1. Junzhou Huang, Shaoting Zhang, Dimitris Metaxas CBIM, Dept. Computer Science, Rutgers University Efficient MR Image Reconstruction for Compressed MR Imaging

2. 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

3. Introduction: Compressive Sensing Compress X p k p k X p n p n Random Measurement y=Rx Traditional Data Acquisition Compressive Sensing Data Acquisition Sample Decompress Receive Transmit Receive Compressed Reconstruction Compressive sensing is very important k<<p O(k (p/k))

4. Introduction: Compressive Sensing MRI [Magnetic Resonance in Medicine, 2007] If image is Sparsely represented by Wavelet WT Compressed MRI Reconstruction Key problem of MRI: reducing the imaging & reconstructing time

5. 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 is the wavelet transform α and β are two positive weight parameters Loss function f(x), convex smooth Total variation norm g 1 (x), convex non-smooth L1 norm g 2 (x), convex non-smooth

6. Related Work Related work on compressed MRI reconstruction Conjugate Gradient (SparseMRI) [Lustig, MRM07] Operator Splitting (TVCMRI) [Ma, CVPR08] Variable Splitting (RecPF) [Yang, JSTSP09] Related work on general optimization Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) [Beck, JIS09] 1 st order gradient algorithm with best convergence rate O(1/k 2 )

7. Problem: min{ F(x)=f(x)+g(x) } f(x) convex and smooth g(x) convex and non-smooth Theorem 1: Suppose {x k } are obtained by FISTA, Error Bound: = F(x k )-F(x*) ~ O(1/k 2 ) FISTA [Beck, SIAM-JIS09] Bottleneck: Step2 g(x)= ||x|| TV, [Beck, TIP09] g(x)= || x|| 1 [Beck, JIS09] g(x)= || x|| TV + || x|| 1 Proximal gradient descent O(p) O(plog(p))

8. Solution for Step 2: Where: g(x)=|| x|| TV +||x|| 1 Average two independent solutions for TV and L1 norms Theorem 2: Suppose {x j } 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

9. 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 {x k } are obtained by FCSA, Error bound: =F(x k )-F(x*) ~ O(1/k 2 ) proved by combining the Theorem 1 and Theorem 2 (Refer to our papers for details of proofs)

10. FCSA for MRI Reconstruction In the k th iteration: x 1 k =argmin x {||x-x g || 2 + 4 ||x|| TV } x 2 k =argmin x {||x-x g || 2 + 4 || x|| 1 } x k =(x 1 k +x 2 k )/2 O(plog(p)) O(p) O(plog(p)) O(p) CSA, without acceleration step: ~ O(1/k) Total computations O(plog(p)) Gradient Descent Proximal gradient according to TV norm Proximal gradient according to L1 norm Averaging Acceleration Step FCSA,with acceleration step: ~ O(1/k 2 )

11. Experiments Implementation MATLAB, 2.4GHz PC Codes for others are downloaded from their websites Comparisons with Conjugate Gradient (CG) [Lustig, MRM07] Operator Splitting (TVCMRI) [Ma, CVPR08] Variable Splitting (RecPF) [Yang, JSTSP09] Sampling Randomly sampling in the frequency domain White color denotes being sampled (20%)

12. 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 CG8.71db TVCMRI12.12db RecPF12.40db CSA 18.68db FCSA20.35db

13. 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 CG11.73db TVCMRI15.49db RecPF16.05db CSA22.27db FCSA23.70db

14. (a) Artery image (b) Brain image Comparisons (CPU-Time vs. SNR) Statistical results after 100 runs SNR(db) CPU-time(s) SNR(db) CPU-time(s)

15. Visual Comparisons on Full Body MR Image (a) Original(b) TVCMRI (c) RecPF (d) CSA(e) FCSA 1024 x 256, sampling ratio 25%

16. Comparisons with Different Sampling Ratios Exp I: 20%Exp II: 25%Exp III: 36% TVCMRI10.88db12.67db15.82db RecPF11.06db13.02db16.12db CSA16.36db18.07db21.98db FCSA17.82db19.28db23.66db All methods run 50 iterations

17. Contributions We proposed a new algorithm for compressed MRI reconstruction. It theoretically converges with accuracy ε ~ O(1/k 2 ) 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

18. Thank You! Any Questions?