Transcript Markus Strohmeier Sparse MRI: The Application of Compressed Sensing for Rapid MRI Michael Lustig, David Donoho, John M.

Markus Strohmeier
Sparse MRI: The Application of
Compressed Sensing for Rapid MRI
Michael Lustig, David Donoho, John M. Pauly
Outline
Overview of MRI imaging
 Motivation for Compressed Sensing
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Signal constraints for CS, Sparsity, PSF
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Sampling Schemes and Data Processing
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Results of Sparse MRI
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Outlook
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Overview of MRI imaging (1)
The sample is exposed to a static magnetic field B0 which
polarizes the protons along a certain direction.
In the B0-field, the protons show a resonance behavior when
excited by a microwave which can be seen by a receiver coil.
By applying a spatial gradient to the static B-field, one
changes the resonance frequency as a function of the spatial
coordinate.
B x = B0 G x x
Limiting factors are: Slew rate and amplitude of gradient
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Overview of MRI imaging (2)
Magnetic Resonance Imaging samples the frequency space
of the human body -> Data set consists of Fourier
Coefficients
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Overview of MRI imaging (3)
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Motivation for Compressed Sensing
Most images can be compressed with some transform algorithm
(JPEG or JPEG2000), as the most important information is
carried by only a fraction of the Fourier coefficients.
Neglecting the high frequency coefficients (they carry only little
energy) doesn't degrade the image noticeable enough for the
human eye.
QUESTION:
If we throw away "most" of the image information anyway,
why do we have to acquire it at all in the first place?
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Motivation for Compressed Sensing
This approach does not work for images captured in the spatial
domain: Which and how much pixels should be omitted?
However, since MRI captures frequency information, CS has the
potential to reduce the necessary amount of acquired data to
reconstruct the image.
→ Reduced acquisition time makes a scan
shorter and less stressful for the patient.
→ MRI scanners would be able operate more economically
since more patients can be examined in the same time
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Signal Constraints for CS
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Signal has to be sparse in a domain, that is it has to
be compressible by a transform algorithm.
Under-sampling artifacts must be incoherent. Then they appear
in the reconstructed data like noise and can be thresholded.
Knowing the Point-Spread-Function is a measure of
the incoherence.
The image needs to be reconstructed by a non-linear algorithm
in order to enforce sparsity and keep the consistency of
the acquired samples with the reconstructed image (see later).
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Signal Constraints for CS
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Signal Constraints for CS
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Point Spread Function & Coherence
The peak side-lobe ratio contains incoherence information .
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Point Spread Function & Coherence
The peak side-lobe ratio is a measure of the incoherence.
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Sampling Schemes
Incoherence has to be preserved when sampling the k-space.
→ No equispaced under-sampling, but random under-sampling!!
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"Randomness is too important to be left to Chance!"
→ The (random) sampling is controlled in the sense that different
regions of the k-space are sampled with different densities.
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Monte-Carlo Incoherent Sampling Design is an approach
to try to optimize the random under-sampling.
→ Iterative procedure in order to avoid "bad" point spread
functions which would destroy incoherence.
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Sampling Schemes
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For simplicity reasons, mostly
Cartesian coordinates to sample
the k-space were used up to now.
However, w.r.t. variable density
sampling, spiral or radial
trajectories have been
successfully tested.
Those schemes are just slightly
less coherent compared to
random 2D sampling
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∥∥
Reconstruction of Images

Basic image reconstruction algorithm is the following
minimization problem, based on minimizing the L1-norm:
minimize
m
1
such that:
F u m− y
2
= operator, transforming from pixel to sparse representation
m
= reconstructed image
F u = undersampled Fourier transform
y
= measured k-space data
= parameter, that assures accuracy between
reconstruction and measured data
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Reconstruction of Images
Simulated phantom serves as an input
for the reconstruction algorithms.
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Image size: 100x100 pixels.
5.75 % of the pixels are non zero,
18 objects with 3 distinct intensities
and 6 different sizes:
→ Sparse image, similar to
angiogram or brain scan.
Interested in how the artifacts
evolve as the data is under-sampled

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Reconstruction of Images
Generally, CS gives
the best results:
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Reconstruction of Images
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Reconstruction of Images
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Reconstruction of Images
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Reconstruction of Images
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Reconstruction of Images
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Reconstruction of Images
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Reconstruction Results
Blood flow due to bypass is only visible with 5x CS an Nyquist sampling
Nyquist sampled
reconstruction
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Low resolution
reconstruction
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ZF w/dc
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Summary & Outlook
It was shown that for an appropriate data set, compressed
sensing has the capability to perform a "random" sub-Nyquist
sampling and still recover the image to a large extent without
noticeable visual artifacts.
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Depending on the respective demands, a extreme sub-sampling
is possible without losing significant amounts of information.
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With increasing computing power and code optimization, it might
be possible in the (near) future to implement CS into commercially
available scanners

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Thank you...
... the end!
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