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APPLICATIONS OF COMPRESSED SENSING TO MAGNETIC RESONANCE IMAGING Speaker: Lingling Pu

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Acknowledgements 2 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

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

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

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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. 5

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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-1 DB-2 DB-3 DB-4 DB-5DB-6 6

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

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SPArse Reconstruction using a ColLEction of bases (SPARCLE)† 8 Incoherence between Ψ and F Ω Undersampling artifacts accumulate incoherently in Ψ Small coefficients in Ψ Our approach: Enforce sparsity in a collection of bases Ψ i, i=1,…,N Each basis Ψ i 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.

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SPArse Reconstruction using a ColLEction of bases (SPARCLE)† 9 Measurement Space (Fourier) Sparsity Space Ψ 1 Project Threshold to remove small coefficients Assert consistency with measured data Sparsity Space Ψ 2 Project Repeat for the next Sparsity basis

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Results 10 Radial-FSE dataset (TR=4.5s, FOV=26cm and ETL=4, 256x256 acquisition) retrospectively subsampled to 64 radial views Originall1-min DB6SPARCLE

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

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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 12

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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 Parent and child Neighborhood 13 parent v 1 v 2 v 3 v 4 v c v 5 v 6 v 7 v 8

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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 σ 2 I, accounting for aliasing artifacts Bayes least squares estimate for wavelet coefficients 14 †: J. Portillat et al. “Image Denoising Using Scale Mixtures of Gaussians in the Wavelet Domain,” IEEE Tran. On Image Processing, 2003

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

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

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Results 2.59 dB improvement on average 17

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Results 18

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

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A Voxel-based Morphometry (VBM) Study† 20 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 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.

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A Voxel-based Morphometry (VBM) Study 21 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.

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

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

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VBM Results 24 ROI 1ROI 2ROI 3ROI 4ROI 5ROI 6ROI 7ROI 8ROI 9ROI ROI 11ROI 12ROI 13ROI 14ROI 15ROI 16ROI 17ROI 18ROI 19 mean Correlation coefficients

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