磁共振頻譜造影(Magnetic Resonance Spectroscopic Imaging, MRSI)的技術能夠一次性地獲得大範圍的代謝物濃度變化，讓醫生較容易地得到疾病的位置，因此在醫學及臨床實驗上面相當受到歡迎。然而要獲得空間解析度較佳的頻譜資訊，需要耗費相當久的掃描時間來完成，造成在臨床使用上的瓶頸。
Michael Lustig 團隊於 2008年提出了將壓縮感知理論應用在磁共振影像上，壓縮感知的理論，主要為了解決取樣時間過長的問題。他們提出讓取樣點數僅有原本的一半或者更少，並利用適當的演算法重建出完整的灰階圖像。由於MRSI 與 MRI 的掃描方式類似，因此壓縮感知的理論也應當可以應用在MRSI上面，可大幅減少掃描時間，有利於針對病患進行檢測。但是重建演算法所要耗費的時間相當的冗長，對於一些需要立刻得到頻譜造影的病患相當不利，所以一定要再重建的過程做一些改變。
因此，本論文主要以壓縮感知演算法針對 MRSI 進行深度的探討。主要研究分成三大階段: 第一階段針對MRSI設計隨機取樣的方式; 第二階段是針對 CS 重建的方式以及採用的演算法進行改進; 最後一階段是以 Nvidia 提供的 CUDA 程式庫,使用 GPU 做平行化運算,更進一步加速重建頻譜造影所需要的時間。

MRSI can be used to obtained metabolic information in large area of human brain in vivo. By means of this technique, medical doctors can resolve the location of the diseases easily. Therefore, MRSI has become popular in clinical applications. However, the limitation of spatial resolution and acquisition time remains the bottleneck for its usage.
In order to solve the problem of long acquisition time in MRI, Michael Lustig's team proposed the theory of Compressed Sensing (CS) in 2008, and used it in magnetic resonance imaging (MRI) successfully. They proposed that the original signal can be recovered by the reconstruction algorithm with undersampling. This technique uses only half or fewer sample points to reconstruct a complete grayscale image. Since the MRSI scanning method are similar with MRI, the theory of CS should be able to applied to MRSI. The advantage of reduced scanning time allows patients to have a better examination. However, the reconstruction algorithm requires lots of computation time, which may not be suitable to produce the resultant spectra in real time for diagnosis.
Therefore, in this study, we investigate the feasibility of the MRSI using CS. The main study is divided into three phases: Firstly, some undersampling schemes have been designed for MRSI and compared their performance. Secondly, CS MRSI and reconstruction algorithms were investigated and developed. Then we applied parallel computing with GPU process to improve the computation time for CS reconstruction.

第一章 簡介 1
1.1 背景 1
1.2 文獻回顧 3
1.3 研究動機 11
1.4 組織架構 12
第二章 原理 13
2.1 磁共振頻譜造影 13
2.2 壓縮感知條件 16
2.3 壓縮感知的最佳化重建 21
2.4 頻譜的後處理 23
2.5 圖形處理器 26
第三章 實驗環境和方法 29
3.1 實驗環境 29
3.2 實驗方法 30
第四章 實驗結果 37
4.1 MRI重建結果 37
4.2 MRSI重建結果 39
4.3 CPU與GPU重建時間比較 54
第五章 結論 56
參考文獻 57

[1] Michael Lusting, David L. Donoho, and John M. Pauly, Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging. Magnetic Resonance in Medicine, 2007. 58(6): p. 1182-1195.
[2] Michael Lusting, David L. Donoho, Juan M. Santos, and John M. Pauly, Compressed Sensing MRI. IEEE Signal Procsessing Magazine, 2008. 25(2): p. 72-82.
[3] Ching-Hua Chang, Xiangdong Yu, and Jim X. Ji, Compressed Sensing MRI Reconstruction from 3D Multichannel Data Using GPUs. Magnetic Resonance in Medicine, 2017. 78(6): p. 2265-2274.
[4] E. J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Information Theory Society, 2006. 52(2): p. 489-509.
[5] Simon Hu, Michael Lustig, Albert P. Chen, Jason Crane, Adam Kerr, Douglas A.C. Kelley, Ralph Hurd, John Kurhanewicz, Sarah J. Nelson, John M. Pauly, and Daniel B. Vigneron, Compressed sensing for resolution enhancement of hyperpolarized (_^13)C flyback 3D-MRSI. Journal of Magnetic Resonance, 2008. 192(2): p. 258-264.
[6] Geethanath S, Baek HM, Ganji SK, Ding Y, Maher EA, Sims RD, Choi C, Lewis MA, and Kodibagkar VD, Compressive sensing could accelerate (_^1)H MR metabolic imaging in the clinic. Radiology, 2012. 262(3): p. 985-994.
[7] Shankar RV, Agarwal S, Geethanath S, and Kodibagkar VD, Rapid MR spectroscopic imaging of lactate using compressed sensing. SPIE Medical Imaging, 2015. 9417: 94171J 1-9.
[8] Furuyama JK, Wilson NE, Burns BL, Nagarajan R, Margolis DJ, and Thomas MA, Application of compressed sensing to multidimensional spectroscopic imaging in human prostate. Magnetic Resonance in Medicine, 2012. 67(6): p. 1499-1505.
[9] Sarma MK, Nagarajan R, Macey PM, Kumar R, Villablanca JP, Furuyama J, and Thomas MA, Accelerated echo-planar J-resolved spectroscopic imaging in the human brain using compressed sensing: a pilot validation in obstructive sleep apnea. American Journal of Neuroradiology, 2014. 35(6): S. 81-89
[10] Heikal AA, Wachowicz K, and Fallone BG, Correlation between k-space sampling pattern and MTF in compressed sensing MRSI. Medical Physics, 2016. 43(10): p. 5626-5634.
[11] Itthi Chatnuntawech, Borjan Gagoski, Berkin Bilgic, Stephen F. Cauley, Kawin Setsompop, and Elfar Adalsteinss, Accelerated (_^1)H MRSI using randomly undersampled spiral-based k-space trajectories. Magnetic Resonance in Medicine, 2015. 74(1): p. 13-24
[12] Daubechies I, Defrise M, and Mol C D, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint[J]. Communications on Pure & Applied Mathematics,2004, 57(11):1413–1457.
[13] Filip Jiru, Introduction to post-processing techniques. European Journal of Radiology, 2008. 67(2): p. 202–217.
[14] E. Cabanes, S. Confort-Gouny, Y. Le Fur, G. Simond, and P. J. Cozzone, Otimization of Residual Water Signal Removal by HLSVD on Simulated Short Echo Time Proton MR Spectra of the Human Brain. Journal of Magnetic Resonance 2001. 150: p. 116–125.
[15] NVIDIA Corporation, 2788 San Tomas Expressway, Santa Clara, CA 95051. Cuda C Programming Guide version 9.1, Dec 2017. http://www.nvidia.com.
[16] SparseMRI software package (http://people.eecs.berkeley.edu/~mlustig/Software.html).