Reconstruction of Self-sparse 2D NMR Spectra from Undersampled Data

Xiaobo Qu^{1,2}, Di Guo^{2}, Xue Cao^{3}, Shuhui Cai^{2} and Zhong Chen^{2},*

1 Department of Communication Engineering, Fujian Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen 361005 China;

2 Department of Electronic Science, Fujian Key Laboratory of Plasma and Magnetic Resonance, Xiamen 361005, China;

3 School of Software, Shanghai Jiao Tong University, Shanghai 200240, China;

Xiaobo Qu's Email: quxiaobo <at> xmu.edu.cn or quxiaobo2009 <at> gmail.com.

This work was presented on May 1-7, 2010 at *18th* ISMRM Annual Meeting, here is the full information:

Xiaobo Qu, Xue Cao, Di Guo, Zhong Chen. Compressed Sensing for Sparse Magnetic Resonance Spectroscopy, *International Society for Magnetic Resonance in Medicine 18th Scientific Meeting*. Stockholm, Sweden, May 1-7, 2010, p 3371.

Citations: **Xiaobo Qu**, Di Guo, Xue Cao, Shuhui Cai, Zhong Chen. Reconstruction of self-sparse 2D NMR spectra from undersampled data in the indirect dimension, Sensors, 11(9):8888-8909,2011.

[Note: This paper was submitted on 4 July 2011]

Download the paper here or from Sensors

Download the code: Toolbox_SparseMRS

**Abstract: **

Reducing the acquisition time for two-dimensional nuclear magnetic resonance (2D NMR) spectra is important. One way to achieve this goal is reducing the acquired data. In this paper, under the framework of compressed sensing, we proposed to undersample the data in the indirect dimension for a type of self-sparse 2D NMR spectra, that is, only a few meaningful spectral peaks occupy partial locations, while the rest locations own very small or even no peaks. The spectrum is reconstructed by enforcing its sparsity in an identity matrix domain with *Lp *(*p *= 0.5) norm optimization algorithm. Both theoretical analysis and simulation results show that the proposed method can reduce the reconstruction error compared with the wavelet-based *L1* norm optimization.

**Keywords: **NMR; spectral reconstruction; sparsity; undersampling; compressed sensing.

The spectra is reconstructed from undersampled FID by solving

where y is the sampled FID data, is the operator to undersmpling the FID, and x the spectra to be reconstructed.

Main results:

For the 1H-1H COSY spectra,

With the L1 norm minimization, all the peaks are recovered successfully by using identity matrix (Figure 9(d,f)), while some peaks are lost by using wavelets (Figure 9(c,e)). Since the contours for the marked peaks look faint, we also plot the 1D slices along the indirect dimension in Figure 10. The height of one peak in the wavelet-based reconstruction in Figure 10(a) are much lower than those in the fully sampled spectrum, leading to the peak lost in the contour plots in Figure 9(c,e). Furthermore, the nonlinear operation on wavelet coefficients induces the artifacts labeled in Figure 9(c,e). This phenomenon is also observed in the 1D slices shown in Figure 10(a), where wavelet reconstruction generates illusive peaks. With the L0.5 norm minimization, the spectrum are reconstructed better for both wavelets and identity matrix. One can still observe the reduced peak height and artifacts in wavelet-based reconstruction, but identity matrix performs very well (Figure 10(c)).

For the 1H-13C COSY spectra,

Some peaks are obviously lost in the reconstructed spectra using wavelets with both L1 norm and L0.5 norm minimization (Figure 11(c,e)). These lost peaks are found in the identity matrix-based reconstruction spectra (Figure 11(d,f)). With the L0.5 norm minimization, the intensities of the peaks marked with arrow in Figure 11(f) are more consistent to the fully sampled spectra in Figure 11(b) than those in the reconstructed spectra with the L1 norm minimization (Figure 11(d,f)).

Simulation results demonstrate that wavelet-based reconstruction obviously induces the loss of some peaks in the crowded 1H-13C COSY spectrum and loss of some weak peaks in the less crowded 1H-1H COSY spectrum. Wavelet may even worsen the reconstructed spectra. Thus, it is not a good choice to use wavelet for the self-sparse spectra discussed in this paper.

**Figure 1.** Simulated *free induction decay *(FID) data in time domain **(a)** and its corresponding 1D NMR spectrum **(b)**.

**Figure 8. **Sampling pattern used in simulation. **(a)** Cartesian sampling pattern with sampling rate 0.20 for the 2D 1H-1H COSY spectrum ( N1=256 points) in Figure 9(a);

**(b)** Cartesian sampling pattern with sampling rate 0.25 for the 2D 1H-13C COSY spectrum (128 points) in Figure 11(a).

(a) (b)

**Figure 9.** CS reconstruction of a 2D 1H-1H COSY spectrum using wavelet and identity matrix. **(a,b)** reconstructed spectra using fully sampled FID and undersampled FID with zero filling, respectively;

**(c,d) **reconstructed spectra using wavelets and identity matrix with IST-based L1 norm, respectively;** (e,f) **reconstructed spectra using wavelets and identity matrix with PSOCA-based L0.5 norm, respectively.

**Figure 10.** 1D slices along the indirect dimension for the chemical shift of 8.2 ppm (**a**-**b**) or 7.2 ppm (**c**) in the direct dimension (The full 2D spectra is shown in Fig. 9).

**(a)** Spectra reconstructed with IST-based L1 norm;** (b) **spectra reconstructed with PSOCA-based L0.5 norm; **(d) **spectra reconstructed with PSOCA-based 0.5 norm.

**Figure 11.** CS reconstruction of a 2D 1H-13 COSY spectrum using wavelet and identity matrix. **(a,b)** spectra reconstructed using fully sampled FID (N1=128 points) and undersampled FID with zero filling, respectively;

**(c,d) **spectra reconstructed using wavelets and identity matrix with IST-based L1 norm, respectively;**(e,f) **spectra reconstructed using wavelets and identity matrix with PSOCA-based L0.5 norm, respectively.

**Acknowledgements:**

This work was partially supported by the NNSF of China under Grant 10974164, and the Research Fund for the Doctoral Program of Higher Education of China under Grant 200803840019. Xiaobo Qu and Di Guo would like to acknowledge the fellowship of Postgraduates’ Oversea Study Program for Building High-Level Universities from the China Scholarship Council. The authors also thank the reviewers for their thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of this article.

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