Ricardo Otazo, PhD [email protected]
G16.4428 – Practical Magnetic Resonance Imaging II Sackler Institute of Biomedical Sciences New York University School of Medicine
Compressed Sensing
Compressed sensing: The big picture • Exploit image sparsity/compressibility to reconstruct
undersampled data
– Nyquist: same FOV, same number of samples – Common sense: fewer non-zero pixels, fewer samples
Original Gradient (sparse)
Which image would require fewer samples?
Image compression • Essential tool for modern data storage and transmission • Exploit pixel correlations to reduce number of bits • First reconstruct, then compress
Fully-sampled acquisition (Nyquist rate)
Sparsifying transform (e.g. wavelets)
Store or transmit non-zero coefficients only
Recover image from sparse coefficients
10-fold compression
3 MB 300 kB
Image compression is inefficient • Question for image compression
– Why do we need to acquire samples at the Nyquist rate if we are going to throw away most of them?
• Answer
– We don’t. Do compressed sensing instead – Build data compression in the acquisition – First compress, then reconstruct
Candès E,Romberg J,Tao T. IEEE Trans Inf Theory 2006; 52(2): 489-509 Donoho D. IEEE Trans Inf Theory 2006; 52(4): 1289-1306.
Compressed sensing components • Sparsity
– Represent images with a few coefficients – Transform: wavelets, gradient, etc.
• Incoherence – Sparsity is preserved – Noise-like aliasing artifacts
• Non-linear reconstruction – Denoise aliasing artifacts
Simple compressed sensing example k-space Image space
Regular undersampling
Random undersampling
No undersampling
Coherent aliasing
Incoherent aliasing
Courtesy of Miki Lustig
Simple compressed sensing example
Threshold
- Threshold
Final sparse solution
Undersampled signal
Courtesy of Miki Lustig
Sparsifying transforms • Finite differences
( ) ( ) ( )1−−= nxnxny
−−
−=
=
1111
11:
D
Dxyform matrix In
• Total variation
( ) ( ) ( )∑=
−−=N
nnxnxxTV
2
1 ( )1
minmin DxxTV =
Sparsifying transforms • Wavelets
– Space-frequency localization – More efficient than sines and cosines to represent
discontinuities
Sparsifying transforms • Wavelets
– Multiresolution image representation – Recursive application of the wavelet function modified by the
scaling function (each resolution is twice of that of the previous scale) Daubechies 4-tap wavelet transform Brain image
Incoherence • Incoherence property: the acquisition domain and the
sparse domain must have low correlation E: acquisition matrix W: sparsifying transform matrix <ai,wj> is small for all basis functions
Transform Point Spread Function (TPSF) Ems =
Wpm =
• Encoding model:
• Representation model: (p is sparse)
• Transform point spread function for position r
EWps =
( )rHHrTPSF EWEW=)(
Transform Point Spread Function (TPSF) • Tool to check incoherence in the sparse domain • Computation
– Apply inverse FFT to sampling mask (1=sampled, 0=otherwise) – Apply sparsifying transform
• Incoherence = ratio of the main peak to the std of the pseudo-noise (incoherent artifacts)
Sampling mask PSF TPSF (Wavelet)
Incoherence of k-space trajectories
Lustig M, Donoho DL, Santos JM, Pauly JM. Compressed Sensing. MRI, IEEE Signal Processing Magazine, 2008; 25(2): 72-82
Incoherent sampling patterns ? • What are the best samples?
Sampling pattern Image domain Sparse domain
Random undersampling
(Cartesian)
Variable-density random
undersampling (Cartesian)
Regular undersampling
(radial)
Dimensionality and incoherence
64x64 space
Sampling pattern (R=4) PSF
I=71
128x128 space
I=140
Sparsity Data consistency
Compressed sensing reconstruction
• Acquisition model: s = Em s : acquired data E : undersampled Fourier transform m : image to reconstruct
• Sparsifying transform: W
sEmWmm
= subject to min1
∑=
=N
nix
11 x (l1- norm of x)
Compressed sensing reconstruction • Unconstrained optimization (in practice)
• Regularization parameter λ – Trade-off between data fidelity and removal of
aliasing artifacts – High λ: artifact removal and denoising at the expense
of image corruption (blurring, ringing, blocking, etc) – Low λ : no image corruption, but residual aliasing
1
2
2min WmsEm
mλ+−
Compressed sensing reconstruction • Gradient descent
– Cost function:
– Iterations: 1
2
2)( WmsEmm λ+−=C
)(1 nnn C mmm ∇−=+ α
( ) nH
nH
nC WmMWsEmEm 1)( −+−=∇ λ
M is a diagonal matrix:
Classwork: compute the gradient of C(m)
Hint: µ+= xxx *
Compressed sensing reconstruction • Proximal gradient descent (iterative soft-thresholding)
– Soft-thresholding operation
– Our gradient gradient-descent algorithm becomes
( )
>−
≤=
λλ
λλ
xxxx
xxS
if ,
if ,0),(
-5 0 5 1 2 3 4 -1 -2 -3 -4 -4
-3
-2
-1
0
1
2
3
4
x
S(x
, 1 ) ( )[ ]( )[ ]λ,1
1 sEmEmWWm −−= −+ n
Hnn S
Original
Iterative soft-thresholding
With noise
• Sparse signal denoising
Original
Iterative soft-thresholding • Sparse signal denoising
After 5 iterations
Original
Iterative soft-thresholding • Sparse signal denoising
After 15 iterations
Iterative soft-thresholding in Matlab function m=cs_ista(s,lambda,nite) % W and Wi are the forward and inverse sparsifying % transforms m = ifft2_mri(s); % initial solution sm = s~=0; % sampling mask for ite=1:nite, % enforce sparsity m = Wi (SoftT(W(m),lambda)); % enforce data consistency m = m – ifft2_mri( (fft2_mri(m).*sm – s).*sm ); end % soft-thresholding function function y=SoftT(x,p) y = (abs(x)-p).*x./abs(x).*(abs(x)>p); end
Iterative soft-thresholding
Iterations Initial solution
Inverse FT of the zero-filled k-space
After soft-thresholding
),( λiwS
k-space representation
Sparse representation
After data consistency
Iterative soft-thresholding (R=3) Initial solution
Inverse FT of the zero-filled k-space
After 30 iterations
Iterative soft-thresholding
How many samples do we need in practice? • ~4K samples for a K-sparse image
N = 6.5x104 K = 6.5x103
Compression ratio C = 10
R = 2.5 (safe undersampling factor)
• Rule of thumb: about 4 incoherent samples for each sparse coefficient
Image quality in compressed sensing • SNR is not a good metric
• Loss of small coefficients in the sparse domain
– Loss of contrast – Blurring – Blockiness – Ringing – Images look more synthetic
R=2 R=4
Combination of compressed sensing and parallel imaging
Why would CS & PI make sense? • Image sparsity and coil-sensitivity encoding are
complementary sources of information
• Compressed sensing can regularize the inverse problem in parallel imaging
• Parallel imaging can reduce the incoherent aliasing artifacts
Why would CS & PI make no sense? • CS requires irregular k-space sampling while PI requires
regular k-space sampling Non-linear reconstruction in CS can prevent high g-
factors due to irregular sampling
• Incoherence along the coil dimension is very limited because the sampling pattern for all coils is the same
It does not matter, because we are not exploiting incoherence along the coil dimension
Approaches for CS&PI • CS with SENSE parallel imaging model
– Muticoil radial imaging with TV minimization Block T et al. MRM 2007
– SPARSE-SENSE Liang D et al. MRM 2009 Otazo R et al. MRM 2010
• CS with GRAPPA parallel imaging model
– l1-SPIRiT Lustig M et al. MRM 2010
CS with SENSE parallel imaging model • Enforce joint sparsity on the multicoil system
Coil 1
Coil 2
Coil N
s1
s2
sN
s CS & PI m
W
sparsifying transform
E
coil sensitivities
{ }2
2 1min λ+
mEm - s Wm
SENSE data
consistency
Joint sparsity m represents
the contribution from all coils
How does PI help CS? • Reduces incoherent aliasing artifacts
1 coil 8 coils
y y
Point spread function (PSF) Sampling pattern (R=4)
ky
kx
How does PI help CS? • Reduces incoherent aliasing artifacts
1 coil 8 coils
Which one would be easier to reconstruct?
Limits of Performance • Compressed sensing alone
– ~4K samples for a K-sparse image
• Compressed sensing & parallel imaging – Close to K samples for large number of coils
Otazo et al. ISMRM 2009; 378
Noise-free simulation K = 32 Nc = number of coils
UISNR coil geometry
CS & PI for 2D imaging
• Siemens 3T Tim Trio • 12-channel matrix coil array • 4-fold acceleration
GRAPPA Coil-by-coil CS Joint CS & PI
Application to musculoskeletal imaging
• Conventional clinical protocol for knee imaging
• New compressed sensing protocol (Sparse-SPACE)
3D-FSE
4-5 minutes
NYU-Siemens collaboration – to be presented at RSNA 2014
Multislice 2D-FSE (axial)
Multislice 2D-FSE (coronal)
Multislice2D-FSE (sagittal)
10-12 minutes
Application to musculoskeletal imaging
• First clinical study
– 50 patients
– >90% agreement between conventional 2D-FSE and Sparse-SPACE
2D-FSE (10:56 min)
Sparse-SPACE (4:36 min)
NYU-Siemens collaboration – to be presented at RSNA 2014
Summary • Compressed sensing
– New sampling theorem • Information rate (sparsity) rather than pixel rate (bandwidth)
– Ingredients • Sparsity • Incoherence • Non-linear reconstruction
• Fast imaging tool for MRI
– MR images are naturally compressible – Data acquisition in k-space facilitates incoherence – Can be combined with parallel imaging
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