,
  standard pseudoinverse can be used
• Or use SVD to define BUDVH and then
• Or use an inverse fitting procedure (as described
  in the Taylored SMASH paper)
• SENSE implements a noise adjusted pseudoinverse
• SVD conditioning reduces noise, but retains
  details from small structures that are removed
  from post-processing sensitivity maps

B-1(BHB)-1BH
B-1VD-1UH
nBT and
nTB-1
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VI. Numerical Conditioning

• Too much overlap between coils makes rows in
  matrix nearly identical, making it more sensitive
  to noise and random errors in the coil
  sensitivities
• Also, results in small eigenvalues which produce
  large weighting factors, degrading SNR
• Thresholding eigenvalues can help prevent this
  noise amplification

SENSE w/out body reference or sensitivity
processing
Unconditioned SENSE with post-processed
sensitivities
Conditioned SENSE using SVD with 10 threshold
Full-encoded Reference
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VII. Coil sensitivity calibration

• Can include an additional term to multiply with
  the encoding matrix ? (s -1 B-1) S, where s
  reflects any relaxation or sequence specific
  effects
• It doesnt matter at what point in the recon this
  multiplication occurs.
• In SENSE it represents the body coil image
  division and polynomial fitting routines
• It is also possible to multiply by a sum of
  squares of the component coils afterwards.
• Since unstable regions in the processed coil
  sensitivity maps generally correspond to regions
  with small eigenvalues in the encoding matrix,
  eigenvalue conditioning will result in automatic
  thresholding and an additional body acquisition
  is no longer needed for SENSE recon.