EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging. Martin Uecker

Transcription

1 EE290T: Advanced Reconstruction Methods for Magnetic Resonance Imaging Martin Uecker

2 Tentative Syllabus 01: Jan 27 Introduction 02: Feb 03 Parallel Imaging as Inverse Problem 03: Feb 10 Iterative Reconstruction Algorithms : Feb 17 (holiday) 04: Feb 24 Non-Cartesian MRI : Mar 03 (cancelled) 05: Mar 10 GRAPPA/SPIRiT 06: Mar 17 Nonlinear Inverse Reconstruction : Mar 24 (spring recess) 08: Mar 31 SAKE/ESPIRiT 09: Apr 07 Model-based Reconstruction 10: Apr 14 Compressed Sensing 11: Apr 21 Compressed Sensing 12: Apr 28 Final Project: Presentations

3 Outline Review of last lecture Compressed Sensing (and Parallel Imaging) IEEE Eta Kappa Nu - Survey

4 Nyquist-Shannon Sampling Theorem Theorem 1: If a function f (t) contains no frequencies higher than W cps, it is completely determined by giving its ordinates at a series of points spaced 1/2W seconds apart. 1 Band-limited function Regular sampling Linear sinc-interpolation 1. CE Shannon. Communication in the presence of noise. Proc Institute of Radio Engineers; 37:10 21 (1949)

5 A Puzzling Numerical Experiment 1 Exact recovery of Shepp-Logan phantom from incomplete radial Fourier samples: (Figure: Block et al. 2007) 1. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006)

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7 Compressed Sensing Ingredients: Sparsity Incoherence Non-linear reconstruction 1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc; 45: (1997) 2. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52: (2006)

8 Compressed Sensing Ingredients: Sparsity Incoherence Non-linear reconstruction 1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc; 45: (1997) 2. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52: (2006)

9 Sparsity Definition: vector x R n k-sparse: at most k non-zero entries Example: Notation: Number of non-zero entries x 0 (this is not a norm)

10 Sparsity x R 2 ( ) ( ) Set of sparse vectors is a (non-convex) union of subspaces

11 Denoising Sparse vector

12 Denoising Sparse vector and random noise

13 Denoising Sparse vector and random noise Densoising by hard-thresholding

14 Denoising Sparse vector and random noise Densoising by hard-thresholding Densoising by soft-thresholding (shrinkage)

15 Compressed Sensing Ingredients: Sparsity Incoherence Non-linear reconstruction 1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc; 45: (1997) 2. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52: (2006)

16 Compressed Sensing Ingredients: Sparsity Incoherence Non-linear reconstruction 1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc; 45: (1997) 2. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52: (2006)

17 Regular Under-Sampling: Point-Spread-Function Regular under-sampling in Fourier domain Coherent aliasing in the time domain Point-Spread-Function

18 Coherent Aliasing Regular under-sampling in Fourier domain Coherent aliasing in the time domain Signal

19 Random Sampling: Point-Spread-Function Random sampling in Fourier domain Incoherent aliasing in time domain noise-like artifacts Point-Spread-Function

20 Incoherent Aliasing Random sampling in Fourier domain Incoherent aliasing in time domain Signal

21 Linear Measurements Ax = y x R n, y R m Measurements: m >= n Reconstruction: x = A y Matrix A should be nearly orthogonal. Example: Fourier Matrix A A H A

22 Incoherent Linear Measurements Ax = y x R n and k-sparse, y R m Measurements: k log(n) <= m <= n Reconstruction:? Matrix A should be nearly orthogonal (restricted isometry property) Example: Fourier Matrix with some rows removed A = PF A A H A

23 Restricted Isometry Property A n p matrix and 1 s p s-restricted isometry property: There is a constant δ s such that for every s-sparse vector y: (1 δ s ) y 2 2 Ay 2 2 (1 + δ s ) y 2 2

24 Compressed Sensing Ingredients: Sparsity Incoherence Non-linear reconstruction 1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc; 45: (1997) 2. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52: (2006)

25 Compressed Sensing Ingredients: Sparsity Incoherence Non-linear reconstruction 1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc; 45: (1997) 2. EJ Candès, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans Inform Theory; 52: (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52: (2006)

26 L 1 -Norm and Sparsity ( 0 1) Set of vectors with x 0 1 not convex! L 1 instead of L 0

27 L 1 -Norm and Sparsity ( 0 1) Set of vectors with x 0 1 not convex! L 1 instead of L 0

28 Linear Reconstruction L 2 -regularization: argmin x Ax y α Wx 2 2 Explicit Solution: ( A H A + αw H W ) 1 A H y

29 Nonlinear Reconstruction L 1 -regularization: In general: no explicit solution! argmin x Ax y α Wx 1

30 L 2 -Norm vs L 1 -Norm 1 x 2 = x 1 =

31 L 1 -Norm and Sparsity Minimize x p p subject to Ax = y Ax = y Ax = y x 2 2 x 1 1

32 Inverse Problem with L 1 -Regularization Minimize x 1 subject to Ax y 2 ɛ Ax y 2 = ɛ x 1 1

33 Linear Reconstruction x = argmin z z y 2 + λ z 2 1 x = λ y

34 Soft-Thresholding x = argmin z z y 2 + λ z x λ x > λ η λ (x) = 0 x λ x + λ x < λ 1 0 λ λ

35 Joint Thresholding Shrink magnitude but keep phase/direction complex values: vectors: η λ (x) = η λ (x) = { ηλ ( x ) x x x 0 0 x = 0 { ηλ ( x 2 ) x x 2 x 0 0 x = 0

36 Joint Thresholding Shrink magnitude but keep phase/direction

37 Iterative Soft-Thresholding (IST) Landweber 1 : x n+1 = x n + µa H (y Ax n ) Iterative Soft-Thresholding 2 : z n = x n + µa H (y Ax n ) x n+1 = η λ (z n ) 1. L Landweber. An iteration formula for Fredholm integral equations of the first kind. Amer J Math; 73: (1951) 2. I Daubechies, M Defrise, C De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pure Appl Math; 57: (2004)

38 Nonlinear Reconstruction: Iterative Soft-Thresholding 1. Data consistency: z n = x n + µa H (y Ax n ) 2. Soft-thresholding: x n+1 = η λ (z n ) iteration 0

39 Nonlinear Reconstruction: Iterative Soft-Thresholding 1. Data consistency: z n = x n + µa H (y Ax n ) 2. Soft-thresholding: x n+1 = η λ (z n ) iteration 1

40 Nonlinear Reconstruction: Iterative Soft-Thresholding 1. Data consistency: z n = x n + µa H (y Ax n ) 2. Soft-thresholding: x n+1 = η λ (z n ) iteration 2

41 Nonlinear Reconstruction: Iterative Soft-Thresholding 1. Data consistency: z n = x n + µa H (y Ax n ) 2. Soft-thresholding: x n+1 = η λ (z n ) iteration 9

42 Algorithms FOCUSS 1 Iterative Soft-Thresholding (IST) 2 Fast iterative Soft-Thresholding Algorithm 3 Split Bregman 4 Nonlinear Conjugate Gradients IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm. IEEE Trans Sig Proc 45: (1997) 2. I Daubechies, M Defrise, C De Mol. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pure Appl Math; 57: (2004) 3. A Beck, M Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci; 2: (2009) 4. T Goldstein and S Osher. The Split Bregman Method for L1-Regularized Problems. SIAM J Imaging Sci; 2: (2009)

43 Statistical Model Linear measurements contaminated by noise: Gaussian white noise: y = Ax + n p(n) = N (0, σ 2 ) with N (µ, σ 2 ) = 1 σ (x µ) 2 2π e 2σ 2 Probability of an outcome (measurement) given the image x: p(y A, x, λ) = N (Ax, σ 2 )

44 Bayesian Prior L 2 -Regularization: Ridge Regression N (µ, σ 2 ) = 1 σ (x µ) 2 2π e 2σ 2 Gaussian prior L 1 -Regularization: LASSO p(x µ, b) = 1 x µ 2b e b Laplacian prior L 2 and L 1 : Elastic net 1 1. H Zou, T Hastie. Regularization and variable selection via the elastic net. J R Statist Soc B. 67: (2005)

45 Compressed Sensing in Magnetic Resonance Imaging Sparsity in medical imaging Incoherent sampling Iterative reconstruction Combination with parallel imaging M Lustig, D. Donoho, JM Pauly. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn Reson Med; 58: (2007)

46 Sparsity Transform Medical images are usually not sparse Need to apply sparsity transform Sparsity: Wavelet Transform Total Variation Temporal Constraints Prior images Adapted dictionaries Low rank...

47 Wavelet Transform Orthonormal basis (or almost) Multi-scale transform Localized in frequency and space Compresses many signals/images into few coefficients Efficient computation: O(N) But: not shift-invariant (cycle spinning) L 1 -regularization term in wavelet domain; W wavelet transform R(x) = Wx 1

48 Wavelet Transform brain image wavelet transform Signal concentrated in few coefficients!

49 Cycle spinning Problem: Not shift-invariant Solution: Cycle spinning (or random shifting)

50 Artifacts blurring blocky artifacts good quality Problem: Not shift-invariant Solution: Random shifting (cycle spinning)

51 Total Variation Definition: For a function f L 1 (Ω) with Ω an open set Ω R n, the total variation of f is: { } TV {f } = sup dx f divφ : φ Cc 1 (Ω, R n ), φ L (Ω) 1 Denoising 1 Image Reconstruction 2 1. LI Rudin, S Osher, E Fatemi. Nonlinear total variation based noise removal algorithms. Physica D; 60: (1992) 2. D Geman, C Yang. Nonlinear image recovery with half-quadratic regularization. IEEE T Image Processing; 4: (1995)

52 Total Variation For differentiable function in one variable: TV {f } = dx f (x) For differentiable function in many variables: TV {f } = dx f (x) 2 Sparsity of the partial derivatives!

53 Total Variation f (x) f (x) 2 = 1 f (x) f (x) 2

54 Total Variation: Approximations Anisotropic Total Variation: TV {f } = dx 1 f (x) f (x) 2 dx 1 f (x) + 2 f (x) Finite differences (backward): h 1 Pixel 1 f (x 1, x 2 ) f (x 1, x 2 ) f (x 1 h, x 2 ) h 2 f (x 1, x 2 ) f (x 1, x 2 ) f (x 1, x 2 h) h

55 Total Variation: Staircase Artifacts Staircase artifacts (sparse differences) Solution: use of higher-order derivatives 1 Total Generalized Variation 2,3 1. D Geman, C Yang. Nonlinear image recovery with half-quadratic regularization. IEEE T Image Processing; 4: (1995) 2. K Bredies, K Kunisch, T Pock. Total generalized variation. SIAM J Imaging Sci; 3: (2010) 3. F Knoll, K Bredies, T Pock, R Stollberger. Second order total generalized variation (TGV) for MRI. Magn Reson Med; 65: (2011)

56 Total Variation in Time Domain f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 f 9 f 10 t TV t {f } dx l f l (x) f l 1 (x) HFL Chandarana, T Block, AB Rosenkrantz, R Lim, D Chu, DK Sodickson, R Otazo. Free-breathing dynamic contrast-enhanced MRI of the liver with radial golden-angle sampling scheme and advanced compressed-sensing reconstruction. Proc. 20th ISMRM (2012)

57 GRASP: Compressed-Sensing Reconstruction Narrow data window Few spokes Flickering streak artifacts Prior knowledge: Contrast uptake occurs smoothly and continuously Dynamic image series Temporal differences Ground truth 13 spokes Ground truth 13 spokes CS approach: Find solution that matches data in all windows has lowest flickering CG SENSE-type reconstruction with temporal Total Variation (TV) constraint Otazo et al, MRM 2010: 64 Courtesy of Tobias Block, NYU

58 Example: GRASP Liver Imaging Free-breathing scan over 5 min Contrast injection after 20 s Retrospective selection of temporal resolution Example: 13 spokes 2 s resolution Enables free-breathing liver perfusion imaging Here: 384 x 384 x 30 matrix Spatial resolution 1.0 x 1.0 x 3.0 mm 3 Temporal resolution 1.5 s Chandarana et al, ISMRM 2012: 5529 Top: Gridding Bottom: GRASP Courtesy of Tobias Block, NYU

59 Difference to Prior Image Sparse difference to reference image g R(x) = x x 0 1 Prior image: x 0 composite image, previous frame, GH Chen, J Tang, S Leng. Prior image constrained compressed sensing (PICCS): a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets. Med Phys; 35: (2008) 2. A Fischer, F Breuer, M Blaimer, N Seiberlich, PM Jakob. Accelerated dynamic imaging by reconstructing sparse differences using compressed sensing. Proc 16th ISMRM (2008)

60 Dictionary Learning Patch Reconstruction Dictionary Learning Example-based... =a +b +c = Courtesy of Patrick Virtue, UC Berkeley

61 Spatio-temporal Dictionaries Dictionary based reconstruction of dynamic complex MRI data. Jose Caballero, Anthony Price, Daniel Rueckert, and Joseph V. Hajnal. ISMRM 13 Courtesy of Jose Caballero, Imperial College London

62 Low-rank Approximation Data matrix M C t s, e.g. time space Singular-Value-Decomposition: M = UΣV H Low rank: rank M < K Decomposition into K (temporal and spatial) basis functions: M = K u k σ k v H k 1. Z Bo, JP Haldar. C Brinegarm ZP Liang. Low rank matrix recovery for real-time cardiac MRI. ISBI; (2010) 2. JP Haldar, ZP Liang. Spatiotemporal imaging with partially separable functions: A matrix recovery approach. ISBI (2010) 3. SG Lingala, H Yue, E DiBella, M Jacob. Accelerated Dynamic MRI Exploiting Sparsity and Low-Rank Structure: k-t SLR. IEEE Trans Med Imag; 30: (2011) 4. R Otazo, E Candes, DK Sodickson. Low-rank and sparse matrix decomposition for accelerated DCE-MRI with background and contrast separation. ISMRM Workshop on Data Sampling and Image Reconstruction. Sedona (2013)

63 Low-rank + Sparse Reconstruction of Cardiac Cine 6-fold acceleration (ky-t random undersampling) Temporal resolution: 40 ms Spatial resolution: 1.3x1.3x3 mm 3 Std. CS with temporal FFT CS L+S L S Courtesy of Ricardo Otazo, NYU

64 Parallel MRI Goal: Reduction of measurement time I Subsampling of k-space I Simultaneous acquisition with multiple receive coils I I Coil sensitivities provide spatial information Compensation for missing k-space data 1. DK Sodickson, WJ Manning. Simultaneous acquisition of spatial harmonics (SMASH): Fast imaging with radiofrequency coil arrays. Magn Reson Med; 38: (1997) 2. KP Pruessmann, M Weiger, MB Scheidegger, P Boesiger. SENSE: Sensitivity encoding for fast MRI. Magn Reson Med; 42: (1999) 3. MA Griswold, PM Jakob, RM Heidemann, M Nittka, V Jellus, J Wang, B Kiefer, A Haase. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson Med; 47: (2002)

65 Parallel MRI: Undersampling Undersampling Aliasing k phase k read k partition k phase

66 Parallel MRI as Inverse Problem Signal from multiple coils (image x, sensitivities c j ): s j (t) = d r x( r)c j ( r)e i r k(t) Assumption: known sensitivities c j linear relation between image x and data y Image reconstruction is a linear inverse problem: V Ax = y 1. JB Ra and CY Rim, Magn Reson Med 30: (1993) 2. KP Pruessmann, M Weiger, MB Scheidegger, P Boesiger. Magn Reson Med 4: (1999)

67 Parallel MRI: Regularization General problem: bad condition Noise amplification during image reconstruction L 2 regularization (Tikhonov): argmin x Ax y α x 2 2 (A H A + αi )x = A H y Influence of the regularization parameter α: small medium large

68 Parallel MRI: Nonlinear Regularization Good noise suppression Edge-preserving Sparsity, nonlinear regularization argmin x Ax y αr(x) Regularization: R(x) = TV (x), R(x) = Wx 1, JV Velikina. VAMPIRE: variation minimizing parallel imaging reconstruction. Proc. 13th ISMRM; 2424 (2005) 2. G Landi, EL Piccolomini. A total variation regularization strategy in dynamic MRI, Optimization Methods and Software; 20: (2005) 2. B Liu, L Ying, M Steckner, J Xie, J Sheng. Regularized SENSE reconstruction using iteratively refined total variation method. ISBI; (2007) 3. A Raj, G Singh, R Zabih, B Kressler, Y Wang, N Schuff, M Weiner. Bayesian parallel imaging with edge-preserving priors. Magn Reson Med; 57:8 21 (2007) 4. M Uecker, KT Block, J Frahm. Nonlinear Inversion with L1-Wavelet Regularization - Application to Autocalibrated Parallel Imaging. ISMRM 1479 (2008) 5....

69 Nonlinear Inversion with Non-Quadratic Regularization Iteratively Regularized Gauss Newton Method (IRGNM) x n+1 x n = argmin δx DF H (x n )δx + F (x n ) y α n R(δx + x n ) Previously: Image regularized with L 2 -norm R(x) = ρ (1 + s k 2 ) l FTc j 2 2 Now: Different regularization terms R(x) = R(ρ) + (1 + s k 2 ) l FTc j 2 2 Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).

70 Nonlinear Inversion Quality of the reconstructed images can be improved Acceleration: 3 x 2 L1-Wavelet: Cohen-Daubechies-Feauveau 9/7

71 Compressed Sensing and Parallel Imaging Parallel imaging Sparsity, nonlinear regularization Incoherent sampling 1. KT Block, M Uecker, J Frahm. Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint. Magn Reson Med; 57: (2007) 2. C Zhao, T Lang, J Ji. Compressed Sensing Parallel Imaging. Proc. 16th ISMRM; 1478 (2008) 3. B Wu, RP Millane, R Watts, P Bones. Applying compressed sensing in parallel MRI. Proc. 16th ISMRM; 1480 (2008) 4. KF King. Combining compressed sensing and parallel imaging. Proc. 16th ISMRM; 1488 (2008) 5. B Liu, FM Sebert, YM Zou, L Ying. SparseSENSE: randomly-sampled parallel imaging using compressed sensing. Proc. 16th ISMRM; 3154 (2008) 6. B Liu, K King, M Steckner, J Xie, J Sheng, L Ying. Regularized sensitivity encoding (SENSE) reconstruction using bregman iterations. Magn Reson Med 61: (2009) 7....

72 Sampling Schemes uniform random Poisson-disc Poisson-disc sampling: Minimum distance to exploit parallel imaging Incoherence for compressed sensing M Murphy, K Keutzer, SS Vasanawala, M Lustig. Clinically feasible reconstruction time for L1-SPIRiT parallel imaging and compressed sensing MRI. Proc 18th ISMRM; 4854 (2010)

73 Variable-Density Sampling variable-density Poisson-disc radial Advantages: Auto-calibration for parallel imaging Graceful degradation

74 Compressed Sensing and Parallel Imaging Regularized SENSE: arg min x PFCx y R(x) P projection onto samples, F Fourier transform, C coil sensitivities, R(x) regularization Simple (but slow): IST with R(x) = Wx 1 z n = x n + µc H F H P(y PFCx n ) x n+1 = W 1 η λ (Wz n )

75 Compressed Sensing and Parallel Imaging Linear reconstruction: R(x) = 0 No regularization R(x) = x x Tikhonov Nonlinear reconstruction: R(x) = TV (x) Total Variation R(x) = Wx 1 L 1 -Wavelet R(x) = x x 0 1 Prior Image Incoherent Sampling Compressed Sensing

76 Example: Undersampled Radial with Total Variation KT Block, M Uecker, J Frahm. Undersampled Radial MRI with Multiple Coils. Iterative Image Reconstruction Using a Total Variation Constraint. Magn Reson Med 57: (2007)

77 Nonlinear Inverse Reconstruction with Variational Penalties Experiments: Siemens Tim Trio 3 T, 12-channel head coil 3D FLASH, acceleration: R = 4 (pseudorandom sampling) Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).

78 l 1 -SPIRiT Robust auto-calibrating parallel MRI Calibration of coil-by-coil operator G in k-space Optimization problem: arg min x α Px y }{{} β (G Id)x 2 2 }{{} + γr(x) }{{} data consistency calibration consistency regularization x estimated k-space, y data, P Projection onto samples, G SPIRiT operator, R(x) regularization M Lustig, JM Pauly. SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space. Magn Reson Med; 64: (2010)

79 l 1 -SPIRiT POCS-type algorithm (α = β = ): y n = y + (Id P)x n z n = Gy n x n+1 = W 1 η λ (Wz n ) data consistency calibration consistency soft-thresholding

80 l 1 -ESPIRiT ESPIRiT: Flexibility and efficiency of SENSE Robustness of GRAPPA/SPIRiT Algorithm: Calibration of coil-by-coil operator in k-space Sensitivity maps from eigendecomposition Extended ( soft ) SENSE reconstruction M Uecker, P Lai, MJ Murphy, P Virtue, M Elad, JM Pauly, SS Vasanawala, M Lustig. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magn Reson Med. Epub (2013)

81 ESPIRiT: Reconstruction with Multiple Maps Relaxed ( soft ) SENSE using multiple maps. Simultaneous reconstruction of multiple images m j Data consistency: N y i PF i=1 M S j i mj 2 2 j=1 } {{ } soft SENSE + Q(m 1,, m M ) m j images, S j multiplication with maps, F Fourier transform, P sampling operator, y data, Q regularization Image combination (e.g. root of sum of squares)

82 Example: l 1 -ESPIRiT M Uecker, P Lai, MJ Murphy, P Virtue, M Elad, JM Pauly, SS Vasanawala, M Lustig. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magn Reson Med. Epub (2013)

83 Conclusion Compressed Sensing: Theory Sparsity Incoherent sampling Nonlinear reconstruction Application to Magnetic Resonance Imaging Sparsity transform (adapted to application) Incoherent sampling schemes Efficient iterative reconstruction Combination with parallel imaging

84 Software agile-gpu-image-reconstruction-library

85 Acknowledgements Michael Lustig, University of California, Berkeley Tobias Block, New York University Ricardo Otazo, New York University Patrick Virtue, University of California, Berkeley Jose Caballero, Imperial College London

86 Final Projects Presentation: 10 minutes, 28th April Report