Markus Strohmeier

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

2. Overview of MRI imaging Motivation for Compressed Sensing Signal constraints for CS, Sparsity, PSF Sampling Schemes and Data Processing Results of Sparse MRI Outlook Outline 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 M. Strohmeier

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

5. Overview of MRI imaging (3) M. Strohmeier

6. 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. Motivation for Compressed Sensing QUESTION: If we throw away "most" of the image information anyway, why do we have to acquire it at all in the first place? M. Strohmeier

7. 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 Motivation for Compressed Sensing M. Strohmeier

8. 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). Signal Constraints for CS M. Strohmeier

9. Signal Constraints for CS M. Strohmeier

10. Signal Constraints for CS M. Strohmeier

11. Point Spread Function & Coherence The peak side-lobe ratio contains incoherence information . M. Strohmeier

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

13. 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. 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 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 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 M. Strohmeier

17. Reconstruction of Images Generally, CS gives the best results: M. Strohmeier

18. Reconstruction of Images M. Strohmeier

19. Reconstruction of Images M. Strohmeier

20. Reconstruction of Images M. Strohmeier

21. Reconstruction of Images M. Strohmeier

22. Reconstruction of Images M. Strohmeier

23. Reconstruction of Images M. Strohmeier

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

26. Thank you... ... the end! M. Strohmeier