# Random Noise Suppression of Magnetic Resonance Sounding Data with Intensive Sampling Sparse Reconstruction and Kernel Regression Estimation

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## Abstract

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## 1. Introduction

## 2. MRS Signal Analysis and the Classical Stacking Method

#### 2.1. MRS Signal Analysis

#### 2.2. Classic Stacking Method

## 3. Intensive Sampling Sparse Reconstruction and Kernel Regression Estimation

#### 3.1. Overall Approach of ISSR-KRE for Suppressing Random Noise

#### 3.2. Intensive Sampling Sparse Reconstruction for Suppressing Random Noise

#### 3.2.1. Basic Frequency of Sparse Reconstruction

#### 3.2.2. ISSR Implementation Process

#### 3.2.3. Rectangular Sparse Reconstruction

_{VI}is SNR for voltage prior to sparse reconstruction, SNR

_{VO}is SNR for voltage after sparse reconstruction, $SNI{R}_{V(\mathrm{Re}\mathrm{ct})}$ denotes signal to noise improvement ratio for voltage of rectangular sparse reconstruction, and the unit of $SNI{R}_{V(\mathrm{Re}\mathrm{ct})}$ is one.

#### 3.2.4. Trapezoidal Sparse Reconstruction

_{V}, is derived for evaluating noise elimination effect of trapezoidal sparse reconstruction.

#### 3.2.5. Simpson Sparse Reconstruction

#### 3.3. Kernel Regression Estimation for Suppressing Random Noise

_{i}, then the Taylor series expansion of the regression function $f({x}_{i})$ at the time $x$ can be obtained based on Taylor series expansion theory [29].

## 4. Numerical Simulations

^{2}.

#### 4.1. Simulations of Intensive Sampling Sparse Reconstruction for Random Noise Suppression

#### 4.1.1. Sparse Reconstruction Simulation

^{2}. The blue lines display simulation experimental value, and the red lines display the theoretical value.

#### 4.1.2. Comparison of Three Sparse Reconstructions

#### Comparison of SNR, SNIR and MSE

#### Waveform Comparison

#### Different ${f}_{prop}$ Effect on Sparse Reconstruction

#### 4.2. Comparison of Intensive Sampling Sparse Reconstruction and Classical Stacking Method

#### 4.2.1. Simulations of Classical Stacking for Random Noise Suppression

^{2}. The blue lines display simulation experimental value, and the red lines display theoretical value of the power signal-to-noise ratio. SNR is a non-logarithmic representation of the power signal to noise ratio.

#### 4.2.2. Comparison of Intensive Sampling Sparse Reconstruction and Classical Stacking Method

#### 4.3. Simulation of Kernel Regression Estimation

#### 4.3.1. Window Size and Smoothing Factor Effect on the Estimation Result

#### Window Size Effect

#### Smoothing Factor Effect

^{2}.

#### 4.3.2. Waveform Comparison of Kernel Regression Estimation

## 5. Field Experiments

#### 5.1. Laboratory Experiment

#### 5.2. Processing Experiment of Noisy MRS Data

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Sparse Reconstruction

## References

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**Figure 2.**Flowchart for implementation of the intensive sampling sparse reconstruction and kernel regression estimation (ISSR-KRE) approach.

**Figure 5.**Simulation case: Comparison of suppressing random noise effect of rectangular sparse reconstruction by different sampling frequencies ${f}_{{s}_{H}}$. Time-series and spectra of the sampling frequencies 8${f}_{prop}$, 16 ${f}_{prop}$, 32 ${f}_{prop}$, and 64 ${f}_{prop}$, respectively.

**Figure 6.**The curves of signal to noise ratio (SNR), signal-to-noise improvement ratio (SNIR), mean squared error (MSE) with increasing sampling frequency by rectangular sparse reconstruction.

**Figure 7.**Comparison of three sparse reconstruction methods (rectangular sparse reconstruction, trapezoidal sparse reconstruction and Simpson sparse reconstruction): (

**a**) SNR comparison, (

**b**) SNIR comparison, (

**c**) MSE comparison.

**Figure 8.**Comparison of rectangular sparse reconstruction, trapezoidal sparse reconstruction and Simpson sparse reconstruction for processing signals. (

**a**) Time domain, (

**b**) frequency domain.

**Figure 9.**Different ${f}_{prop}$ effect on sparse reconstruction. The first row shows the comparison of SNR in different ${f}_{prop}$ cases, respectively. The second row shows the comparison of SNIR and MSE in different ${f}_{prop}$ cases, respectively.

**Figure 10.**Simulation case: Time-series and spectra of 8, 16, 32, 64 stacks at one pulse moment, respectively.

**Figure 12.**The effect of kernel regression parameters on estimation results. (

**a**) The effect of window size. (

**b**) The effect of the smoothing factor h.

**Figure 13.**Waveform comparison of the data processed by the local quadratic estimator (window size 19 and smoothing factor $h=5/{f}_{prop}$ ). (

**a**) Comparison in time domain. (

**b**) Comparison in frequency domain. The green lines display the data by the rectangular sparse reconstruction method. The red lines display the ideal signal. The blue lines display the data by the rectangular sparse reconstruction method and the local quadratic kernel regression estimation. In other words, the blue lines display the ISSR-KRE data.

**Figure 14.**Comparison of traditional low frequency sampling and the ISSR method. The gray lines display traditional low frequency sampling recording. The blue lines display the data processed by the ISSR method. (

**a**) Comparison in time domain. (

**b**) Comparison in frequency domain.

**Figure 16.**Comparison of low frequency sampling result and the ISSR method results. The green lines display low frequency sampling recording (its sampling frequency is 8330 Hz) and the standard deviation is 1200.80 nV. The blue lines display the data by the ISSR method (its sampling frequency is 50 kHz, ${f}_{prop}$ = 8333.3 Hz) and the standard deviation is 570.01 nV. (

**a**) Comparison in time domain. (

**b**) Comparison in frequency domain.

**Table 1.**SNR comparison of three ISSR methods in the experiment. The estimated parameters, $SN{R}_{\mathrm{Re}ct}$, $SN{R}_{Trap}$ and $SN{R}_{Simp}$ are reported based on the synthetic MRS signal with ${U}_{0}^{}$ = 40.46 nV, ${T}_{2}^{*}$ = 500 ms, ${f}_{Lamor}$ = 2341.7 Hz, ${f}_{prop}$ = 24${f}_{Lamor}$ and initial SNR = −20 dB.

${\mathit{f}}_{{\mathit{s}}_{\mathit{H}}}(\mathbf{Hz})$ | $\mathbf{8}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{16}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{32}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{64}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{128}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{256}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{512}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ |
---|---|---|---|---|---|---|---|

$SN{R}_{\mathrm{Re}ct}-SN{R}_{Trap}\text{}(dB)$ | 0.2519 | 0.0999 | 0.0610 | 0.0487 | 0.0253 | 0.0032 | 0.0027 |

$SN{R}_{\mathrm{Re}ct}-SN{R}_{Simp}\text{}(dB)$ | 0.3419 | 0.3650 | 0.3974 | 0.4625 | 0.4689 | 0.4814 | 0.4970 |

**Table 2.**The classical stacking method. The estimated parameters, SNR and SNIR are reported based on the synthetic noisy MRS signal with ${U}_{0}^{}$ = 40.46 nV, ${T}_{2}^{*}$ = 500 ms, ${f}_{Larmor}$ = 2341.7 Hz, and initial SNR = −20 dB.

${\mathit{N}}_{\mathit{S}}$ | 1 | 8 | 16 | 32 | 64 |
---|---|---|---|---|---|

$SNR\text{}(dB)$ | −20 | −10.9502 | −7.8772 | −4.9493 | −1.9423 |

$SNIR$ | 1 | 7.8387 | 15.9964 | 32.3185 | 63.2067 |

**Table 3.**The intensive sampling rectangular sparse reconstruction. The estimated parameters, SNR and SNIR are reported based on the synthetic MRS signal with ${U}_{0}^{}$ = 40.46 nV, ${T}_{2}^{*}$ = 500 ms, ${f}_{Larmor}$ = 2341.7 Hz, ${f}_{prop}$ = 24${f}_{Larmor}$ and SNR = −20 dB.

${\mathit{f}}_{{\mathit{s}}_{\mathit{H}}}(\mathbf{Hz})$ | ${\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{8}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{16}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{32}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ | $\mathbf{64}{\mathit{f}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{p}}$ |
---|---|---|---|---|---|

$SNR\text{}(\mathrm{dB})$ | −20 | −10.9537 | −7.8869 | −4.9097 | −1.9680 |

$SNIR$ | 1 | 7.7996 | 15.8032 | 31.3668 | 61.7511 |

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**MDPI and ACS Style**

Yao, X.; Zhang, J.; Yu, Z.; Zhao, F.; Sun, Y.
Random Noise Suppression of Magnetic Resonance Sounding Data with Intensive Sampling Sparse Reconstruction and Kernel Regression Estimation. *Remote Sens.* **2019**, *11*, 1829.
https://doi.org/10.3390/rs11151829

**AMA Style**

Yao X, Zhang J, Yu Z, Zhao F, Sun Y.
Random Noise Suppression of Magnetic Resonance Sounding Data with Intensive Sampling Sparse Reconstruction and Kernel Regression Estimation. *Remote Sensing*. 2019; 11(15):1829.
https://doi.org/10.3390/rs11151829

**Chicago/Turabian Style**

Yao, Xiaokang, Jianmin Zhang, Zhenyang Yu, Fa Zhao, and Yong Sun.
2019. "Random Noise Suppression of Magnetic Resonance Sounding Data with Intensive Sampling Sparse Reconstruction and Kernel Regression Estimation" *Remote Sensing* 11, no. 15: 1829.
https://doi.org/10.3390/rs11151829