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

2. 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
March 29th, 2011
M. Strohmeier

3. 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.
Limiting factors are: Slew rate and amplitude of gradient
March 29th, 2011
M. Strohmeier

4. Overview of MRI imaging (2)
Magnetic Resonance Imaging samples the frequency space of the human body -> Data set consists of Fourier Coefficients
March 29th, 2011
M. Strohmeier

5. Overview of MRI imaging (3)
March 29th, 2011
M. Strohmeier

6. 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?
March 29th, 2011
M. Strohmeier

7. 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
March 29th, 2011
M. Strohmeier

8. 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).
March 29th, 2011
M. Strohmeier

9. Signal Constraints for CS
March 29th, 2011
M. Strohmeier

10. Signal Constraints for CS
March 29th, 2011
M. Strohmeier

11. Point Spread Function & Coherence
The peak side-lobe ratio contains incoherence information .
March 29th, 2011
M. Strohmeier

12. Point Spread Function & Coherence
The peak side-lobe ratio is a measure of the incoherence.
March 29th, 2011
M. Strohmeier

13. "Randomness is too important to be left to Chance!"
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.
March 29th, 2011
M. Strohmeier

14. 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
March 29th, 2011
M. Strohmeier

15. Reconstruction of Images
Basic image reconstruction algorithm is the following
minimization problem, based on minimizing the L1-norm:
minimize
such that:
= 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
March 29th, 2011
M. Strohmeier

16. Reconstruction of Images
Simulated phantom serves as an input
for the reconstruction algorithms.
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
March 29th, 2011
M. Strohmeier

17. Reconstruction of Images
Generally, CS gives
the best results:
March 29th, 2011
M. Strohmeier

18. Reconstruction of Images
March 29th, 2011
M. Strohmeier

19. Reconstruction of Images
March 29th, 2011
M. Strohmeier

20. Reconstruction of Images
March 29th, 2011
M. Strohmeier

21. Reconstruction of Images
March 29th, 2011
M. Strohmeier

22. Reconstruction of Images
March 29th, 2011
M. Strohmeier

23. Reconstruction of Images
March 29th, 2011
M. Strohmeier

24. Reconstruction Results
Blood flow due to bypass is only visible with 5x CS an Nyquist sampling
Nyquist sampled
reconstruction
Low resolution
reconstruction
ZF w/dc CS
March 29th, 2011
M. Strohmeier

25. 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
March 29th, 2011
M. Strohmeier

26. Thank you...
... the end!
March 29th, 2011
M. Strohmeier