Applications of Compressed Sensing to Magnetic Resonance Imaging

1. Applications of Compressed Sensing to Magnetic Resonance Imaging Speaker: Lingling Pu

2. Acknowledgements Ali Bilgin, Ted Trouard, Maria Altbach, Yookyung Kim, Lee Ryan Department of Biomedical Engineering, University of Arizona, Tucson, AZ Department of Radiology, University of Arizona, Tucson, AZ Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ Department of Psychology, University of Arizona, Tucson, AZ Onur Guleryuz Department of Electrical Engineering, Polytechnic Institute of NYU, Brooklyn, NY Mariappan Nadar Siemens Corporation, Corporate Research, Princeton, NJ

3. Outline SPArse Reconstruction using a ColLEction of bases (SPARCLE) Wavelet Information Assisted Model-based CS Reconstruction Voxel-based Morphometry Study Based on SPARCLE-CS Reconstructed T1-weighted images

4. Compressed Sensing CS theory has demonstrated that MR images can be reconstructed from a small number of k-space measurements. minimizations: Fourier measurements image consistency sparsity Sparsity transform Undersampled Fourier measurement matrix

5. Selection of Sparsity Basis • Two considerations for selection of the sparsity transform Ψ • Sparse signal representation • Incoherency with measurement basis • Ex: Orthonormal wavelet transforms • Usually no strong preference to select a particular wavelet basis. • Many wavelets yield qualitatively and quantitatively similar reconstructions.

6. Selection of Sparsity Basis minimization: • T2-weighted axial brain data set, radially undersampled in k-space. • Ψ: Orthonormal Daubechies wavelets with different number of vanishing moments (1-6). DB-2 DB-1 DB-3 DB-5 DB-6 DB-4

7. Selection of Sparsity Basis Question: Can we somehow benefit from the fact that the reconstruction artifacts are (slightly) different in different bases? DB-1 DB-2 DB-3 • Observations: • Qualitatively no significant difference between reconstructions. • Reconstruction artifacts are slightly different. DB-4 DB-5 DB-6

8. SPArse Reconstruction using a ColLEction of bases (SPARCLE)† Incoherencebetween Ψ and FΩ Undersampling artifacts accumulate incoherently in Ψ Small coefficients in Ψ Our approach: Enforce sparsity in a collection of bases Ψi , i=1,…,N Each basis Ψi provides a sparse representation. In addition, the undersampling artifacts are different in each basis. A large coefficient due to undersampling artifacts in one basis is likely to result in small coefficients in the other basis. By requiring that the result be sparse in multiple bases, a significantly larger portion of the undersampling artifacts can be removed. †: A. Bilgin et al, “SPArse Reconstruction using a ColLEction of bases (SPARCLE),” in Proc. of 2009 Meeting of ISMRM, 2009.

9. SPArse Reconstruction using a ColLEction of bases (SPARCLE)† Sparsity Space Ψ1 Project Measurement Space (Fourier) Sparsity Space Ψ2 Assert consistency with measured data Threshold to remove small coefficients Project Repeat for the next Sparsity basis Project

10. Results Radial-FSE dataset (TR=4.5s, FOV=26cm and ETL=4, 256x256 acquisition) retrospectively subsampled to 64 radial views Original l1-min DB6 SPARCLE

11. Outline SPArse Reconstruction using a ColLEction of bases (SPARCLE) Wavelet Information Assisted Model-based CS Reconstruction Voxel-based Morphometry Study Based on SPARCLE-CS Reconstructed T1-weighted images

12. Motivation • CS assumes that transform coefficients are independent • Correlation between wavelet coefficients → We exploit statistical dependencies of the wavelet coefficients by modeling them as Gaussian Scale Mixture (GSM) in the CS framework

13. Statistics in Wavelet Domain • Marginal distribution of wavelet coefficients exhibits leptokurtotic behavior. • Dependencies between coefficients • Correlated with coefficients of similar position, orientation and scale • Parent and child • Eight spatially adjacent neighbors v1 v2 v3 v4 vc v5 v6 v7 v8 parent Parent and child Neighborhood

14. Bayes Least Squares-Gaussian Scale Mixtures†(BLS-GSM) • GSM model • u: zero-mean Gaussian vector • z: positive hidden multiplier • Signal model for a reconstructed coefficient: • y : a neighborhood vector from reconstructed wavelet coefficients • e : a Gaussian random vector with covariance σ2I, accounting for aliasing artifacts • Bayes least squares estimate for wavelet coefficients †: J. Portillat et al. “Image Denoising Using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Tran. On Image Processing, 2003

15. Iterative Hard Thresholding (IHT)† • IHT • Mo-Sparse problem • Solved by the iterative algorithm where HMo is the element-wise hard thresholding operator that retains the Mo largest coefficients • BLS-GSM IHT • IHT is used to generate signal estimates • BLS-GSM model is imposed to re-estimate the signal • Impose Mo sparsity †: T. Blumensath, M. E. Davies,"Normalised Iterative Hard Thresholding; guaranteed stability and performance.” 2009.

16. Results 100V Original Test images: T2-weighted radial-FSE (256 radial views x 256 points ) BLS-GSM IHT IHT 17.58 dB 23.88 dB 20.87 dB

17. Results 2.59 dB improvement on average

18. Results

19. Outline SPArse Reconstruction using a ColLEction of bases (SPARCLE) Wavelet Information Assisted Model-based CS Reconstruction Voxel-based Morphometry Study Based on SPARCLE-CS Reconstructed T1-weighted images

20. A Voxel-based Morphometry (VBM) Study† • VBM • Investigates local differences in brain anatomy, after discounting the large-scale anatomical differences • Enables classical inferences about the regionally-specific effects • Participants • 69 females (ages 52-92 years) living independently, normal memory and executive function. • Two groups: • Anti-inflammatory (AI) drug users • Control (non-AI drug users) • Investigate • Correlation between gray matter volume changes and age. • Identify brain regions where age-related volume decreases were significantly greater in one group compared to the other. †: K. Walther et al, “Anti-inflammatory drugs reduce age-related decreases in brain volume in cognitively normal older adults,” in Neurobiology of Aging, 2009.

21. A Voxel-based Morphometry (VBM) Study • Images • T1-weighted images of the whole brain with a section thickness of 0.7mm (TR = 5.1 ms, TE = 2 ms, TI = 500 ms; flip angle = 15◦; matrix = 256×256; FOV= 260mm×260 mm). • Image reconstructions • SPARCLE CS • General linear model (GLM) was used to carry out the multiple regression analysis.

22. VBM Results Original full data SPARCLE CS reconstruction from 25% data NUFFT reconstruction from 25% data

23. VBM Results • SPM result based on the original data. • Define Region-of-Interest (ROI): • - centered at each of the peaked voxel • - radius 10 mm sphere

24. VBM Results Correlation coefficients