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Efficient MR Image Reconstruction for
Compressed MR Imaging
Junzhou Huang, Shaoting Zhang, Dimitris Metaxas
CBIM, Dept. Computer Science, Rutgers University
Outline
Introduction
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
Compressive sensing is very important
Traditional Data Acquisition
X
Sample
p
p
k
Compress
k
Decompress
Compressive Sensing Data Acquisition
X
p
p
k<<p
Transmit
Receive
O(k㏒(p/k))
Random Measurement y=Rx
n
Transmit
Compressed Reconstruction
n
Receive
Introduction: Compressive Sensing MRI
[Magnetic Resonance in Medicine, 2007]
Key problem of MRI: reducing the imaging & reconstructing time
Fu
Fu
1
WT
If image is
Sparsely represented
by Wavelet
Compressed MRI
Reconstruction
Compressed MRI Reconstruction
Problem 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
L1 norm
g2(x),
Loss function
f(x),
Total
variation norm
g1(x),
convex non-smooth
convex smooth
convex non-smooth
Related Work
Related 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)
FISTA [Beck, SIAM-JIS’09]
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)
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||1
O(p)
O(plog(p))
Proximal
gradient descent
Our Contribution:
Composite Splitting Denoising (CSD)
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
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 Reconstruction
In the kth iteration: Total computations O(plog(p))
O(plog(p))
Gradient Descent
Proximal gradient
according to TV norm
Proximal gradient
according to L1 norm
xk=(x1k+x2k)/2
Averaging
O(p)
x1k=argminx {||x-xg||2+ 4𝝆𝜶||x||TV
}
k=argmin {||x-x ||2+ 4𝝆𝜷||𝜱x||
O(p)
x
2
x
g
1
CSA, without acceleration step:
𝜺 ~ O(1/k)
FCSA ,with acceleration
step:
𝜺 ~ O(1/k2)
O(plog(p))
Acceleration
Step
O(p)
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
256 x 256
SNR
(a) Original
(b) CG [Lustig07]
CG
8.71db
TVCMRI
12.12db
RecPF
12.40db
(c) TVCMRI [Ma08]
CSA
FCSA
(d) RecPF [Yang09] (e) CSA(proposed)
(f) FCSA(proposed)
18.68db
20.35db
Comparisons on Artery MR Image
256 x 256
SNR
(a) Original
(b) CG [Lustig07]
(d) RecPF [Yang09] (e) CSA(proposed)
CG
11.73db
TVCMRI
15.49db
RecPF
16.05db
CSA
22.27db
FCSA
23.70db
(c) TVCMRI [Ma08]
(f) FCSA(proposed)
Comparisons (CPU-Time vs. SNR)
Statistical results after 100 runs
24
16
22
14
18
12
SNR
SNR(db)
SNR
SNR(db)
20
16
10
14
CG [Lustig 07]
TVCMRI [Ma 08]
RecPF [Yang 09]
CSA [Huang 10]
FCSA [Huang 10]
12
10
8
6
0
0.5
1
1.5
2
2.5
CPU Time (s)
CPU-time(s)
(a) Artery image
3
CG [Lustig 07]
TVCMRI [Ma 08]
RecPF [Yang 09]
CSA [Huang 10]
FCSA [Huang 10]
8
6
3.5
4
0
0.5
1
1.5
2
CPU
Time (s)
CPU-time(s)
(b) Brain image
2.5
3
Visual Comparisons on Full Body MR Image
1024 x 256, sampling ratio 25%
(a) Original
(b) TVCMRI
(c) RecPF
(d) CSA
(e) FCSA
Comparisons with Different Sampling Ratios
All methods run 50 iterations
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
Contributions
We
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?