Junzhou Huang, Shaoting Zhang, Dimitris Metaxas

CBIM, Dept. Computer Science, Rutgers University

Efficient MR Image Reconstruction for Compressed MR Imaging

OutlineIntroduction

Compressed MR Image Reconstruction

Related Work Different algorithms for this problem

Proposed Algorithms Fast Composite Splitting Algorithm (FCSA)

Experimental Results Visual and Statistical Comparisons

Conclusions

Introduction: Compressive Sensing

CompressXp k

p k

Xp n

p n

Random Measurement y=R�x

Traditional Data Acquisition

Compressive Sensing Data Acquisition

Sample

Decompress Receive

Transmit

Transmit

ReceiveCompressed Reconstruction

Compressive sensing is very important

k<<p

O(k ㏒ (p/k))

Introduction: Compressive Sensing MRI[Magnetic Resonance in Medicine, 2007]

uF1

uF

If image is Sparsely represented

by Wavelet

WT

Compressed MRI Reconstruction

Key problem of MRI: reducing the imaging & reconstructing time

Compressed MRI ReconstructionProblem Formulation

Where x is the unknown MR image to be reconstructed R is a partial Fourier transform b is the under-sampled Fourier measurements 𝝓 is the wavelet transform α and β are two positive weight parameters

Loss function f(x), convex

smooth

Total variation norm g1(x), convex non-

smooth

L1 norm g2(x), convex non-smooth

Related WorkRelated work on compressed MRI reconstruction

Conjugate Gradient (SparseMRI) [Lustig, MRM’07] Operator Splitting (TVCMRI) [Ma, CVPR’08] Variable Splitting (RecPF) [Yang, JSTSP’09]

Related work on general optimization Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)

[Beck, JIS’09] 1st order gradient algorithm with best convergence rate

O(1/k2)

Problem: min{ F(x)=f(x)+g(x) } f(x) convex and smooth g(x) convex and non-smooth

Theorem 1: Suppose {xk} are obtained by FISTA, Error Bound:

𝜺=F(xk)-F(x*) ~ O(1/k2)

FISTA [Beck, SIAM-JIS’09]

Bottleneck: Step2 g(x)=𝜶||x||TV , [Beck, TIP’09]

g(x)=𝜷||𝜱x||1 [Beck, JIS’09]

g(x)=𝜶||x||TV+𝜷||𝜱x||1Proximal gradient descent

O(p)

O(plog(p))

Solution for Step 2:

Where: g(x)=𝜶||x||TV+𝜷||𝜱x||1 Average two independent

solutions for TV and L1 norms

Theorem 2: Suppose {xj} are obtained by CSD, It will strongly converge to

true solution Refer to our papers for

details of proofs

Our Contribution:Composite Splitting Denoising (CSD)

Compute proximal gradient with TV norm

and L1 norm independently

Averaging two independent

solutions

Additional Contribution:Fast Composite Splitting Algorithm

(FCSA)Compressed MRI reconstruction

FCSA: We modify the FISTA to obtain the FCSA by using the CSD

algorithm instead of Step 2 of the FISTA

Theorem 3: Suppose {xk} are obtained by FCSA , Error bound: 𝜺=F(xk)-F(x*) ~ O(1/k2) proved by combining the Theorem 1 and Theorem 2

(Refer to our papers for details of proofs)

FCSA for MRI ReconstructionIn the kth iteration:

x1k=argminx {||x-xg||2+ 4𝝆𝜶||x||TV}

x2k=argminx {||x-xg||2+ 4𝝆𝜷||𝜱x||1}

xk=(x1k+x2

k�)/2

O(plog(p))

O(p)

O(p)

O(plog(p))

O(p)

CSA, without acceleration step: 𝜺 ~ O(1/k)

Total computations O(plog(p))

Gradient Descent

Proximal gradientaccording to TV norm

Proximal gradient according to L1 norm

Averaging

AccelerationStep

FCSA ,with acceleration step: 𝜺 ~ O(1/k2)

Experiments

Implementation MATLAB, 2.4GHz PC Codes for others are downloaded from their websites

Comparisons with Conjugate Gradient (CG) [Lustig, MRM’07] Operator Splitting (TVCMRI) [Ma, CVPR’08] Variable Splitting (RecPF) [Yang, JSTSP’09]

Sampling Randomly sampling in the frequency domain White color denotes being sampled (20%)

Comparisons on Brain MR Image

(a) Original (b) CG [Lustig07] (c) TVCMRI [Ma08]

(d) RecPF [Yang09] (e) CSA(proposed) (f) FCSA(proposed)

256 x 256

SNR

CG 8.71db

TVCMRI 12.12db

RecPF 12.40db

CSA18.68db

FCSA 20.35db

Comparisons on Artery MR Image

(a) Original (b) CG [Lustig07] (c) TVCMRI [Ma08]

(d) RecPF [Yang09] (e) CSA(proposed) (f) FCSA(proposed)

256 x 256

SNR

CG 11.73db

TVCMRI 15.49db

RecPF 16.05db

CSA 22.27db

FCSA 23.70db

0 0.5 1 1.5 2 2.5 34

6

8

10

12

14

16

CPU Time (s)

SN

R

CG [Lustig 07]TVCMRI [Ma 08]RecPF [Yang 09]CSA [Huang 10]FCSA [Huang 10]

(a) Artery image (b) Brain image

Comparisons (CPU-Time vs. SNR)Statistical results after 100 runs

0 0.5 1 1.5 2 2.5 3 3.56

8

10

12

14

16

18

20

22

24

CPU Time (s)

SN

R

CG [Lustig 07]TVCMRI [Ma 08]RecPF [Yang 09]CSA [Huang 10]FCSA [Huang 10]

SN

R(d

b)

CPU-time(s)

SN

R(d

b)

CPU-time(s)

Visual Comparisons on Full Body MR Image

(a) Original (b) TVCMRI (c) RecPF (d) CSA (e) FCSA

1024 x 256, sampling ratio 25%

Comparisons with Different Sampling Ratios

Exp I: 20% Exp II: 25% Exp III: 36%

TVCMRI 10.88db 12.67db 15.82db

RecPF 11.06db 13.02db 16.12db

CSA 16.36db 18.07db 21.98db

FCSA 17.82db 19.28db 23.66db

All methods run 50 iterations

ContributionsWe proposed a new algorithm for compressed MRI

reconstruction. It theoretically converges with accuracy ε ~ O(1/k2) after k iterations.

The computation complexity is only O(plog(p)) for each iteration of the proposed algorithm, where p is the dimension of MR images

The proposed algorithm is very efficient in practice and impressively outperforms previous methods. It is fast enough to be used in MRI scanners.

Offers near future potential of real time image reconstruction

HUGE IMPACT

Patent filed on method and MATLAB code

Thank You!

Any Questions?