Compressed Sensing - CAI2R · – Recursive application of the wavelet function modified by the...

Ricardo Otazo, PhD ricardo.otazo@nyumc.org

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