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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?