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

Signal constraints for CS, Sparsity, PSF

Sampling Schemes and Data Processing

Results of Sparse MRI

Outlook
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Overview of MRI imaging (1)
The sample is exposed to a static magnetic field B
0 polarizes the protons along a certain direction.
which
In the B
0
-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 = B
0
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

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!!
"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.

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

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 L
1
-norm: minimize
∥ m ∥
1 such that:
∥ F u m
− y ∥
2 m
F u y
= operator, transforming from pixel to sparse representation
= reconstructed image
= undersampled Fourier transform
= 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
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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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CS

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.

Depending on the respective demands, a extreme sub-sampling is possible without losing significant amounts of information.

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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