# Efficient Lidar Signal Denoising Algorithm Using Variational Mode Decomposition Combined with a Whale Optimization Algorithm

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## Abstract

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## 1. Introduction

## 2. Brief Description of the VMD Algorithm

_{k}is the kth mode, ω

_{k}is the center frequency, ${\partial}_{t}$ represents the gradient with respect to t, t is the time script, δ(t) denotes the impulse function, ‖…‖

_{2}is the norm, ${\Vert {\partial}_{t}[(\delta \left(t\right)+\frac{j}{\pi t})\times {u}_{k}\left(t\right)]{e}^{-j{\omega}_{k}t}\Vert}_{2}^{2}$ represents the bandwidth of each mode, and f is the original signal to be decomposed.

_{k}and ω

_{k}in two directions is assumed to help realize the analysis process of VMD, and the solutions for them are as follows:

Algorithm 1. Pseudocode for VMD |

1: Initialize ${\widehat{u}}_{k},{\omega}_{k}\to 0$ |

2: Update ${\widehat{u}}_{k}$ $\to $ ${\widehat{u}}_{k}^{n+1}\left(\omega \right)=\frac{\widehat{f}\left(\omega \right)-{\displaystyle {\sum}_{i\ne k}{\widehat{u}}_{i}\left(\omega \right)+\frac{\widehat{\lambda}\left(\omega \right)}{2}}}{1+2\alpha {\left(\omega -{\omega}_{k}\right)}^{2}}$ |

3: Update ${\omega}_{k}$ $\to $ ${\omega}_{k}^{n+1}=\frac{{\displaystyle {\int}_{0}^{\infty}\omega {\left|{\widehat{u}}_{k}(\omega )\right|}^{2}d\omega}}{{\displaystyle {\int}_{0}^{\infty}{\left|{\widehat{u}}_{k}(\omega )\right|}^{2}d\omega}}$ |

4: Update ${\widehat{\lambda}}^{n}$ $\to $ ${\widehat{\lambda}}^{n+1}\left(\omega \right)={\widehat{\lambda}}^{n}\left(\omega \right)+\tau \left(\widehat{f}\left(\omega \right)-{\displaystyle {\sum}_{k}{\widehat{u}}_{k}^{n+1}}\left(\omega \right)\right)$ until convergence ${{\displaystyle {\sum}_{k}\Vert {\widehat{u}}_{k}^{n+1}-{\widehat{u}}_{k}^{n}\Vert}}_{2}^{2}/\Vert {\widehat{u}}_{k}^{n}\Vert <e$. |

## 3. Principle of the VMD-WOA Model for Noise Reduction

#### 3.1. Optimization of VMD Parameters Based on the WOA

_{rand}. The mathematical model is as follows:

_{k}/E is the percentage of the energy of the kth mode in the total signal energy, ${E}_{k}={\displaystyle \sum _{t}{u}_{k}{(t)}^{2}}$, and $E={\displaystyle \sum _{k}{E}_{k}}$.

Algorithm 2. Pseudocode for WOA |

Initialize the whale population position X |

Calculate the fitness of each search agent |

X* is the best search agentwhile (i < maximum iteration number)for each search agent Update a, A, C, l, and p if1 (p < 0.5)if2 (|A| < 1)Update the position of the current search agent by the Equation (9) else if2 (|A| ≥ 1)Select a random search agent (X _{rand})Update the position of the current search agent by the Equation (14) end if2else if1 (p ≥ 0.5)Update the position of the current search by the Equation (12) end if1 |

end forCheck if any search agent goes beyond the search space and amend it Calculate the fitness of each search agent Update X* if there is a better solution i = i + 1 end whilereturn X* |

#### 3.2. Identification of Relevant Modes

_{i}, C(i), is defined as follows:

_{r}= i.

#### 3.3. Proposed VMD-WOA Methodology

## 4. Results and Discussion

#### 4.1. Experiments with Simulated Signals

_{in}= 5 dB and signal length N = 2048. The blue lines are the true signals, and the red lines represent the noisy signals. We compared the performance of the proposed algorithm with that of different denoising methods, namely, NeighCoeff-db4 WT (WT-db4), EMD direct thresholding (EMD-DT), EMD combined with soft thresholding and a roughness penalty (EMD-STRP), and VMD method based on the decomposition level of EMD (EMD-VMD).

_{out}) and the root-mean-square error (RMSE), were adopted and defined as follows:

_{2}and BLIMF

_{3}was the maximum. Thus, BLIMF

_{1}and BLIMF

_{2}were considered the relevant modes and could be used for signal reconstruction to achieve noise filtering.

_{out}of 14.27 dB. The EMD-DT, EMD-STRP, and EMD-VMD methods were better, but the denoised signals still lost many useful signal components, and the methods generated a large-amplitude distortion at both the beginning and the end of the filtered signal. The result of EMD-VMD was superior to that of EMD-based denoising. Thus, VMD is superior to EMD and can overcome the modal mixing problems in EMD.

_{in}values varying from −4 dB to 11 dB is displayed in Figure 7. The VMD-WOA method obtained the best results, as expected, under different SNR

_{in}. Even in the worst-case scenario, with the lowest SNR

_{in}of −4 dB, VMD-WOA still guaranteed a high SNR

_{out}up to 10.3 dB. In addition, the performance comparisons in terms of RMSE at different SNR

_{in}revealed that VMD-WOA can achieve the minimum error in reconstructing the Bumps signal.

_{out}/RMSE as a function of the SNR

_{in}, obtained by applying WT-db4, EMD-DT, EMD-STRP, EMD-VMD, and VMD-WOA to the three other synthetic signals. The entries that correspond to the best observed SNR

_{out}/RMSE are highlighted in bold. The table shows that the proposed VMD-WOA outperformed all the other methods for the Heavy Sine, Blocks, and Doppler signals in the whole range of the SNR

_{in}values. VMD-WOA showed the best performance, particularly for the Heavy Sine signal. When the SNR

_{in}value was 11 dB, the SNR

_{out}improved to 25.59 dB and the RMSE was reduced to 0.16 for VMD-WOA. The superiority and reliability of the proposed method were further proved by these experiments.

#### 4.2. Experiments on a Lidar Echo Signal

## 5. Conclusions

_{in}showed that the VMD-WOA method outperformed other denoising methods (WT-db4, EMD-IT, EMD-STRP, and EMD-VMD). Experimentally, the performance of VMD-WOA was compared with that of the other four methods on a lidar echo signal. The signal-to-noise ratio of the denoised signal was increased to 23.92 dB, and the detection range was extended from 6 to 10 km. Experiments on different aerosols were also carried out, and the aerosol extinction coefficient retrieved from the denoised signal showed good smoothness.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flowchart of the proposed variational mode decomposition (VMD)-whale optimization algorithm (WOA). PDF is probability density function.

**Figure 5.**Bhattacharyya distance between modes and the Bumps signal. BLIMF is band-limited intrinsic mode function.

**Figure 6.**Comparisons of different denoising techniques for the noisy Bumps signal. WT is wavelet transform, EMD is empirical mode decomposition, EMD-DT is EMD-direct thresholding, EMD-STRP is EMD combined with soft thresholding and a roughness penalty, EMD-VMD is the VMD method based on the decomposition level of EMD, and VMD-WOA is the VMD method combined with the whale optimization algorithm (WOA).

**Figure 8.**Comparison of denoising results from the proposed VMD-WOA and the other methods. GOF is geometric overlap factor.

**Table 1.**Denoising performance with different SNR

_{in}for the Heavy Sine, Blocks, and Doppler signals.

SNR_{in} | −4 dB | −1 dB | 2 dB | 5 dB | 8 dB | 11 dB |
---|---|---|---|---|---|---|

(a) Heavy Sine | ||||||

WT-db4 | 5.04/1.73 | 7.69/1.27 | 10.40/0.93 | 14.31/0.59 | 16.86/0.44 | 20.19/0.30 |

EMD-DT | 8.67/1.14 | 11.38/0.83 | 13.60/0.85 | 17.06/0.43 | 21.29/0.27 | 22.93/0.22 |

EMD-STRP | 9.03/1.10 | 11.49/1.04 | 13.87/0.82 | 17.36/0.42 | 21.39/0.26 | 23.02/0.22 |

EMD-VMD | 11.30/0.84 | 14.10/0.61 | 17.89/0.39 | 19.61/0.32 | 22.39/0.23 | 24.35/0.19 |

VMD-WOA | 12.59/0.73 | 16.80/0.45 | 19.25/0.34 | 21.78/0.24 | 24.23/0.19 | 25.59/0.16 |

(b) Blocks | ||||||

WT-db4 | 4.76/1.72 | 8.16/1.16 | 10.48/0.89 | 13.93/0.60 | 15.72/0.49 | 18.14/0.37 |

EMD-DT | 7.75/1.22 | 10.19/0.92 | 12.43/0.71 | 14.76/0.54 | 15.67/0.49 | 16.37/0.45 |

EMD-STRP | 7.87/1.20 | 10.30/0.91 | 12.50/0.70 | 14.68/0.55 | 15.40/0.50 | 17.55/0.39 |

EMD-VMD | 10.90/0.85 | 11.93/0.75 | 13.81/0.61 | 15.26/0.51 | 16.08/0.47 | 17.80/0.38 |

VMD-WOA | 11.02/0.83 | 12.30/0.72 | 14.01/0.59 | 15.85/0.48 | 16.27/0.46 | 18.25/0.36 |

(c) Doppler | ||||||

WT-db4 | 4.98/0.167 | 8.62/0.109 | 10.81/0.082 | 13.25/0.064 | 15.06/0.052 | 18.38/0.035 |

EMD-DT | 6.74/0.127 | 10.38/0.089 | 12.61/0.069 | 14.89/0.053 | 16.28/0.045 | 18.57/0.034 |

EMD-STRP | 6.68/0.132 | 10.55/0.087 | 11.43/0.079 | 13.55/0.062 | 14.89/0.053 | 17.69/0.038 |

EMD-VMD | 9.76/0.095 | 11.70/0.076 | 12.35/0.071 | 14.91/0.053 | 16.54/0.044 | 18.45/0.035 |

VMD-WOA | 10.28/0.089 | 12.30/0.071 | 13.45/0.062 | 15.04/0.052 | 17.01/0.041 | 19.41/0.031 |

**Table 2.**The SNRs of the geometric overlap factor (GOF)-corrected signal processed by the different denoising techniques.

WT-db4 | EMD-DT | EMD-STRP | EMD-VMD | VMD-WOA | |
---|---|---|---|---|---|

SNR (dB) | 20.67 | 20.36 | 22.25 | 22.61 | 23.92 |

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**MDPI and ACS Style**

Li, H.; Chang, J.; Xu, F.; Liu, Z.; Yang, Z.; Zhang, L.; Zhang, S.; Mao, R.; Dou, X.; Liu, B.
Efficient Lidar Signal Denoising Algorithm Using Variational Mode Decomposition Combined with a Whale Optimization Algorithm. *Remote Sens.* **2019**, *11*, 126.
https://doi.org/10.3390/rs11020126

**AMA Style**

Li H, Chang J, Xu F, Liu Z, Yang Z, Zhang L, Zhang S, Mao R, Dou X, Liu B.
Efficient Lidar Signal Denoising Algorithm Using Variational Mode Decomposition Combined with a Whale Optimization Algorithm. *Remote Sensing*. 2019; 11(2):126.
https://doi.org/10.3390/rs11020126

**Chicago/Turabian Style**

Li, Hongxu, Jianhua Chang, Fan Xu, Zhenxing Liu, Zhenbo Yang, Luyao Zhang, Shuyi Zhang, Renxiang Mao, Xiaolei Dou, and Binggang Liu.
2019. "Efficient Lidar Signal Denoising Algorithm Using Variational Mode Decomposition Combined with a Whale Optimization Algorithm" *Remote Sensing* 11, no. 2: 126.
https://doi.org/10.3390/rs11020126