Low rank Hankel matrix-based magnetic resonance spectroscopy reconstruction in fast NMR (中文，English)

(Recover missing data of exponential signals by enforcing the low-rank constraint on its the Hankel matrix)

Xiaobo Qu^{1,#,*}, Maxim Mayzel^{2}, Jian-Feng Cai^{3}, Zhong Chen^{1}, Vladislav Orekhov^{2,*}

1 Department of Electronic Science, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen, China

2 Swedish NMR Centre, University of Gothenburg, Sweden

3
Department of Mathematics, University of Iowa, USA

# Emails: quxiaobo <at> xmu.edu.cn or quxiaobo2009 <at> gmail.com

Citations: Xiaobo Qu*, Maxim Mayzel, Jian-Feng Cai, Zhong Chen, Vladislav Orekhov*. Accelerated NMR spectroscopy with low-rank reconstruction, *Angewandte Chemie International Edition*, vol. 54, no. 3, pp. 852-854, 2015.

Synopsis:

Accelerated multi-dimensional NMR spectroscopy is a prerequisite for studying short-lived molecular systems, monitoring chemical reactions in real-time, high-throughput applications, etc. Non-uniform sampling is a common approach to reduce the measurement time. Here, we introduce a new method for high quality spectra reconstruction from non-uniformly sampled data, which is based on recent developments in the field of signal processing theory and utilizes the so far unexploited general property of the NMR signal, its low rank of Hankel matrix converted from time-domain signals, which are assumed to be composed of complex exponential functions. Using experimental and simulated data, we demonstrate that the low-rank reconstruction is a viable alternative to the current state-of-the-art technique compressed sensing. In particular, the low-rank approach is good in preserving of low intensity broad peaks, and thus increases the effective sensitivity in the reconstructed spectra.

Method:

1. Low rank Hankel matrix (LRHM) of exponential signals

It is common to model a signal as the sum of complex exponential functions. When the signal is converted into Hankel matrix, its rank is equal to the number of exponential functions as shown in Fig.1. The procedure of converting exponential signals into a Hankel matrix is shown in Fig. 2. If the number of exponential functions is much smaller than the size of the Hankel matrix, the Hankel matrix will be in low rank. This property has been shown to be very useful in many fields, e.g. nuclear magnetic resonance (NMR) spectroscopy [1,2] and magnetic resonance imaging [3,4].

Fig. 1. Lorentizian peaks in the frequency domain and the rank of its Hankel matrix of the time domain signal.

Fig. 2. The procedure of converting exponential signals into a Hankel matrix.

2. Low rank Hankel matrix (LRHM) for non-uniformly sampling NMR spectrascopy

The duration of a multi-dimensional NMR experiment is proportional to the number of measured data points and increases rapidly with spectral resolution and dimensionality. The Non-Uniformly Sampling (NUS) approach offers a general solution for a dramatic reduction in measurement time. Additional constraints on the signal in the time or frequency domains is of great importance to reconstruct a high-quality spectrascopy.

When the number of functions is much smaller than the length of the data, it is possible to recover the missing entries by enforcing the low rank of the Hankel matrix converted from the signal. In the work [1], we proposed to recover the undersampled time-domain data and reconstruct a full spectra in the non-uniformly sampled nuclear magnetic resonance (NMR) spectroscopy as shown in Fig.3. This new approach seeks for a spectrum with the least number of spectral peaks. Notably, the rank is independent of the line width of the peaks in the spectrum as shown in Fig. 1. This nice property will produces correct reconstruction of line shapes for both sharp and broad peaks as shown in Fig. 4.

Fig. 3. Low rank Hankel matrix-based NMR spectrascopy reconstruction.

Fig.4 shows a comparison between a simulated fully sampled reference spectrum and its NUS reconstructions obtained using the compressed sensing with L1 norm minimization (CS-L1) on the spectrum [5-10] and Low rank Hankel matrix (LRHM) on the time-domain signal [1]. The spectrum contains five peaks with the same integrals but different line widths. Both NUS processing methods successfully recover the narrowest peak to the right in Fig. 1a-c. The broadest peak to the left is faithfully recovered by the LRHM approach but is seriously distorted by the CS-L1. For the three middle peaks with moderate line width, the CS produces clearly visible line shape distortions as shrinkage of the peaks. Correlation analysis of the spectral intensities, shown in Fig. 4, indicates that, for the NUS level in the range 10% - 20%, the two broadest peaks are recovered systematically better using the LRHM than by the CS-L1. For the three remaining narrower peaks, the LRHM and CS-L1 provide comparable results. These observations imply that the LRHM, while performing similarly for the narrow peaks, outperforms CS-L1 when the peaks are relatively broad.

Fig. 4. Reconstructions of the synthetic spectrum containing five peaks with different line widths.

Fig.5 shows a NUS 2D 1H-15N HSQC spectrum of the intrinsically disordered cytosolic domain of human CD79b protein from the B-cell receptor. Similarity between the LRHM reconstruction in Fig. 5a and the fully sampled reference spectrum in Fig. 5b illustrates the high quality of the LRHM reconstruction obtained from only 35% of the traditionally acquired spectrum. This qualitative observation corroborates with the faithful reproduction of the peak intensities shown in the inset of Fig. 5b. Similar results are obtained for a 2D NOESY spectrum of ubiquitin (see Supporting Information [1]). The quality of the CS-L1 reconstruction obtained from the same NUS HSQC data (not shown) is generally as good, with the majority of the peaks reproduced equally well by the CS-L1 and LRHM. This is illustrated for the amide group of Gly11 in Fig. 5c. Nevertheless, several low intensity peaks are notably compromised in the CS-L1 spectrum as shown in Fig. 5d-f. While peaks for Thr37 and Gly45 show clear line shape shrinkage, the peak of Gly38 is completely lost. The opposite situation, when a true peak is present in the CS-L1 but is missing in the LRHM reconstruction, never occurred in our spectra. It should be also noted that the virtual-echo pre-processing used for all of the CS-L1 reconstructions in this work improves quality of the spectra but requires prior knowledge about the signal phase [10]. In general, when the phase is unknown, the virtual-echo cannot be used and the comparison between the CS-L1 and LRHM would be even more in favour of the LRHM method.

The experimental results in Fig. 5 are fully consistent with the simulations shown in Fig. 4 and lead us to the conclusion that the LRHM produces at least as good spectral reconstructions as the CS-L1 and often outperforms it for broadest and weakest peaks. Effective sensitivity of a spectrum reconstruction method from NUS data is defined as a possibility to detect weak peaks and discriminate them from eventual false signals. Thus, the observed good reconstructions of the low intensity peaks by the LRHM indicate high sensitivity of the new method.

Fig. 5. Reconstructions of 2D 1H-15N HSQC spectrum of the cytosolic domain of CD79b with 35% FID data. Note: The inset shows correlation of the peak intensities between the reference and the LR spectra; the correlation coefficient equals to 0.99.

3. Numerical Algorithm: Alternating Direction Method of Multipliers for Low rank Hankel matrix (LRHM) reconstruction

How to numerically solve the reconstruction efficiently is still challenging. An Alternating Direction Method of Multipliers (ADMM) method was developed to solve the signal recovery problem with low rank contraint on the Hankel matrix. This algorithm is fast and stable. It was reported in the Appendix of [2]. Readers can freely access the derivation and psudo-code of this algorithm at http://onlinelibrary.wiley.com/doi/10.1002/anie.201409291/suppinfo or here . The ADMM algorithm developed in the appendix is aslo very useful since reconstruction the Hankel matrix may be of great interest for other fields [11-13], although this paper focus on proposing the model of low rank Hankel matrix to recover the missing data in NMR spectroscopy and evaluating the performance on data acquisition.

4. Our related publications on Low Rank Hankle Matrix Reconstruction for NMR Spsectroscopy

A.
Jiaxi Ying, Hengfa Lu, Qingtao Wei, Jian-Feng Cai, Di Guo, Jihui Wu, Zhong Chen, Xiaobo Qu*. Hankel matrix nuclear norm regularized tensor completion for *N*-dimensional exponential signals, * IEEE Transactions on Signal Processing*, 65(14): 3702-3717, 2017.

B. Hengfa Lu, Xinlin Zhang, Tianyu Qiu, Jian Yang, Jiaxi Ying, Di Guo, Zhong Chen, Xiaobo Qu*, Low rank enhanced matrix recovery of hybrid time and frequency data in fast magnetic resonance spectroscopy, * IEEE Transactions on Biomedical Engineering*, 10.1109/TBME.2017.2719709, 2017. [Code]

C. Di Guo, Hengfa Lu, Xiaobo Qu*, A fast low rank Hankel matrix factorization reconstruction method for non-uniformly sampled magnetic resonance spectroscopy, IEEE Access, 5: 16033-16039, 2017.

**References**:

[1] X. Qu, M. Mayzel, J.-F. Cai, Z. Chen, V. Orekhov, "Accelerated NMR Spectroscopy with Low-Rank Reconstruction,"* Angewandte Chemie International Edition*, vol. 54, no. 3, pp. 852-854, 2015. (Supplement: http://onlinelibrary.wiley.com/doi/10.1002/anie.201409291/suppinfo)

[2] J. C. Hoch, A. S. Stern, NMR data processing: Wiley-Liss, 1996.

[3] Z.-P. Liang, Spatiotemporal imaging with partially separable functions, *The 4th IEEE International Symposium on Biomedical Imaging: From Nano to Macro-ISBI'07*, April 12, 2007 - April 15, Arlington, VA, USA, pp. 988-991, 2007.

[4] H. M. Nguyen, X. Peng, M. N. Do, Z.-P. Liang, Denoising MR spectroscopic imaging data with low-rank approximations, *IEEE Transactions on Biomedical Engineering*, vol. 60, no. 1, pp. 78-89, 2013.

[5] X. Qu, X. Cao, D. Guo, Z. Chen. Compressed Sensing for Sparse Magnetic Resonance Spectroscopy, * 18th Scientific Meeting on International Society for Magnetic Resonance in Medicine-ISMRM'10*. Stockholm, Sweden, p 3371, 2010.

[6] X. Qu, D. Guo, X. Cao, S. Cai, Z. Chen. Reconstruction of self-sparse 2D NMR spectra from undersampled data in the indirect dimension, *Sensors*, 11(9):8888-8909, 2011. [Code]

[7] I. Drori, Fast l1 minimization by iterative thresholding for multidimensional NMR Spectroscopy. Eurasip J. Adv. Sig. Pr., Article ID 20248, 2007.

[8] K. Kazimierczuk, V. Orekhov. Accelerated NMR spectroscopy by using compressed sensing. *Angewandte Chemie International Edition*, vol. 50, pp. 5556-5559, 2011.

[9] D.J. Holland, M. J. Bostock, L.F. Gladden, D. Nietlispach. Fast multidimensional NMR spectroscopy using compressed sensing. *Angewandte Chemie International Edition*, vol. 50, pp. 6548-6551, 2011

[10] M. Mayzel, K. Kazimierczuk, V. Orekhov, The causality principle in the reconstruction of sparse NMR spectra, *Chemical Communications*, vol. 50, pp. 8947-8950, 2014.

[11] M. Fazel, T. Pong, D. Sun, P. Tseng, Hankel matrix rank minimization with applications to System Identification and Realization, *SIAM Journal on Matrix Analysis and Applications*, vol. 34, no. 3, pp. 946-977, 2013.

[12] Jian-Feng Cai, Xiaobo Qu, Weiyu Xu, Gui-Bo Ye, Robust recovery of complex exponential signals from random Gaussian projections via low rank Hankel matrix reconstruction, *Applied and Computational Harmonic Analysis*, vol. 41, no. 2, pp. 470-490, 2016.

[13] Y. Chen, Y. Chi. Robust spectral compressed sensing via structured matrix completion, *IEEE Transactions on Information Theory*, vol. 60, no. 10, pp. 6576 – 6601, 2014.