Practical Magnetic Resonance Imaging II Sackler Institute...

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: sparse image would require fewer samples

Original Gradient

Which image would require fewer samples?

Compressed sensing: The big picture

• Ingredients

– Sparsity/compressibility

– Incoherence

– Non-linear reconstruction

Compressed sensing in the news (Wired magazine)

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 sparse coefficients only

Recover image from sparse coefficients

10-fold compression

3 MB 300 kB

Compressed sensing

• 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

k-space Image

How to undersample

k-space?

How to reconstruct the sparse

representation?

Sparse domain

CS will reconstruct the sparse representation

Intuitive example: incoherence

k-space Image space

Regular undersampling

Random undersampling

No undersampling

Coherent aliasing

Incoherent aliasing

Courtesy of Miki Lustig

Intuitive example: non-linear reconstruction

Threshold

- Threshold

Final sparse solution

Undersampled signal

Courtesy of Miki Lustig

More on incoherence

• Incoherence property: the acquisition domain and the sparse domain must be almost uncorrelated

A: acquisition matrix

W: sparsifying transform matrix

<ai,wj> is small for all basis functions

• In most of the cases, this is satisfied if the acquisition matrix has an autocorrelation matrix AHA very close to an identity matrix

More on incoherence

• MRI case: A is the DFT matrix

Full DFT matrix AHA

More on incoherence

• Regular undersampling

AHA

A

• Random undersampling

Bad CS acquisition matrix! Good CS acquisition matrix!

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

What kind of random undersampling pattern?

Undersampling pattern Image domain

Uniform random

distribution

Variable density

distribution

Fully-sampled k-space IFT

IFT

It matches k-space signal intensity distribution

Dimensionality and incoherence

64x64 space

Sampling pattern (R=4) PSF

I=71

128x128 space

I=140

Enforcing a sparse solution

• Minimum l2-norm (signal energy)

– Smooth, non-sparse solution

• Minimum l0-norm (number of non-zero coefficients)

– Sparse solution

– K+1 samples for a K-sparse signal

– Computationally intractable (combinatorial search)

• Minimum l1-norm (sum of absolute values)

– Sparse solution

– About Klog(N/K) samples for a K-sparse signal

– Computationally tractable (convex problem)

Practical reconstruction algorithms

• Unconstrained optimization

– Non-linear conjugate gradient and many others

• Projection onto convex sets (POCS)

– Iterative soft-thresholding

• Greedy methods

– Orthogonal matching pursuit (OMP)

1

2

2minargˆ WmsFmm

Iterative soft-thresholding

Original


• POCS-type method

• Constraints

– Data consistency

– Sparsity (soft-thresholding)

xx

x

x

x

xS 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 )

With noise



• Constraints



xx

x

x

x

xS 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 )


Original

After 5 iterations



• Constraints



xx

x

x

x

xS 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 )


Original

After 15 iterations


Iterations Initial solution

Inverse FT of the zero-filled k-space

After soft-thresholding

),( iwS

k-space representation

Sparse representation

After data consistency

Insert the acquired data back

Iterative soft-thresholding (R=3)

Initial solution

Inverse FT of the zero-filled k-space

After 30 iterations


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

~2.5-fold reduction in the number of samples

• 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 (CS) and coil-sensitivity encoding (PI) are complementary sources of information

• CS might serve as a regularizer for the PI inverse problem

• PI might produce more incoherent samples for CS

Why would CS & PI make no sense?

• CS requires irregular (random) sampling of k-space while PI requires regular sampling of k-space

The noise regularization feature of CS reconstruction 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. Combining multiple coil data adds more k-space samples which reduce incoherent aliasing artifacts

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

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

CS & PI for multislice imaging

• Multislice 2D imaging

• 3D wavelet transform

• Different ky random undersampling pattern per slice

Coil-by-coil CS Joint CS & PI

z=5

z=6

z=7

CS & PI 3D MR angiography (MRA)

• Angiograms are very sparse

• 3D scan: ky-kz acceleration

Coil-by-coil CS Joint CS & PI

x10 Undersampled

MIP x

y

z

y

Summary • Compressed sensing (Donoho) / compressive sampling (Candes)

– Data compression directly in the acquisition

– Exploit compressibility/sparsity to reduce number of samples

– Ingredients

• Sparsity

• Incoherence

• Non-linear reconstruction

• Fast imaging tool for MRI

– MR images are inherently compressible

– Incoherence is easily achieved since acquisition is performed in the Fourier domain

Summary

• Combination of compressed sensing and parallel imaging

– Synergy

– This is the way to go, since we have coil arrays everywhere

• One of the hottest topics of MRI research

– It will change the way we do MRI today (Dan Sodickson)

• Tons of research opportunities

– Sparsifying transforms (dictionary learning)

– Incoherent acquisitions (non-Cartesian)

– Higher dimensional imaging (more compressible)

– Fast implementation of reconstruction algorithms

Practical Magnetic Resonance Imaging II Sackler Institute...

Documents

Transcript of Practical Magnetic Resonance Imaging II Sackler Institute...