# Enhancing Compression Level for More Efficient Compressed Sensing and Other Lessons from NMR Spectroscopy

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## Abstract

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## 1. Introduction

## 2. Theory

- highly probable exact recovery of a sparse signal based on limited information about it;
- highly probable approximately exact recovery of a compressible signal based on limited information about it.

## 3. Results and Discussion

#### 3.1. Lesson 1: Reduce the Number of Peaks

^{1}H-

^{13}C Heteronuclear Single-Quantum Correlation spectrum (HSQC), making CH

_{2}and CH

_{3}peaks invisible. Similarly, one can make the transfer selective by exploiting the differences in J between various pairs of nuclei. In 3D HNCA, the basic experiment used to establish sequential connectivities in spectra of proteins, the excitation is initially transferred from amide hydrogens to amide nitrogens. Then, from each

^{15}N nucleus the transfer may go two-fold—to $\alpha $ carbon of the same (i) and preceding ($i-1$) amino acid residue. This is caused by the fact that

^{H}N-C

_{α}coupling constants for both ways are similar (typically 11 Hz for N

_{i}-C

_{αi}and 7 Hz for N

_{i}-C

_{αi−1}) and $\Delta $ can be set to average value in-between. However, the variants of the experiment with an exclusive transfer to one C

_{α}also exist and have been used in combination with non-uniform sampling [43], also due to better compressibility of the spectrum and thus the sampling.

#### 3.2. Lesson 2: Minimize Dynamic Range

#### 3.3. Lesson 3: Pre-Processing

_{α}-C

_{β}coupling) or HC-CH TOCSY spectra [63] (coupling between carbon atoms belonging to methyl and a neighboring group).

#### 3.4. Lesson 4: Match Sampling with the Decay

#### 3.5. Lesson 5: Non-Stationarity

#### 3.6. Practical Example

^{13}C HSQC experiment characterized by a different number of peaks in a spectrum. The acquired 2D NMR signals for each HSQC variant were artificially sub-sampled by taking out random points from the full data in the t

_{1}(

^{13}C) dimension and reconstructed back to the original size. The reconstructed spectra from the corresponding sub-sampled HSQC experiments are depicted in Figure 6: standard unedited

^{13}C HSQC (Figure 6b),

^{13}C HSQC with CH-only editing (Figure 6c) and

^{13}C HSQC with CPMG filter (Figure 6d). A fully sampled, unedited

^{13}C HSQC spectrum is also depicted in Figure 6a) and stands as a quality reference for the reconstructed spectra (Figure 6b–d). The

^{13}C HSQC NMR experiments used in this study employ the same core HSQC pulse sequence [88], which allows observing single-quantum

^{1}H–

^{13}C correlation signals. The use of appropriate filters (multiplicity-editing—Figure 6c and CPMG—Figure 6d) to the core HSQC sequence (Figure 6a,b) allowed us to reduce the number of components in the signal. The filters were chosen concerning the physicochemical properties of the substances being measured. The sample used for experiments was a mixture of sucrose and heparin dissolved in D

_{2}O. Both compounds are saccharides, but their molecular weights (MW) differ significantly, as heparin is a polysaccharide of MW in the range from 6000 up to 20,000 g/mol, while sucrose is a disaccharide of MW = 342.3 g/mol. We used this fact to suppress signals of fast-relaxing nuclei belonging to large heparin molecules by means of CPMG relaxation-filter (Figure 6d). We also used the fact that structures of sucrose and heparin consist mainly of CH and CH

_{2}chemical sites, which are the source of

^{1}H–

^{13}C single-quantum correlation signals. We employed the multiplicity-editing block to suppress signals that arise from CH

_{2}chemical sites, thus, only the signals corresponding to CH sites were visible. The unedited

^{13}C HSQC spectrum experiment (Figure 6a—reference and Figure 6b—reconstructed spectrum) show all the single-bond

^{1}H–

^{13}C correlations regardless of the molecular size and type of chemical site.

^{1}H–

^{13}C correlation signals in the unedited

^{13}C HSQC spectrum (Figure 6a) were poorly reconstructed using 24 out of 256 t

_{1}sub-samples (Figure 6b). The effect is visible on heparin signals near 3.55/75.0 ppm and 3.65/80.0 ppm (marked with the black arrows in Figure 6). A reduction of the number of peaks in a spectrum allowed for more reliable reconstruction using the same 24-points sampling level for

^{13}C HSQC with CH-only editing (Figure 6c) and

^{13}C HSQC with CPMG filter (Figure 6d).

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CPMG | Carr–Purcell–Meiboom–Gill |

CS | Compressed Sensing |

DOAJ | Directory of open access journals |

FID | Free Induction Decay |

FT | Fourier Transform |

HSQC | Heteronuclear Single-Quantum Correlation |

IPAP | In-phase anti-phase |

IRLS | Iteratively Re-weighted Least Squares |

IST | Iterative Soft Thresholding |

MDPI | Multidisciplinary Digital Publishing Institute |

MW | Molecular weight |

NMR | Nuclear Magnetic Resonance |

NOESY | Nuclear Overhauser Effect Spectroscopy |

OMP | Orthogonal matching pursuit |

PS | Pure-shift (NMR) |

PSF | Point spread function |

RF | Radio frequency |

RIP | Restricted isometry property |

ROESY | Rotating Frame Overhauser Effect Spectroscopy |

SNR | Signal-to-noise ratio |

TOCSY | Total Correlation Spectroscopy |

UUP | Uniform Uncertainty Principle |

VD | Virtual Decoupling |

VE | Virtual Echo |

## References

- Simpson, J.H. Organic Structure Determination Using 2-D NMR Spectroscopy; Academic Press: San Diego, CA, USA, 2012. [Google Scholar] [CrossRef]
- Forseth, R.; Schroeder, F. NMR-spectroscopic analysis of mixtures: From structures to function. Curr. Opin. Chem. Biol.
**2012**, 15, 38–47. [Google Scholar] [CrossRef][Green Version] - Dass, R.; Koźmiński, W.; Kazimierczuk, K. Analysis of complex reacting mixtures by time-resolved 2D NMR. Anal. Chem.
**2015**, 87, 1337–1343. [Google Scholar] [CrossRef] - Dinges, S.S.; Hohm, A.; Vandergrift, L.A.; Nowak, J.; Habbel, P.; Kaltashov, I.A.; Cheng, L.L. Cancer metabolomic markers in urine: Evidence, techniques and recommendations. Nat. Rev. Urol.
**2019**, 16, 339–362. [Google Scholar] [CrossRef] - Sattler, M.; Heidelberg, E. Introduction to biomolecular NMR spectroscopy. Science
**2004**, 1–18. [Google Scholar] - Ernst, R.R.; Anderson, W.A. Application of Fourier Transform Spectroscopy to Magnetic Resonance. Rev. Sci. Instrum.
**1966**, 37, 93–102. [Google Scholar] [CrossRef] - Jeener, J. AMPERE International Summer School. Basko Polje Yugoslavia
**1971**, 197. [Google Scholar] - Ying, J.; Barnes, C.A.; Louis, J.M.; Bax, A. Importance of time-ordered non-uniform sampling of multi-dimensional NMR spectra of Aβ1–42 peptide under aggregating conditions. J. Biomol. NMR
**2019**, 73, 429–441. [Google Scholar] [CrossRef] [PubMed] - Nyquist, H. Certain topics in telegraph transmission theory. Trans. Am. Inst. Electr. Eng.
**1928**, 47, 617–644. [Google Scholar] [CrossRef] - Szántay, C. NMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity. IV. Uncertainty. Concept. Magn. Reson. A
**2008**, 32A, 373–404. [Google Scholar] [CrossRef] - Mobli, M.; Hoch, J.C. Nonuniform sampling and non-Fourier signal processing methods in multidimensional NMR. Prog. Nucl. Mag. Res. Spectrosc.
**2014**, 83, 21–41. [Google Scholar] [CrossRef][Green Version] - Matsuki, Y.; Konuma, T.; Fujiwara, T.; Sugase, K. Boosting protein dynamics studies using quantitative nonuniform sampling NMR spectroscopy. J. Phys. Chem. B
**2011**, 115, 13740–13745. [Google Scholar] [CrossRef] [PubMed] - Qu, X.; Mayzel, M.; Cai, J.F.; Chen, Z.; Orekhov, V. Accelerated NMR spectroscopy with low-rank reconstruction. Angew. Chem. Int. Ed. Engl.
**2015**, 54, 852–854. [Google Scholar] [CrossRef] [PubMed] - Kazimierczuk, K.; Orekhov, V. Accelerated NMR spectroscopy by using compressed sensing. Angew. Chem. Int. Ed. Engl.
**2011**, 50, 5556–5559. [Google Scholar] [CrossRef] [PubMed] - Holland, D.J.; Bostock, M.J.; Gladden, L.F.; Nietlispach, D. Fast multidimensional NMR spectroscopy using compressed sensing. Angew. Chem. Int. Ed. Engl.
**2011**, 50, 6548–6551. [Google Scholar] [CrossRef] [PubMed] - Qu, X.; Guo, D.; Cao, X.; Cai, S.; Chen, Z. Reconstruction of self-sparse 2D NMR spectra from undersampled data in the indirect dimension. Sensors
**2011**, 11, 8888–8909. [Google Scholar] [CrossRef][Green Version] - Rani, M.; Dhok, S.B.; Deshmukh, R.B. A Systematic Review of Compressive Sensing: Concepts, Implementations and Applications. IEEE Access
**2018**, 6, 4875–4894. [Google Scholar] [CrossRef] - Holland, D.J.; Gladden, L.F. Less is more: How compressed sensing is transforming metrology in chemistry. Angew. Chem. Int. Ed. Engl.
**2014**, 53, 13330–13340. [Google Scholar] [CrossRef] - Hyberts, S.; Milbradt, A.; Wagner, A.; Arthanari, H.; Wagner, G. Application of iterative soft thresholding for fast reconstruction of NMR data non-uniformly sampled with multidimensional Poisson Gap scheduling. J. Biomol. NMR
**2012**, 52. [Google Scholar] [CrossRef][Green Version] - Kazimierczuk, K.; Orekhov, V.Y. A comparison of convex and non-convex compressed sensing applied to multidimensional NMR. J. Magn. Reson.
**2012**, 223, 1–10. [Google Scholar] [CrossRef] - Shchukina, A.; Kasprzak, P.; Dass, R.; Nowakowski, M.; Kazimierczuk, K. Pitfalls in compressed sensing reconstruction and how to avoid them. J. Biomol. NMR
**2017**, 68, 79–98. [Google Scholar] [CrossRef][Green Version] - Coggins, B.E.; Venters, R.A.; Zhou, P. Radial sampling for fast NMR: Concepts and practices over three decades. Prog. Nucl. Mag. Res. Spectrosc.
**2010**, 57, 381–419. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brüschweiler, R.; Zhang, F. Covariance nuclear magnetic resonance spectroscopy. J. Chem. Phys.
**2004**, 120, 5253–5260. [Google Scholar] [CrossRef] [PubMed] - Koehl, P. Linear prediction spectral analysis of NMR data. Prog. Nucl. Mag. Res. Spectrosc.
**1999**, 34, 257–299. [Google Scholar] [CrossRef] - Foroozandeh, M.; Jeannerat, D. Reconstruction of full high-resolution HSQC using signal split in aliased spectra. Magn. Reson. Chem.
**2015**, 53, 894–900. [Google Scholar] [CrossRef] - Candes, E.J.; Romberg, J.; Tao, T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory
**2006**, 52, 489–509. [Google Scholar] [CrossRef][Green Version] - Donoho, D.L. Compressed sensing. IEEE Trans. Inform. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Foucart, S.; Rauhut, H. A Mathematical Introduction to Compressive Sensing; Wiley: Hoboken, NJ, USA, 2010; p. 526. [Google Scholar]
- Candes, E.J. The restricted isometry property and its implicationsfor compressed sensing. C. R. Math.
**2008**, 346, 589–592. [Google Scholar] [CrossRef] - Candès, E.J.; Romberg, J.K.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math.
**2006**, 59, 1207–1223. [Google Scholar] [CrossRef][Green Version] - Rovnyak, D.; Sarcone, M.; Jiang, Z. Sensitivity enhancement for maximally resolved two-dimensional NMR by nonuniform sampling. Magn. Reson. Chem.
**2011**, 483–491. [Google Scholar] [CrossRef] - Palmer, M.R.; Suiter, C.L.; Henry, G.E.; Rovnyak, J.; Hoch, J.C.; Polenova, T.; Rovnyak, D. Sensitivity of nonuniform sampling NMR. J. Phys. Chem. B
**2015**, 119, 6502–6515. [Google Scholar] [CrossRef][Green Version] - Zangger, K. Pure shift NMR. Prog. Nucl. Mag. Res. Spectrosc.
**2015**, 86–87, 1–20. [Google Scholar] [CrossRef] [PubMed][Green Version] - Castañar, L. Pure shift 1H NMR: What is next? Magn. Reson. Chem.
**2017**, 55, 47–53. [Google Scholar] [CrossRef] [PubMed][Green Version] - Aguilar, J.A.; Kenwright, A.M. Compressed NMR: Combining compressive sampling and pure shift NMR techniques. Magn. Reson. Chem.
**2018**, 56, 983–992. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ndukwe, I.E.; Shchukina, A.; Kazimierczuk, K.; Butts, C.P. Rapid and safe ASAP acquisition with EXACT NMR. Chem. Commun.
**2016**, 52, 12769–12772. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ndukwe, I.; Shchukina, A.; Kazimierczuk, K.; Cobas, C.; Butts, C. EXtended ACquisition Time (EXACT) NMR—A Case for ’Burst’ Non-Uniform Sampling. ChemPhysChem
**2016**. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ndukwe, I.; Shchukina, A.; Zorin, V.; Cobas, C.; Kazimierczuk, K.; Butts, C. Enabling Fast Pseudo-2D NMR Spectral Acquisition for Broadband Homonuclear Decoupling: The EXACT NMR Approach. ChemPhysChem
**2017**, 18, 2081–2087. [Google Scholar] [CrossRef] [PubMed][Green Version] - Shchukina, A.; Kaźmierczak, M.; Kasprzak, P.; Davy, M.; Akien, G.R.; Butts, C.P.; Kazimierczuk, K. Accelerated acquisition in pure-shift spectra based on prior knowledge from 1H NMR. Chem. Commun.
**2019**, 55, 9563–9566. [Google Scholar] [CrossRef] - Mobli, M.; Miljenović, T.M. Framework for and evaluation of bursts in random sampling of multidimensional NMR experiments. J. Magn. Reson.
**2019**, 300, 103–113. [Google Scholar] [CrossRef] - Davis, D.G. Improved multiplet editing of proton-detected, heteronuclear shift-correlation spectra. J. Magn. Reson. (1969)
**1991**. [Google Scholar] [CrossRef] - Kay, L.E.; Bax, A. Separation of NH and NH2 resonances in 1H-detected heteronuclear multiple-quantum correlation spectra. J. Magn. Reson. (1969)
**1989**. [Google Scholar] [CrossRef] - Jaravine, V.A.; Zhuravleva, A.V.; Permi, P.; Ibraghimov, I.; Orekhov, V. Hyperdimensional NMR Spectroscopy with Nonlinear Sampling. J. Am. Chem. Soc.
**2008**, 130, 3927–3936. [Google Scholar] [CrossRef] [PubMed] - Dötsch, V.; Wagner, G. Editing for amino-acid type in CBCACONH experiments based on the 13 Cβ- 13 Cγ coupling. J. Magn. Reson. Ser. B
**1996**, 111, 310–313. [Google Scholar] [CrossRef] [PubMed] - Grzesiekt, S.; Bax, A. Correlating Backbone Amide and Side Chain Resonances in Larger Proteins by Multiple Relayed Triple Resonance NMR. J. Am. Chem. Soc.
**1992**, 114, 6291–6293. [Google Scholar] [CrossRef] - Piai, A.; Gonnelli, L.; Felli, I.; Pierattelli, R.; Kazimierczuk, K.; Grudziaz, K.; Koźmiński, W.; Zawadzka-Kazimierczuk, A. Amino acid recognition for automatic resonance assignment of intrinsically disordered proteins. J. Biomol. NMR
**2016**, 64, 239–253. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lin, M.; Shapiro, M.J.; Wareing, J.R. Diffusion-Edited NMR-Affinity NMR for Direct Observation of Molecular Interactions. J. Am. Chem. Soc.
**1997**, 119, 5249–5250. [Google Scholar] [CrossRef] - Vega-Vázquez, M.; Cobas, J.C.; Oliveira De Sousa, F.F.; Martin-Pastor, M. A NMR reverse diffusion filter for the simplification of spectra of complex mixtures and the study of drug receptor interactions. Magn. Reson. Chem.
**2011**, 49, 464–468. [Google Scholar] [CrossRef] - Carr, H.Y.; Purcell, E.M. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev.
**1954**. [Google Scholar] [CrossRef] - Kazimierczuk, K.; Zawadzka, A.; Koźmiński, W. Optimization of random time domain sampling in multidimensional NMR. J. Magn. Reson.
**2008**, 192, 123–130. [Google Scholar] [CrossRef] - Hyberts, S.G.; Arthanari, H.; Wagner, G. Applications of non-uniform sampling and processing. Top. Curr. Chem.
**2012**, 316, 125–148. [Google Scholar] [CrossRef][Green Version] - Kazimierczuk, K.; Orekhov, V. Non-uniform sampling: Post-Fourier era of NMR data collection and processing. Magn. Reson. Chem.
**2015**, 53, 921–926. [Google Scholar] [CrossRef] - Zambrello, M.A.; Craft, D.L.; Hoch, J.C.; Rovnyak, D.; Schuyler, A.D. The influence of the probability density function on spectral quality in nonuniformly sampled multidimensional NMR. J. Magn. Reson.
**2020**, 311, 106671. [Google Scholar] [CrossRef] [PubMed] - Diercks, T.; Truffault, V.; Coles, M.; Millet, O. Diagonal-free 3D/4D HN,HN-trosy-noesy-trosy. J. Am. Chem. Soc.
**2010**, 132, 2138–2139. [Google Scholar] [CrossRef] [PubMed] - Stanek, J.; Augustyniak, R.; Koźmiński, W. Suppression of sampling artefacts in high-resolution four-dimensional NMR spectra using signal separation algorithm. J. Magn. Reson.
**2012**, 214, 91–102. [Google Scholar] [CrossRef] [PubMed] - Wen, J.; Zhou, P.; Wu, J. Efficient acquisition of high-resolution 4-D diagonal-suppressed methyl-methyl NOESY for large proteins. J. Magn. Reson.
**2012**, 218, 128–132. [Google Scholar] [CrossRef][Green Version] - Stanek, J.; Nowakowski, M.; Saxena, S.; Ruszczyńska-Bartnik, K.; Ejchart, A.; Koźmiński, W. Selective diagonal-free 13 C, 13 C-edited aliphatic-aromatic NOESY experiment with non-uniform sampling. J. Biomol. NMR
**2013**, 56, 217–226. [Google Scholar] [CrossRef][Green Version] - Werner-Allen, J.W.; Coggins, B.E.; Zhou, P. Fast acquisition of high resolution 4-D amide-amide NOESY with diagonal suppression, sparse sampling and FFT-CLEAN. J. Magn. Reson.
**2010**, 204, 173–178. [Google Scholar] [CrossRef][Green Version] - Amir, A.; Zuk, O. Bacterial community reconstruction using compressed sensing. J. Comput. Biol.
**2011**, 18, 1723–1741. [Google Scholar] [CrossRef] - Morris, G.A. NMR Data Processing. Encycl. Spectrosc. Spectrom.
**2017**, 125–133. [Google Scholar] [CrossRef] - Mayzel, M.; Kazimierczuk, K.; Orekhov, V.Y. The causality principle in the reconstruction of sparse NMR spectra. Chem. Commun.
**2014**, 50, 8947–8950. [Google Scholar] [CrossRef] [PubMed][Green Version] - Shimba, N.; Stern, A.S.; Craik, C.S.; Hoch, J.C.; Dötsch, V. Elimination of 13Cα splitting in protein NMR spectra by deconvolution with maximum entropy reconstruction. J. Am. Chem. Soc.
**2003**, 125, 2382–2383. [Google Scholar] [CrossRef] [PubMed] - Kerfah, R.; Hamelin, O.; Boisbouvier, J.; Marion, D. CH3-specific NMR assignment of alanine, isoleucine, leucine and valine methyl groups in high molecular weight proteins using a single sample. J. Biomol. NMR
**2015**, 63, 389–402. [Google Scholar] [CrossRef] [PubMed] - Robson, S.A.; Takeuchi, K.; Boeszoermenyi, A.; Coote, P.W.; Dubey, A.; Hyberts, S.; Wagner, G.; Arthanari, H. Mixed pyruvate labeling enables backbone resonance assignment of large proteins using a single experiment. Nat. Commun.
**2018**, 9. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ottiger, M.; Delaglio, F.; Bax, A. Measurement of J and Dipolar Couplings from Simplified Two-Dimensional NMR Spectra. J. Magn. Reson.
**1998**, 131, 373–378. [Google Scholar] [CrossRef] [PubMed][Green Version] - Andersson, P.; Weigelt, J.; Otting, G. Spin-state selection filters for the measurement of heteronuclear one-bond coupling constants. J. Biomol. NMR
**1998**, 12, 435–441. [Google Scholar] [CrossRef] - Stern, A.S.; Hoch, J.C. A new approach to compressed sensing for NMR. Magn. Reson. Chem.
**2015**, 53, 908–912. [Google Scholar] [CrossRef] - Jaravine, V.; Ibraghimov, I.; Orekhov, V.Y. Removal of a time barrier for high-resolution multidimensional NMR spectroscopy. Nat. Methods
**2006**, 3, 605–607. [Google Scholar] [CrossRef] - Liu, Y.; Li, M.; Pados, D.A. Motion-aware decoding of compressed-sensed video. IEEE Trans. Circuits Syst. Video Technol.
**2013**, 23, 438–444. [Google Scholar] [CrossRef] - Konar, A.S.; Aiholli, S.; Shashikala, H.C.; Babu, D.R.; Geethanath, S. Application of Region of Interest Compressed Sensing to accelerate magnetic resonance angiography. In Proceedings of the 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBC 2014, Chicago, IL, USA, 26–30 August 2014; pp. 2428–2431. [Google Scholar] [CrossRef]
- Kazimierczuk, K.; Zawadzka, A.; Koźmiński, W.; Zhukov, I. Lineshapes and artifacts in Multidimensional Fourier Transform of arbitrary sampled NMR data sets. J. Magn. Reson.
**2007**, 188, 344–356. [Google Scholar] [CrossRef] - Mitchell, D.P. Generating antialiased images at low sampling densities. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1987, Anaheim, CA, USA, 27–31 July 1987; pp. 65–72. [Google Scholar] [CrossRef]
- Lagae, A.; Dutré, P. A comparison of methods for generating Poisson disk distributions. Comput. Graph. Forum
**2008**, 27, 114–129. [Google Scholar] [CrossRef][Green Version] - Hyberts, S.G.; Takeuchi, K.; Wagner, G. Poisson-gap sampling and forward maximum entropy reconstruction for enhancing the resolution and sensitivity of protein NMR data. J. Am. Chem. Soc.
**2010**, 132, 2145–2147. [Google Scholar] [CrossRef][Green Version] - Barna, J.C.; Laue, E.D.; Mayger, M.R.; Skilling, J.; Worrall, S.J. Exponential sampling, an alternative method for sampling in two-dimensional NMR experiments. J. Magn. Reson. (1969)
**1987**, 73, 69–77. [Google Scholar] [CrossRef] - Paramasivam, S.; Suiter, C.L.; Hou, G.; Sun, S.; Palmer, M.; Hoch, J.C.; Rovnyak, D.; Polenova, T. Enhanced sensitivity by nonuniform sampling enables multidimensional MAS NMR spectroscopy of protein assemblies. J. Phys. Chem. B
**2012**, 116, 7416–7427. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kazimierczuk, K.; Lafon, O.; Lesot, P. Criteria for sensitivity enhancement by compressed sensing: Practical application to anisotropic NAD 2D-NMR spectroscopy. Analyst
**2014**, 139, 2702–2713. [Google Scholar] [CrossRef] [PubMed] - Hyberts, S.G.; Robson, S.A.; Wagner, G. Interpolating and extrapolating with hmsIST: Seeking a t
_{max}for optimal sensitivity, resolution and frequency accuracy. J. Biomol. NMR**2017**, 68, 139–154. [Google Scholar] [CrossRef] [PubMed] - Lee, M.E.; Redner, R.A.; Uselton, S.P. Statistically Optimized Sampling for Distributed Ray Tracing. Comput. Graph. (ACM)
**1985**, 19, 61–67. [Google Scholar] [CrossRef] - Kajiya, J.T. The rendering equation. In Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 1986, Dallas, TX, USA, 18–22 August 1986; pp. 143–150. [Google Scholar] [CrossRef]
- Gołowicz, D.; Kasprzak, P.; Orekhov, V.; Kazimierczuk, K. Fast time-resolved NMR with non-uniform sampling. Prog. Nucl. Mag. Res. Spectrosc.
**2019**. [Google Scholar] [CrossRef] - Dass, R.; Kasprzak, P.; Koźmiński, W.; Kazimierczuk, K. Artifacts in time-resolved NUS: A case study of NOE build-up curves from 2D NOESY. J. Magn. Reson.
**2016**, 265, 108–116. [Google Scholar] [CrossRef] - Bermel, W.; Dass, R.; Neidig, K.P.; Kazimierczuk, K. Two-Dimensional NMR Spectroscopy with Temperature-Sweep. ChemPhysChem
**2014**, 15, 2217–2220. [Google Scholar] [CrossRef] - Dass, R.; Grudzia̧ż, K.; Ishikawa, T.; Nowakowski, M.; Dbowska, R.; Kazimierczuk, K. Fast 2D NMR spectroscopy for in vivo monitoring of bacterial metabolism in complex mixtures. Front. Microbiol.
**2017**, 8, 1306. [Google Scholar] [CrossRef][Green Version] - Wu, Y.; D’Agostino, C.; Holland, D.J.; Gladden, L.F. In situ study of reaction kinetics using compressed sensing NMR. Chem. Commun.
**2014**, 50, 14137–14140. [Google Scholar] [CrossRef][Green Version] - Gołowicz, D.; Kazimierczuk, K.; Urbańczyk, M.; Ratajczyk, T. Monitoring Hydrogenation Reactions using Benchtop 2D NMR with Extraordinary Sensitivity and Spectral Resolution. ChemistryOpen
**2019**, 8, 196–200. [Google Scholar] [CrossRef] [PubMed] - Vasanawala, S.S.; Alley, M.T.; Hargreaves, B.A.; Barth, R.A.; Pauly, J.M.; Lustig, M. Improved pediatric MR imaging with compressed sensing. Radiology
**2010**, 256, 607–616. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kay, L.E.; Keifer, P.; Saarinen, T. Pure absorption gradient enhanced heteronuclear single quantum correlation spectroscopy with improved sensitivity. J. Am. Chem. Soc.
**1992**, 114, 10663–10665. [Google Scholar] [CrossRef] - Orekhov, V.Y.; Jaravine, V.; Mayzel, M.; Kazimierczuk, K. MddNMR—Reconstruction of NMR Spectra from NUS Signal Using MDD and CS. Available online: http://mddnmr.spektrino.com (accessed on 28 February 2020).

**Figure 1.**A simulation showing gains from the enhanced compressibility of the NMR spectrum obtained with pure-shift method. A conventionally sampled signal of 512 points length, containing 7 components (see corresponding triplet and quartet in spectrum (

**A**)) was sub-sampled to 32 random points and reconstructed using 40 iterations of iteratively re-weighted least squares [20] algorithm (

**B**). In this case, the sampling level turned out to be too low resulting in wrong reconstruction. A fully sampled (512 points) pure-shift experiment shows multiplets collapsed into the singlets (spectrum (

**C**)). The corresponding reconstructed spectrum (

**D**) obtained using the same sub–sampling scheme and reconstruction parameters as for (

**B**) reveals to be of good quality. Reduced number of signal components (enhanced compressibility) allowed for reliable signal reconstruction using the same number of sampling points.

**Figure 2.**A simulation demonstrating the quality of the reconstruction of a signal built of two low amplitude components and one very high amplitude component (

**A**,

**B**); and of the reconstruction of the same signal with the highest amplitude component being removed (

**C**,

**D**). Both starting signals (their corresponding spectra—

**A**,

**C**) were sub-sampled to the same 42 random points (out of 512 points) and reconstructed using 40 iterations of the IRLS algorithm. The reconstruction of low amplitude signals in the presence of a high dynamic range of amplitudes was unsuccessful (

**B**). The same sub-sampling level and scheme turned out to be sufficient for the fine reconstruction of the signal in the absence of a high amplitude component (

**D**). The ratio of the largest peak intensity to the small ones in (

**A**) was 160. Notably, the level of noise, artificially added to the FID is not reproduced in the CS reconstruction.

**Figure 3.**The concept of Virtual-Echo enhancing the compression level and improving the reconstruction quality. The upper panels (

**A**–

**C**) correspond to the signal and reconstruction without the use of VE, while the lower panels (

**D**–

**F**) correspond to the signal and reconstruction with VE pre-processing. A starting signal of 256 points length, containing 5 components of different amplitudes was zero-filled to 512 points (

**A**) or processed accordingly to the VE method (

**D**). The spectral representations of both signals, including the real (blue line) and imaginary (red line) parts, are shown in (

**B**,

**E**). A less sparse imaginary part of the starting signal is completely removed when VE is applied (

**E**). Both signals were sub-sampled using the same sampling scheme of 48 random points. Importantly, a sampling scheme also undergoes the operation of VE in the same way as the signal. The missing points were reconstructed using 40 iterations of the CS-IRLS algorithm. The resulting spectra indicate that VE pre-processing leads to a better-quality spectrum (

**F**). At this level of sampling, the spectrum reconstructed without VE pre-processing (

**C**) suffers from characteristic phase distortions (see black arrows in the corresponding panel). The dotted line in panels C and F shows the real part of the fully sampled spectrum.

**Figure 4.**A simulation illustrating the benefit of relaxation-matched non-uniform sampling on signal reconstruction. A signal of 1024 points length containing 2 components of equal amplitudes (

**A**) was artificially contaminated with a white noise such that peaks in a corresponding spectrum (

**B**) were hardly visible. A blue spectrum imposed in (

**B**,

**D**,

**F**) is obtained from the noiseless signal (

**A**) to mark the correct positions of the hidden peaks (for better visualization, the peak intensities in blue spectra are normalized to half-intensity of the maximum peak in the corresponding black spectrum). The same 2-component signal was sub-sampled to 256 random points (

**C**), and 256 points selected according to the relaxation-matched probability (

**E**). A continuous black line in (

**C**,

**E**) stands for the full signal, whereas red markers correspond to sub-sampled points. Both sub-sampled sets of points were used for reconstruction using 40 iterations of the IRLS algorithm. Importantly, the sub-sampled signals (

**C**,

**E**) were injected into a noise being 2 times lower than for signal A. This is due to a fact that 256 points can be acquired with 4 times more scans keeping the same total experimental time, thus SNR of the acquired samples will be 2 times higher. A reconstructed spectrum (

**D**), obtained from random non-uniform sampling strategy (

**C**) shows no improvement, while the spectrum (

**F**) obtained from a relaxation-matched non-uniform sampling strategy (

**E**) indicates a significant improvement of the visibility of peaks. As described above in the text, the relaxation-matched sampling (

**E**) strategy leads to better results in such cases as more samples are collected for the initial part of the signal, where SNR is higher.

**Figure 5.**A simulation showing the benefit of applying NUS for the acquisition of a non-stationary signal (with frequency varying linearly during the experiment). A signal of 512 points length contains two components of equal amplitudes (blue dotted line in each subplot). The frequency of the component corresponding to the left peak changes by 0.05 spectral point with every NUS point acquired. The sampling levels and total frequency change are: 100% and 25.6 pts. (

**A**), 75% and 19.2 pts. (

**B**), 50% and 12.8 pts. (

**C**), 25% and 6.4 pts. (

**D**), 12.5% and 3.2 pts. (

**E**), and 6.25% and 1.6 pts. (

**F**). The sampling schedule is shuffled so the change is not linear in a sampled time, but in a real time of experiment. Thus, the non-stationarity leads to line broadening and additional noise-like artifacts [81,82]. Importantly, the best spectrum is obtained with 12.5 % sampling (E, far better than with full sampling A). All the NUS data sets, except of 100% NUS, were reconstructed with 40 iterations of CS–IRLS algorithm and their corresponding spectra are plotted in black.

**Figure 6.**A reference unedited

^{13}C HSQC spectrum with conventional sampling of 256 t

_{1}(

^{13}C dimension) × 3348 t

_{2}(

^{1}H dimension) points matrix (

**a**) and the reconstructed spectra obtained using only 24 t

_{1}sub-samples from corresponding experiments: unedited

^{13}C HSQC (

**b**),

^{13}C HSQC with CH-only editing (

**c**),

^{13}C HSQC with CPMG filter (

**d**). The missing data for (

**b**–

**d**) was reconstructed with IRLS algorithm based on CS using 40 iterations. The virtual-echo method was applied in all the reconstructions. The processing was performed using mddnmr software [89]. The concentration of each compound in a sample was adjusted to yield similar peak heights in the

^{1}H NMR spectrum (ca. 0.6 mg/mL of sucrose, and 14.6 mg/mL of heparin).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gołowicz, D.; Kasprzak, P.; Kazimierczuk, K. Enhancing Compression Level for More Efficient Compressed Sensing and Other Lessons from NMR Spectroscopy. *Sensors* **2020**, *20*, 1325.
https://doi.org/10.3390/s20051325

**AMA Style**

Gołowicz D, Kasprzak P, Kazimierczuk K. Enhancing Compression Level for More Efficient Compressed Sensing and Other Lessons from NMR Spectroscopy. *Sensors*. 2020; 20(5):1325.
https://doi.org/10.3390/s20051325

**Chicago/Turabian Style**

Gołowicz, Dariusz, Paweł Kasprzak, and Krzysztof Kazimierczuk. 2020. "Enhancing Compression Level for More Efficient Compressed Sensing and Other Lessons from NMR Spectroscopy" *Sensors* 20, no. 5: 1325.
https://doi.org/10.3390/s20051325