Transcript Video Based Face Analysis

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?