EE290T: Advanced Reconstruction Methods for Magnetic...

EE290T: Advanced Reconstruction Methods for MagneticResonance Imaging

Martin Uecker

Tentative Syllabus

I 01: Jan 27 Introduction

I 02: Feb 03 Parallel Imaging as Inverse Problem

I 03: Feb 10 Iterative Reconstruction Algorithms

I –: Feb 17 (holiday)

I 04: Feb 24 Non-Cartesian MRI

I –: Mar 03 (cancelled)

I 05: Mar 10 GRAPPA/SPIRiT

I 06: Mar 17 Nonlinear Inverse Reconstruction

I –: Mar 24 (spring recess)

I 08: Mar 31 SAKE/ESPIRiT

I 09: Apr 07 Model-based Reconstruction

I 10: Apr 14 Compressed Sensing

I 11: Apr 21 Compressed Sensing

I 12: Apr 28 Final Project: Presentations

Outline

I Review of last lecture

I Compressed Sensing (and Parallel Imaging)

I IEEE Eta Kappa Nu - Survey

Nyquist-Shannon Sampling Theorem

Theorem 1: If a function f (t) contains no frequencies higher thanW cps, it is completely determined by giving its ordinates at aseries of points spaced 1/2W seconds apart.1

I Band-limited function

I Regular sampling

I Linear sinc-interpolation

1. CE Shannon. Communication in the presence of noise. Proc Institute of Radio Engineers; 37:10–21 (1949)

A Puzzling Numerical Experiment1

Exact recovery of Shepp-Logan phantom from incomplete radialFourier samples:

(Figure: Block et al. 2007)

1. EJ Candes, J Romberg, T Tao. Robust Uncertainty Principles: Exact Signal Reconstruction From HighlyIncomplete Frequency Information. IEEE Trans Inform Theory; 52:489–509 (2006)

Compressed Sensing

Ingredients:

I Sparsity

I Incoherence

I Non-linear reconstruction

1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc; 45:600–616 (1997) 2. EJ Candes, J Romberg, T Tao. Robust UncertaintyPrinciples: Exact Signal Reconstruction From Highly Incomplete Frequency Information. IEEE Trans InformTheory; 52:489–509 (2006) 3. DL Donoho. Compressed sensing. IEEE Trans Inform Theory; 52:1289-1306 (2006)

Sparsity

Definition:

I vector x ∈ Rn

I k-sparse: at most k non-zero entries

Example:

Notation: Number of non-zero entries ‖x‖0 (this is not a norm)

Sparsity

x ∈ R2

(0

1.3

) (1.51.3

)

Set of sparse vectors is a (non-convex) union of subspaces

Denoising

I Sparse vector

I Densoising by hard-thresholding

I Densoising by soft-thresholding (shrinkage)

Denoising

I Sparse vector and random noise

I Densoising by hard-thresholding

I Densoising by soft-thresholding (shrinkage)

Compressed Sensing

Ingredients:

I Sparsity

I Incoherence



Regular Under-Sampling: Point-Spread-Function

I Regular under-sampling in Fourier domain

I Coherent aliasing in the time domain

Point-Spread-Function

Coherent Aliasing

I Regular under-sampling in Fourier domain

I Coherent aliasing in the time domain

1 12 2 33

Signal

Random Sampling: Point-Spread-Function

I Random sampling in Fourier domain

I Incoherent aliasing in time domain

noise-like artifacts

Point-Spread-Function

Incoherent Aliasing

I Random sampling in Fourier domain

I Incoherent aliasing in time domain

Signal

Linear Measurements

Ax = y x ∈ Rn, y ∈ Rm

Measurements: m >= n

Reconstruction: x = A†y

Matrix A should be nearly orthogonal.Example: Fourier Matrix

A AHA

Incoherent Linear Measurements

Ax = y x ∈ Rn and k-sparse, y ∈ Rm

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 AHA

Restricted Isometry Property

A n × p matrix and 1 ≤ s ≤ p

s-restricted isometry property: There is a constant δs such thatfor every s-sparse vector y :

(1− δs)‖y‖22 ≤ ‖Ay‖2

2 ≤ (1 + δs)‖y‖22

Compressed Sensing

Ingredients:

I Sparsity

I Incoherence



L1-Norm and Sparsity

(01

)

Set of vectors with ‖x‖0 ≤ 1 not convex! ⇒ L1 instead of L0

Linear Reconstruction

L2-regularization:

argminx‖Ax − y‖22 + α‖Wx‖2

2

Explicit Solution: (AHA + αWHW

)−1AHy

Nonlinear Reconstruction

L1-regularization:

argminx‖Ax − y‖22 + α‖Wx‖1

In general: no explicit solution!

L2-Norm vs L1-Norm

−1

0

1

−1 0 1

‖x‖2 = 1

‖x‖1 = 1

L1-Norm and Sparsity

Minimize ‖x‖pp subject to Ax = y

Ax = y

‖x‖22

Ax = y

‖x‖11

Inverse Problem with L1-Regularization

Minimize ‖x‖1 subject to ‖Ax − y‖2 ≤ ε

‖Ax − y‖2 = ε

‖x‖11

Linear Reconstruction

x = argminz |z − y |2 + λ|z |2

x =1

1 + λy

0

0−1

1

−11

Soft-Thresholding

x = argminz |z − y |2 + λ|z |

ηλ(x) =

x − λ x > λ0 |x | ≤ λx + λ x < −λ

0

0−1

1

−11

−λ

λ

Joint Thresholding

Shrink magnitude but keep phase/direction

I complex values:

ηλ(x) =

ηλ(|x |) x

|x | x 6= 0

0 x = 0

I vectors:

ηλ(x) =

ηλ(‖x‖2) x

||x ||2 x 6= 0

0 x = 0

Joint Thresholding

Shrink magnitude but keep phase/direction

Iterative Soft-Thresholding (IST)

Landweber1:

xn+1 = xn + µAH(y − Axn)

Iterative Soft-Thresholding2:

zn = xn + µAH(y − Axn)

xn+1 = ηλ(zn)

1. L Landweber. An iteration formula for Fredholm integral equations of the first kind. Amer J Math; 73:615–624(1951) 2. I Daubechies, M Defrise, C De Mol. An iterative thresholding algorithm for linear inverse problems witha sparsity constraint. Comm Pure Appl Math; 57:1413–1457 (2004)

Nonlinear Reconstruction: Iterative Soft-Thresholding

1. Data consistency: zn = xn + µAH(y − Axn)

2. Soft-thresholding: xn+1 = ηλ(zn)

iteration 0




iteration 1




iteration 2




iteration 9

Algorithms

I FOCUSS1

I Iterative Soft-Thresholding (IST)2

I Fast iterative Soft-Thresholding Algorithm3

I Split Bregman4

I Nonlinear Conjugate Gradients

I ...

1. IF Gorodnitsky, BD Rao. Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimumnorm algorithm. IEEE Trans Sig Proc 45:600–616 (1997) 2. I Daubechies, M Defrise, C De Mol. An iterativethresholding algorithm for linear inverse problems with a sparsity constraint. Comm Pure Appl Math; 57:1413–1457(2004) 3. A Beck, M Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAMJ Imaging Sci; 2:183–202 (2009) 4. T Goldstein and S Osher. The Split Bregman Method for L1-RegularizedProblems. SIAM J Imaging Sci; 2:323–343 (2009)

Statistical Model

Linear measurements contaminated by noise:

y = Ax + n

Gaussian white noise:

p(n) = N (0, σ2) with N (µ, σ2) =1

σ√

2πe−

(x−µ)2

2σ2

Probability of an outcome (measurement) given the image x :

p(y |A, x , λ) = N (Ax , σ2)

Bayesian Prior

I L2-Regularization: Ridge Regression

N (µ, σ2) =1

σ√

2πe−

(x−µ)2

2σ2 Gaussian prior

I L1-Regularization: LASSO

p(x |µ, b) =1

2be−|x−µ|

b Laplacian prior

I L2 and L1: Elastic net1

1. H Zou, T Hastie. Regularization and variable selection via the elastic net. J R Statist Soc B. 67:301–320 (2005)

Compressed Sensing in Magnetic Resonance Imaging

I Sparsity in medical imaging

I Incoherent sampling

I Iterative reconstruction

I Combination with parallel imaging

M Lustig, D. Donoho, JM Pauly. Sparse MRI: The application of compressed sensing for rapid MR imaging. MagnReson Med; 58:1182–1195 (2007)

Sparsity Transform

I Medical images are usually not sparse

I Need to apply sparsity transform

Sparsity:

I Wavelet Transform

I Total Variation

I Temporal Constraints

I Prior images

I Adapted dictionaries

I Low rank

I . . .

Wavelet Transform

I Orthonormal basis (or almost)

I Multi-scale transform

I Localized in frequency and space

I Compresses many signals/images into few coefficients

I Efficient computation: O(N)

I But: not shift-invariant (cycle spinning)

L1-regularization term in wavelet domain;

R(x) = ‖Wx‖1

W wavelet transform

Wavelet Transform

brain image wavelet transform

Signal concentrated in few coefficients!

Cycle spinning

Problem: Not shift-invariantSolution: Cycle spinning (or random shifting)

Artifacts

blurring blocky artifacts good quality

Problem: Not shift-invariantSolution: Random shifting (cycle spinning)

Total Variation

Definition: For a function f ∈ L1(Ω) with Ω an open set Ω ⊂ Rn,the total variation of f is:

TV f = sup

∫dx f divφ : φ ∈ C 1

c (Ω,Rn), ‖φ‖L∞(Ω) ≤ 1

I Denoising1

I Image Reconstruction2

1. LI Rudin, S Osher, E Fatemi. Nonlinear total variation based noise removal algorithms. Physica D; 60:259–268(1992) 2. D Geman, C Yang. Nonlinear image recovery with half-quadratic regularization. IEEE T ImageProcessing; 4:932–946 (1995)

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!

Total Variation

f (x)‖∇f (x)‖2 =√

|∂1f (x)|2 + |∂2f (x)|2

Total Variation: Approximations

Anisotropic Total Variation:

TV f =

∫dx√|∂1f (x)|2 + |∂2f (x)|2

≈∫

dx |∂1f (x)|+ |∂2f (x)|

Finite differences (backward):

∂1f (x1, x2) ≈ f (x1, x2)− f (x1 − h, x2)

|h|

∂2f (x1, x2) ≈ f (x1, x2)− f (x1, x2 − h)

|h|

h ≈ 1 Pixel

Total Variation: Staircase Artifacts

I Staircase artifacts (sparse differences)

I Solution: use of higher-order derivatives1

I Total Generalized Variation2,3

1. D Geman, C Yang. Nonlinear image recovery with half-quadratic regularization. IEEE T Image Processing;4:932–946 (1995) 2. K Bredies, K Kunisch, T Pock. Total generalized variation. SIAM J Imaging Sci; 3:492–526(2010) 3. F Knoll, K Bredies, T Pock, R Stollberger. Second order total generalized variation (TGV) for MRI.Magn Reson Med; 65:480–491 (2011)

Total Variation in Time Domain

f1 f2 f3 f4 f5 f6 f7 f8 f9 f10

t

TVt f ≈∫

dx∑l

|fl(x)− fl−1(x)|

HFL Chandarana, T Block, AB Rosenkrantz, R Lim, D Chu, DK Sodickson, R Otazo. Free-breathing dynamiccontrast-enhanced MRI of the liver with radial golden-angle sampling scheme and advanced compressed-sensingreconstruction. Proc. 20th ISMRM (2012)

GRASP: Compressed-Sensing Reconstruction

Otazo et al, MRM 2010: 64

Narrow data window Few spokes Flickering streak artifacts

Prior knowledge: Contrast uptake occurs “smoothly” and “continuously”

CS approach: Find solution that

CG SENSE-type reconstruction with temporal Total Variation (TV) constraint

has lowest flickering

matches data in all windows

Ground truth Ground truth13 spokes 13 spokes

Dynamic image series Temporal differences

Courtesy of Tobias Block, NYU

Example: GRASP Liver Imaging

Chandarana et al, ISMRM 2012: 5529

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 mm3

Temporal resolution 1.5 s

Top: GriddingBottom: GRASP

Courtesy of Tobias Block, NYU

Difference to Prior Image

Sparse difference to reference image g

R(x) = ‖x − x0‖1

Prior image: x0 composite image, previous frame, ...

1. GH Chen, J Tang, S Leng. Prior image constrained compressed sensing (PICCS): a method to accuratelyreconstruct dynamic CT images from highly undersampled projection data sets. Med Phys; 35:660–663 (2008) 2.A Fischer, F Breuer, M Blaimer, N Seiberlich, PM Jakob. Accelerated dynamic imaging by reconstructing sparsedifferences using compressed sensing. Proc 16th ISMRM (2008)

Dictionary Learning

=

Patch Reconstruction

Dictionary Learning Example-based

...

=a +b +cCourtesy of Patrick Virtue, UC Berkeley

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

Low-rank Approximation

Data matrix M ∈ Ct×s , e.g. time × space

Singular-Value-Decomposition: M = UΣVH

Low rank: rank M < K

Decomposition into K (temporal and spatial) basis functions:

M =∑K

ukσkvHk

1. Z Bo, JP Haldar. C Brinegarm ZP Liang. Low rank matrix recovery for real-time cardiac MRI. ISBI; 996-999(2010) 2. JP Haldar, ZP Liang. Spatiotemporal imaging with partially separable functions: A matrix recoveryapproach. ISBI 716–719 (2010) 3. SG Lingala, H Yue, E DiBella, M Jacob. Accelerated Dynamic MRI ExploitingSparsity and Low-Rank Structure: k-t SLR. IEEE Trans Med Imag; 30:1042–1054 (2011) 4. R Otazo, E Candes,DK Sodickson. Low-rank and sparse matrix decomposition for accelerated DCE-MRI with background and contrastseparation. ISMRM Workshop on Data Sampling and Image Reconstruction. Sedona (2013)

Low-rank + Sparse Reconstruction of Cardiac Cine

I 6-fold acceleration (ky-t random undersampling)

I Temporal resolution: 40 ms

I Spatial resolution: 1.3x1.3x3 mm3

I Std. CS with temporal FFT

CS L+S L S

Courtesy of Ricardo Otazo, NYU

Parallel MRI

Goal: Reduction of measurement time

I Subsampling of k-space

I Simultaneous acquisition with multiple receive coils

I Coil sensitivities provide spatial information

I Compensation for missing k-space data

1. DK Sodickson, WJ Manning. Simultaneous acquisition of spatial harmonics (SMASH): Fast imaging withradiofrequency coil arrays. Magn Reson Med; 38:591–603 (1997) 2. KP Pruessmann, M Weiger, MB Scheidegger,P Boesiger. SENSE: Sensitivity encoding for fast MRI. Magn Reson Med; 42:952–962 (1999) 3. MA Griswold, PMJakob, RM Heidemann, M Nittka, V Jellus, J Wang, B Kiefer, A Haase. Generalized autocalibrating partiallyparallel acquisitions (GRAPPA). Magn Reson Med; 47:1202–10 (2002)

Parallel MRI: Undersampling

Undersampling Aliasing

kread

kphase

kphase

kpartition

Parallel MRI as Inverse Problem

I Signal from multiple coils (image x , sensitivities cj):

sj(t) =

∫Vd~r x(~r)cj(~r)e−i~r ·

~k(t)

I Assumption: known sensitivities cj⇒ linear relation between image x and data y

I Image reconstruction is a linear inverse problem:

Ax = y

1. JB Ra and CY Rim, Magn Reson Med 30:142–145 (1993) 2. KP Pruessmann, M Weiger, MB Scheidegger, PBoesiger. Magn Reson Med 4:952–962 (1999)

Parallel MRI: Regularization

I General problem: bad condition

I Noise amplification during image reconstruction

I L2 regularization (Tikhonov):

argminx‖Ax − y‖22 + α‖x‖2

2 ⇔ (AHA + αI )x = AHy

I Influence of the regularization parameter α:

small medium large

Parallel MRI: Nonlinear Regularization

I Good noise suppression

I Edge-preserving

⇒ Sparsity, nonlinear regularization

argminx‖Ax − y‖22 + αR(x)

Regularization: R(x) = TV (x), R(x) = ‖Wx‖1, . . .

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 andSoftware; 20:545–558 (2005) 2. B Liu, L Ying, M Steckner, J Xie, J Sheng. Regularized SENSE reconstructionusing iteratively refined total variation method. ISBI; 121-123 (2007) 3. A Raj, G Singh, R Zabih, B Kressler, YWang, 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 toAutocalibrated Parallel Imaging. ISMRM 1479 (2008) 5. . . .

Nonlinear Inversion with Non-Quadratic Regularization

Iteratively Regularized Gauss Newton Method (IRGNM)

xn+1 − xn = argminδx‖DFH(xn)δx + F (xn)− y‖22 + αnR(δx + xn)

Previously: Image regularized with L2-norm

R(x) = ‖ρ‖22 + ‖(1 + s|~k|2)lFTcj‖2

2

Now: Different regularization terms

R(x) = R(ρ) + ‖(1 + s|~k |2)lFTcj‖22

Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).

Nonlinear Inversion

I Quality of the reconstructed images can be improved

I Acceleration: 3 x 2

I L1-Wavelet: Cohen-Daubechies-Feauveau 9/7

Compressed Sensing and Parallel Imaging

I Parallel imaging

I Sparsity, nonlinear regularization


1. KT Block, M Uecker, J Frahm. Undersampled radial MRI with multiple coils. Iterative image reconstructionusing a total variation constraint. Magn Reson Med; 57:1086–1098 (2007) 2. C Zhao, T Lang, J Ji. CompressedSensing Parallel Imaging. Proc. 16th ISMRM; 1478 (2008) 3. B Wu, RP Millane, R Watts, P Bones. Applyingcompressed sensing in parallel MRI. Proc. 16th ISMRM; 1480 (2008) 4. KF King. Combining compressed sensingand 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 bregmaniterations. Magn Reson Med 61:145–152 (2009) 7. . . .

Sampling Schemes

uniform random Poisson-disc

Poisson-disc sampling:

I Minimum distance to exploit parallel imaging

I Incoherence for compressed sensing

M Murphy, K Keutzer, SS Vasanawala, M Lustig. Clinically feasible reconstruction time for L1-SPIRiT parallelimaging and compressed sensing MRI. Proc 18th ISMRM; 4854 (2010)

Variable-Density Sampling

variable-density Poisson-disc radial

Advantages:

I Auto-calibration for parallel imaging

I Graceful degradation


Regularized SENSE:

argminx‖PFCx − y‖22 + R(x)

P projection onto samples, F Fourier transform, C coil sensitivities,

R(x) regularization

Simple (but slow): IST with R(x) = ‖Wx‖1

zn = xn + µCHFHP(y − PFCxn)

xn+1 = W−1ηλ(Wzn)


Linear reconstruction:

I R(x) = 0 ⇒ No regularization

I R(x) = ‖x − x0‖22 ⇒ Tikhonov

Nonlinear reconstruction:

I R(x) = TV (x) ⇒ Total Variation

I R(x) = ‖Wx‖1 ⇒ L1-Wavelet

I R(x) = ‖x − x0‖1 ⇒ Prior Image

I · · ·

Incoherent Sampling ⇒ Compressed Sensing

Example: Undersampled Radial with Total Variation

KT Block, M Uecker, J Frahm. Undersampled Radial MRI with Multiple Coils. Iterative Image ReconstructionUsing a Total Variation Constraint. Magn Reson Med 57:1086-1098 (2007)

Nonlinear Inverse Reconstruction with Variational Penalties

Experiments:I Siemens Tim Trio 3 T, 12-channel head coilI 3D FLASH, acceleration: R = 4 (pseudorandom sampling)

Knoll F, Clason C, Bredies K, Uecker M, Stollberger R, Magn Reson Med, 67:34-41 (2012).

`1-SPIRiT

I Robust auto-calibrating parallel MRI

I Calibration of coil-by-coil operator G in k-space

Optimization problem:

arg minx

α‖Px − y‖22︸︷︷︸

data consistency

+ β‖(G − Id)x‖22︸︷︷︸

calibration consistency

+ γR(x)︸︷︷︸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. MagnReson Med; 64:457–471 (2010)

`1-SPIRiT

“POCS-type” algorithm (α = β =∞):

yn = y + (Id − P)xn data consistency

zn = Gyn calibration consistency

xn+1 = W−1ηλ(Wzn) soft-thresholding

`1-ESPIRiT

ESPIRiT:

I Flexibility and efficiency of SENSE

I Robustness of GRAPPA/SPIRiT

Algorithm:

I Calibration of coil-by-coil operator in k-space

I Sensitivity maps from eigendecomposition

I Extended (“soft”) SENSE reconstruction

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

ESPIRiT: Reconstruction with Multiple Maps

Relaxed (“soft”) SENSE using multiple maps.

I Simultaneous reconstruction of multiple images mj

I Data consistency:

N∑i=1

‖yi − PFM∑j=1

S ji m

j‖22︸︷︷︸

“soft” SENSE

+ Q(m1, · · · ,mM)

mj images, S j multiplication with maps, F Fourier transform,

P sampling operator, y data, Q regularization

I Image combination (e.g. root of sum of squares)

Example: `1-ESPIRiT

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

Conclusion

Compressed Sensing: Theory

I Sparsity


I Nonlinear reconstruction

Application to Magnetic Resonance Imaging

I Sparsity transform (adapted to application)

I Incoherent sampling schemes

I Efficient iterative reconstruction

I Combination with parallel imaging

Software

http://www.eecs.berkeley.edu/~mlustig/Software.html

http://www.eecs.berkeley.edu/~uecker/toolbox.html

http://web.eecs.umich.edu/~fessler/code/index.html

http://www.imt.tugraz.at/index.php/research/

agile-gpu-image-reconstruction-library

http://gadgetron.sourceforge.net/

http://codeare.org/

http://impact.crhc.illinois.edu/mri.aspx

http://www.eecs.berkeley.edu/~mlustig/Software.html

http://www.eecs.berkeley.edu/~uecker/toolbox.html

http://web.eecs.umich.edu/~fessler/code/index.html

http://www.imt.tugraz.at/index.php/research/agile-gpu-image-reconstruction-library

http://www.imt.tugraz.at/index.php/research/agile-gpu-image-reconstruction-library

http://gadgetron.sourceforge.net/

http://codeare.org/

http://impact.crhc.illinois.edu/mri.aspx

Acknowledgements

I Michael Lustig, University of California, Berkeley

I Tobias Block, New York University

I Ricardo Otazo, New York University

I Patrick Virtue, University of California, Berkeley

I Jose Caballero, Imperial College London

Final Projects

I Presentation: ≈ 10 minutes, 28th April

I Report

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