Compressed Sensing: A Magnetic Resonance Imaging Perspective

1. Compressed Sensing: A Magnetic Resonance Imaging Perspective D97945003 Jia-Shuo Hsu 2009/12/10

2. Characteristics: Magnetic Resonance Imaging (MRI) Acquires Data from Frequency Other than Image Domain Samples frequency domain then retain image Undersample shortens scan time directly Mostly Fourier Encoding Spatial freq Image Wavelet domain Image domain

3. Sampling Theorem bounds the number of samples required for full signal recovery V.S.

4. Techniques adopted to get around • 1. EfficientSamplingPattern • Ex:OptimizedLatticeGridSampling • 2. Exploit spatio-temporal redundancy • Ex:Short-TimeFTtoaperiodicsignal • 3. Alter characteristics of aliasing • Ex:Variouschoiceoftime-frequencyanalysisthatalterstheshapeofspectrum

5. Fourier Transform 1. Certain undersampling patterns “pack” signals efficiently within given bandwidth • Two different 5-fold undersampling

6. 2.Time-varyingSignalsareRelativelyRedundantinTime-FrequencyDomain2.Time-varyingSignalsareRelativelyRedundantinTime-FrequencyDomain

7. 3.Non-CartesianSamplingDistortsAliasingintoNon-regularPattern3.Non-CartesianSamplingDistortsAliasingintoNon-regularPattern Tsao.et al. Magnetic Resonance in Medicine 55:116–125 (2006)

8. Compressible Signal Suggests “Inhomogeneous” information distribution Tutorial on Compressive Sensing, R. Baraniuk et al. (Feb 2008)

9. Possibility to fully recover highly undersampled signal ?? Normal Reconstruction 512*512 Shepp-Logan Undersampled by 22 radial lines ?????????????????? Emmanuel J. Candès, Justin Romberg, Member, IEEE, and Terence Tao IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 52, NO. 2, FEBRUARY 2006

10. Introduce Compressed Sensing Fulfilling certain criteria, it is possible to fully recover a signal from sampling points much fewer than that defined by Shannon's sampling theorem

11. Compressed Sensing • Given x of length N, only M measurements (M<N) is required to fully recover x when x is K-sparse (K<M<N) • However, three conditions named CS1-3 are to be satisfied for the above statement to be true

12. Three essential criteria • Sparsity: • The desired signal has a sparse representation in a known transform domain • Incoherence • Undersampled sampling space must generate noise-like aliasing in that transform domain • Non-linear Reconstruction • Requires a non-linear reconstruction to exploit sparsity while maintaining consistency with acquired data

13. Sparsity • Number of significant(strictly speaking, nonzero)components is relatively small compared to signal length • Ex: [10100 0 0 0 0…….0 0] • Sparsity Representation: • Lp-Norm: • L0 norm counts the number of non-zero components of x • Ex: if x=[1, 100000, 2, 0], then L0-Norm=3

14. Medical images often demonstrate inherent sparsities

15. Incoherence • Sampling must generate noise-like aliasing in image domain (more strictly, transform domain) • Very loosely speaking, patterns of sampling must demonstrate enough randomness

16. Random results in noise-like while regular equally weights the artifacts U. Gamper et al. Magnetic Resonance in Medicine 59:365–373 (2008)

17. Non-linear Reconstruction • Lacks the linearity of FFT and iFFT • Does not have analytical solution as in STFT, Gabor Transform, WDF….etc • Involves optimizations (often iterative) satisfying certain boundary conditions

18. Conjugate Gradient: non-linear recon with iterative optimization • A multi-dimensional optimization method suitable for non-cartesian sampled images M.S. Hansen et.al Magnetic Resonance in Medicine 55:85–91 (2006)

19. Demo 1: Reconstructing Highly Undersampled Sparse Signal

20. Random sampling generates noise-like artifacts M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007) (a) given that desired signal is sparse (b) different k-space sampling pattern (c)regular undersampling begets regular aliasing (d) random undersampling begets noise-like aliasing, preserving most of the major components

21. M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007) (e) detectedstrong components above the interference level (f) obtain estimates by thresholding (g) convolve (f) with PSF, obtain undersampled version of the signal (f) (h) subtract (g) from (e), thus another major component hindered by noise reveals Signal satisfying CS1-3 are recovered through CS M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

22. CS mathematically • Minimize such that , where x stands for reconstructed signal y’ stands for the estimated measurement y stands for the initial measurement εserves as the boundary condition (usually noise level) • In other words, among all possible solutions of x, find one with the smallest L0-norm(i.e. sparsest) whose estimated measurement y’ remains consistent with the initial measurement y with deviation less than ε

23. Many signals are not as sparse, strictly limiting the application? • Sparsity (i.e. Compressibility) can be generated through sparsifying transform • Signals that are compressible demonstrate sparsities in their sparsifying transform domains

24. Revisit CS mathematically • Minimize such that , where x stands for reconstructed signal stands for sparsifying transform y’ stands for the estimated measurement y stands for the initial measurement εserves as the boundary condition (usually noise level) • Among all possible sparsified solutions, find one with the smallest L0-norm(i.e. sparsest) whose estimated measurement y’ remains consistent with initial measurement y with deviation less than ε • Most use L1-norm, i.e. minimize instead

25. Choice of Sparsifying Transform is Essential to Performance • It’s all about finding the right • STFT, Gabor, WDF, S-Transform, Wavelet Transform……

26. MRI suits CS in certain perspectives • Data is acquired in sampling space • Medical images posses sparsities • Achieved results in angiography, dynamic imaging, MRSI and other potential applications

27. If image is already sparse Non-linear reconstruction CS applies as long as CS1-3 holds in sparsifying transform domain M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

28. Demo 2: CS-reconstructed MR Image

29. If image is already sparse Non-linear reconstruction Challenges and works to be done lie in every aspects of CS procedure M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

30. Criteria of CS remain to be further customized to fit different application • Sparsity: • Representation: What represents sparsity? • Degree: How sparse is enough? • Compressibility: Which sparsifying transform? • Incoherence • Representation: What represents randomness? • Degree: How random is enough? • Non-Linearity • Choice of method and complexity?

31. Representation of Sparsity is essential to required sample number • L0-norm is ideal, yet intractable • Needs only M=K+1 samples for K-sparse signals • is an NP problem when p=0 • L2-norm(i.e.) is well-known, yet inaccurate • p=2 represents Least Mean Square • L1-norm requires more samples than L0, yet is most feasible in its tractability and accuracy • Needs approximately K log(N/K) samples, yet no longer NP • L1-norm minimization is equivalent to a classical convex optimization problem with many well-established approaches

32. Ways to measure and achieve incoherence remains to be developed Fourier Basis • Approaches were taken, yet reliabilities to be verified • Inherent regularity of Fourier basis limits degree of randomness • Randomness doesn’t guarantee performance Non-Fourier

33. Reconstruction involves optimization with unpredictable non-linearity • Complexity of the reconstruction is unpredictable How Long does the loop loops? M.Lustig et.al Magnetic Resonance in Medicine 58:1182–1195 (2007)

34. Summary • Theory of Compressed Sensing: From CS to MRI • Sparsity, incoherence, non-linear reconstruction • Sometimes requires transform (compression) to achieve sparsity • Random sampling of k-space generates noise-like aliasing artifacts • Non-linear reconstruction ties to some well-known optimization problem • Challenges and Focus • Acquisition mechanism of MRI is unfavorable to randomness • Prior knowledge of image on sparsity is required • Criteria of CS and their representations remain to be customized in MRI • Suitable applications are to be further explored

35. Wavelets are no longer the central topic, despite the previous edition’s original title. It is just an important tool, as the Fourier transform is. Sparse representation and processing are now at the core- S. Mallat, 2009 Thanks for Your Attention!!

36. Appendix A: Online Resources • Open Source Softwares • http://sparselab.stanford.edu/ A free matlab toolbox consists of CS algorithms • Collection of current works • http://www.dsp.ece.rice.edu/cs/ • MRI-specific of CS • http://www.stanford.edu/~mlustig/

37. Appendix B: Recommended Literatures • Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging • An MRM publication with many results of CS in MRI • http://www.dsp.ece.rice.edu/cs/CS_notes.pdf • A succinct note on theory of CS • http://www.dsp.ece.rice.edu/~richb/talks/cs-tutorial-ITA-feb08-complete.pdf • A broad view of CS from theory to application