Faster Imaging with Randomly Perturbed, Under sampled Spirals and L1 Reconstruction

1
Faster Imaging with Randomly Perturbed, Under
sampled Spirals and L1 Reconstruction

• Michael Lustig1, Jin-Hyung Lee1, David Donoho2
  and John Pauly1
• 1Electrical Engineering Department, Stanford
  University
• 2Statistics Department, Stanford University
• ISMRM 05

2
Introduction
101000110100
Lossless compression
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Compressed Sensing
100110100110100010011101010100110100
101000110100
Lossless or visually lossless compression
4
Compressible signals

• Compressibility - representation by a few
  coefficients of
  a known transform.

Wavelet transform
Finite Differences
5
A Surprising Experiment
Randomly throw away 83 of samples
FT
?
E.J. Candes, J. Romberg and T. Tao.
6
A Surprising Result
Minimum - norm conventional linear reconstruction
E.J. Candes, J. Romberg and T. Tao.
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A Surprising Result
Minimum - norm conventional linear reconstruction
Min. Total Variation (TV)A convex non-linear
reconstruction
E.J. Candes, J. Romberg and T. Tao.
8
Compressed Sensing

• This work is inspired by
• E.J. Candes, J. Romberg and T. Tao. Robust
  Uncertainty Principles Exact Signal
  Reconstruction from Highly Incomplete Frequency
  Information.http//www.math.ucle.edu/tau/reprint
  s/Exact4.pdf.
• D. Donoho Compressed Sensing http//wwwstat.sta
  nford.edu/donoho/Reports/2004/CompressedSensing091
  604.pdf

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Compressed Sensing

• Basic idea
• Compressible signals can be accurately recovered
  from highly under sampled random Fourier
  coefficients.
• Recovery by solving a non-linear convex
  optimization problem.

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Random k-space sampling

• Random k-space sampling is highly inefficient in
  MR.

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Random k-space sampling

• Random k-space sampling is highly inefficient in
  MR.
• However,
• Spirals (uniform and variable density) are,
• Fast
• HW efficient.
• Irregular sampling pattern.

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Randomly Perturbed Spirals

• Introduce randomness by,
• deviating from analytic spiral.
• Perturbing angle between interleaves.

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Reconstruction Formulation

• minimize ?(m)1
• s.t. Fm-y2 ? ?
• m image
• F Perturbed Spirals Fourier operator
• y k-space measurements
• ? - compression transform

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Reconstruction Formulation

• minimize ?(m)1
• s.t. Fm-y2 lt ?
• m image
• F Perturbed Spirals Fourier operator
• y k-space measurements
• ? - compression transform

Spiral Fourier Transform
Enforces Data Consistency
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Reconstruction Formulation
Compressibility

• minimize ?(m)1
• s.t. Fm-y2 lt ?
• x1?xi - Crucial for the reconstruction!
• m image
• ? - compression transform

transform
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Solving the Optimization

• minimize ?(m)1
• s.t. Fm-y2 lt ?
• Quadratic Program
• Weighted LS
• Non-Linear CG
• Primal-Dual Interior Point Method.
• Min-max nuFFT engine.

Fessler, et al IEEE TSP 200351560-574
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Phantom Simulation

• Analytic phantom
• Perturbed spirals 15/34 itlv

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Phantom Simulation

• Analytic phantom
• Perturbed spirals 15/34 itlv

Finite differences (Total variation)
Least-Norm
Gridding
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Phantom Simulation

• Analytic phantom
• Perturbed spirals 15/34 itlv

Randomness is GOOD!
uniform under-sampled spiral
Perturbed under- sampled spiral
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Phantom Scans
Gridding
Least-Norm
Finite differences (Total Variation)
19/34 interleaves perturbed spiral Nominal FOV
16cm Resolution 1mm 3ms readout, GRE sequence
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Phantom Scans
Gridding
Least-Norm
L1 Wavelet
Finite differences (Total Variation)
19/34 interleaves perturbed spiral Nominal FOV
16cm Resolution 1mm 3ms readout, GRE sequence
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Phantom Scans
Gridding
Least-Norm
L1 Wavelet
Finite differences (Total Variation)
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Coronary Imaging
Gridding
Finite differences (Total Variation)
single breath-hold whole heart at 3T 17 Itlv
Variable density spirals 50 5.6ms
readout Nominal FOV 20 0.8mm resolution
J. Santos, C. Cunningham, M. Lustig, B.
Hargreaves, B. Hu, D. Nishimura, J. Pauly Single
Breath-Hold Whole-Heart MRA Using
Variable-Density Spirals at 3T 2005, Submitted
Mag Med Res
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Extension SENSE

• SENSE reconstruction
• minimize ?(m)1
• s.t. Em-y2 lt ?
• E is encoding matrix that has the coil
  sensitivity information too.
• Use for regularization or further under sampling.

For more info, see abstract 504
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Sense Reconstruction Result

• 7 folded images
• Factor 7 acceleration

Least Squares
Total Variation
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Conclusions

• Pros
• Recon from 50 data
• High-res phase info.
• Outperforms conventional reconstruction methods
• Randomness is good!
• Many applications fast imaging, angiography,
  Time-Resolved imaging
• Cons
• Sensitive to eddy currents and gradient delays.
• Computationally intensive. 256x256 image in 2-5
  minutes.

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The END

• Thank you.
• http//www.stanford.edu/mlustig

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Extension CS with Homodyne detection

• Estimate of low-res phase info and solve for m
• minimize ?(Pm)1 s.t. FPm-y2 lt
  ? m ? 0
• m magnitude image
• P low-res phase estimate

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Phantom Scans
Least-Norm
L1 Wavelet
TV
Gridding
17/34 itlv perturbed spiral Nominal FOV
16cm Resolution 1mm 3ms readout