# Denoising of Magnetocardiography Based on Improved Variational Mode Decomposition and Interval Thresholding Method

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## Abstract

**:**

## 1. Introduction

## 2. Data Model

## 3. Proposed New VMD Scheme

#### 3.1. Eliminate Baseline Drift Noise Using Proposed Formulas

- Compute the associated analytic signal of each mode ${u}_{k}$ by means of the Hilbert transform, that is:$$\text{}{u}_{k,A}\left(t\right)=\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast {u}_{k}\left(t\right)$$
- Mix each mode with an exponential adjustment to the respective estimated center frequency in order to shift the mode spectrum to “baseband”.$$\text{}{\widehat{u}}_{k,A}\left(t\right)=\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast {u}_{k}\left(t\right)\right]{e}^{-j{\omega}_{k}t}$$
- Estimate the bandwidth through the squared ${L}^{2}$-norm of the gradient. The expression of the constrained variational problem is as follows:$$\{\begin{array}{c}\underset{\left\{{u}_{k}\right\},\left\{{\omega}_{k}\right\}}{\mathrm{min}}\left\{{\displaystyle {\displaystyle \sum}_{k}}\Vert {\partial}_{t}\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast {u}_{k}\left(t\right)\right]{e}^{-j{\omega}_{k}t}{\Vert}_{2}^{2}\right\}\\ s.t.{\displaystyle {\displaystyle \sum}_{k=1}^{K}}{u}_{k}\left(t\right)=f\left(t\right)\end{array}$$$$\mathcal{L}\left(\left\{{u}_{k}\right\},\left\{{\omega}_{k}\right\},\lambda \left(t\right)\right)=\alpha {\displaystyle \sum}_{k=1}^{K}\Vert {\partial}_{t}\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)\ast {u}_{k}\left(t\right)\right]{e}^{-j{\omega}_{k}t}{\Vert}_{2}^{2}+\Vert f\left(t\right)-{\displaystyle \sum}_{k=1}^{K}{u}_{k}\left(t\right){\Vert}_{2}^{2}+\lambda \left(t\right),f\left(t\right)-{\displaystyle \sum}_{k=1}^{K}{u}_{k}\left(t\right)$$

#### 3.2. Proposed Adaptive Decomposition

#### 3.3. Iterative Thresholding and Improved VMD Method

- Acquire the MCG signal with noise and initialize the number of modes k, the default of the penalty factor $\alpha $ is 2000, the default of the bandwidth $\tau $ is 0.
- Determine the value of penalty factor by performing the improved VMD method based on the correlation coefficient and obtain the first IMF.
- Eliminate the first IMF that contains baseline drift noise. Decompose the rest signal with the final update formulas and obtain several IMFs.
- Perform the interval thresholding operation on the IMFs obtained from step 3.
- Add all the processed IMFs together and refactor the MCG signal.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**A comparison of the approximately pure magnetocardiography (MCG) signal with the noisy signal; waveform characteristics can be clearly seen in (

**a**); (

**b**) shows the signal waveform with three kinds of noise. The signal is drowned in the noise under high noise circumstances.

**Figure 3.**The noisy signal and the intrinsic mode functions (IMFs) obtained by Ensemble Empirical Mode Decompositioning (EEMD) are shown in (

**a**,

**b**).

**Figure 4.**A comparison of the original signal (dotted line) with the reconstructed signals (solid line) obtained from EEMD based denoising methods with soft and hard thresholding. The panel above is the result of hard thresholding processing and the following panel is the result of soft thresholding processing.

**Figure 5.**The noisy signal and the IMFs obtained from VMD are shown in (

**a**). A comparison of the original signal (dotted line) with the reconstructed signals (solid line) obtained from VMD based denoising methods with soft and hard thresholding is shown in (

**b**). The panel above is the result of hard thresholding processing and the following panel is the result of soft thresholding processing.

**Figure 6.**The first IMF obtained from the improved VMD method (solid line) and baseline drift noise (dotted line). It is seen that the solid line and the dotted line basically coincide. This method can effectively remove baseline drift noise.

**Figure 7.**(

**a**) The noisy signal and the IMFs obtained from the improved VMD method; (

**b**) A comparison of the original signal (dotted line) with the reconstructed signals (solid line) obtained from the improved VMD method with soft and hard thresholding for interval thresholding. The panel above is the result of hard thresholding processing and the following panel is the result of soft thresholding processing.

**Figure 8.**The root-mean-square deviation (RMSE) of the EEMD, the VMD, and the improved VMD methods are revealed, for both soft and hard thresholding of interval thresholding. The improved VMD method outperforms other methods.

**Figure 9.**The variation of the output signal-to-noise ratio (SNR) by EEMD, VMD and the improved VMD methods with soft and hard thresholds for interval thresholding. The improved VMD method outperforms other methods.

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**MDPI and ACS Style**

Liao, Y.; He, C.; Guo, Q.
Denoising of Magnetocardiography Based on Improved Variational Mode Decomposition and Interval Thresholding Method. *Symmetry* **2018**, *10*, 269.
https://doi.org/10.3390/sym10070269

**AMA Style**

Liao Y, He C, Guo Q.
Denoising of Magnetocardiography Based on Improved Variational Mode Decomposition and Interval Thresholding Method. *Symmetry*. 2018; 10(7):269.
https://doi.org/10.3390/sym10070269

**Chicago/Turabian Style**

Liao, Yanping, Congcong He, and Qiang Guo.
2018. "Denoising of Magnetocardiography Based on Improved Variational Mode Decomposition and Interval Thresholding Method" *Symmetry* 10, no. 7: 269.
https://doi.org/10.3390/sym10070269