Atmospheric aerosols are the primary contributors to environmental pollution. As such aerosols are micro-to nanosized particles invisible to the naked eye, it is necessary to utilize LiDAR technology for their detection. The laser radar echo signal is vulnerable to background light and electronic thermal noise. While single-photon LiDAR can effectively reduce background light interference, electronic thermal noise remains a significant challenge, especially at long distances and in environments with a low signal-to-noise ratio (SNR). However, conventional denoising methods cannot achieve satisfactory results in this case. In this paper, a novel adaptive continuous threshold wavelet denoising algorithm is proposed to filter out the noise. The algorithm features an adaptive threshold and a continuous threshold function. The adaptive threshold is dynamically adjusted according to the wavelet decomposition level, and the continuous threshold function ensures continuity with lower constant error, thus optimizing the denoising process. Simulation results show that the proposed algorithm has excellent performance in improving SNR and reducing root mean square error (RMSE) compared with other algorithms. Experimental results show that denoising of an actual LiDAR echo signal results in a 4.37 dB improvement in SNR and a 39.5% reduction in RMSE. The proposed method significantly enhances the ability of single-photon LiDAR to detect weak signals.

Topics
Aerosols, Remote sensing, Environmental pollution, Electronic noise, Signal processing, Signal-to-noise ratio, Wavelet transform, Single-photon detector, Light detection and ranging, Quantum information

  • Adaptive continuous thresholding, which dynamically adjusts the threshold according to the number of wavelet decomposition layers, provides variable thresholds in different layers to effectively differentiate between signal and noise.

  • The continuous threshold function has good continuity and minimizes the deviation of the estimated wavelet coefficients from the actual values, which ensures the accuracy and effectiveness of the denoising process.

  • By denoising real measured LiDAR echo signals, it is demonstrated that this method enhancing the denoising capability while maintaining the integrity of the original signal details.

Topics
Aerosols, Remote sensing, Environmental pollution, Electronic noise, Signal processing, Signal-to-noise ratio, Wavelet transform, Single-photon detector, Light detection and ranging, Quantum information

Aerosols, comprising a mix of airborne solid and liquid particles such as clouds, fog, soot, and PM2.5 and PM10 particulates,1,2 have particle sizes ranging from 0.001 to 100 μm. Often invisible to the naked eye,3,4 their detection then relies on specialized tools such as single-photon LiDAR. This advanced technique combines laser remote sensing with quantum detection for enhanced resolution and accuracy, extending its use to applications in target identification,5,6 3D imaging,7 and atmospheric detection,8,9 especially in monitoring atmospheric elements and analyzing combustion aerosols.

Single-photon LiDAR excels in reducing background noise and boosting signal-to-noise ratio (SNR). However, challenges arise with increasing detection distances, where the echo signal strength decreases inversely with the square of the distance.10 Compounded by solar background light, detector dark current, and amplifier thermal noise, these faint signals become hard to discern.11–14 Electronic thermal noise, inherent in the system’s hardware and exacerbated in prolonged operation or high-temperature conditions, further impacts detection accuracy. Hence, developing sophisticated noise reduction algorithms is crucial for enhancing data quality and accuracy in aerosol detection via LiDAR.

Traditional LiDAR echo signal denoising techniques, including the Fourier transform, moving average, empirical modal decomposition (EMD), variational modal decomposition (VMD), and wavelet transform.15–18 have shown effectiveness in various contexts. However, each method has limitations. The Fourier transform, suitable for linear, smooth signals, may distort LiDAR echo signals during denoising owing to its inadequate handling of signal details.19 The moving average, known for its simplicity, can lead to loss of critical data points, affecting LiDAR’s accuracy.20 EMD, while useful for nonlinear signals, faces modal aliasing issues.21 VMD improves upon EMD, but struggles with parameter determination, impacting noise separation and overall performance.22 

Considering the limitations of the aforementioned methods, the wavelet transform not only addresses the shortcomings of the traditional Fourier transform, but also excels in multiscale analysis and parameter indirection, emerging as a vital alternative for LiDAR signal denoising.23–26 Fang and Huang27 pioneered the application of the wavelet transform to LiDAR signal denoising and noise cancellation, significantly extending the effective detection range of LiDAR. Mao et al.28 leveraged the principles of the wavelet transform algorithm, employing wavelet packet analysis to denoise LiDAR signals. This involved processing and reconstructing both high-frequency and low-frequency components, thereby enabling more accurate extinction coefficient contours, albeit with the use of a uniform threshold. Qin and Mao29 combined wavelet analysis with a neural network adaptive method for denoising LiDAR echo signals, which, despite its effectiveness, faced challenges of high computational complexity and dependence on the quality and quantity of training data. Liu et al.30 introduced a wavelet denoising algorithm based on parameter self-tuning, an improvement over the soft threshold function, but its tuning factor selection rules are vague and require manual adjustment, leaving room for improvement in the denoising process of LiDAR signals. Li et al.31 employed power and exponential functions to modify the threshold function in analyzing the vibration signals of automobile platforms, thereby enhancing the denoising effect. Although these methods proposed innovative threshold functions, mitigating the limitations of hard and soft threshold functions, they also introduced multiple tentative parameters that increased computational load or complexity, thus impeding real-time data output. Moreover, the use of uniform thresholds in the denoising process, given the differing noise levels in each layer of wavelet transform decomposition, inevitably retains some noise, thereby diminishing the subsequent denoising effect.32 

This study introduces an adaptive continuous threshold wavelet denoising (ACT-WD) approach that incorporates a novel threshold and continuous threshold function. The method dynamically adjusts the threshold value based on the wavelet decomposition level, providing a variable threshold across different levels to effectively discriminate between signal and noise, particularly in high-frequency regions. This adaptive strategy enhances the preservation of signal details. The novel threshold function minimizes the deviation of estimated wavelet coefficients from their actual values, ensuring the accuracy and efficacy of the denoising process. The effectiveness of this approach is substantiated through simulations involving three synthesized signals (Bumps, Blocks, and Simulated LiDAR Return Signal) and measured LiDAR echo signals. Comparative results demonstrate the superior performance of this method over three other noise reduction techniques, achieving enhanced denoising capability while preserving the integrity of the original signal details.

A detailed schematic of the single-photon LiDAR system utilized in this study is presented in Fig. 1. It includes a meter-scattering single-photon LiDAR operating at a wavelength of 1550 nm and incorporates a superconducting nanowire single-photon detector. The overall single-photon LiDAR system comprises a laser transmitting system, a laser receiving system, and a control system. The laser transmitting system emits 1550 nm laser pulses, the laser receiving system captures backscattered signals, and the control system executes synchronization and subsequent signal processing. With a remarkable detection efficiency of 88%, and a dark counting rate of 100 pcs in the presence of light with an external light source, reducing to less than 20 pcs without an external light source, these components collaborate to establish a closed-loop control system. This enables the LiDAR to precisely measure and analyze targets.

FIG. 1.
Schematic of single-photon LiDAR system.
View largeDownload slide

Schematic of single-photon LiDAR system.

FIG. 1.
Schematic of single-photon LiDAR system.
View largeDownload slide

Schematic of single-photon LiDAR system.

Close modal

Aerosol LiDAR works on the principle that the laser emitter first generates a monochromatic, well-directed laser beam and shoots it into the atmosphere on a predetermined trajectory. On encountering aerosol particles within the atmosphere, a portion of the photons undergo scattering. The backscattered photons are then captured by a photodetector, and the resultant photoelectric signal is converted by a photoelectric converter. Subsequently, an information processing system facilitates the determination of the concentration, spatial distribution, and optical properties of the aerosol particles.

According to the LiDAR equation, the echo signal can be expressed as33 
(1)
where P(r) is the signal power received at the distance r, P0 is the laser power emitted at time t0, c is the speed of light, τ is the width of the laser pulse, A is the effective receiving area of the receiving telescope, and β(r) and σ(r) are the atmospheric backscattering coefficient and extinction coefficient, respectively.
In single-photon LiDAR systems, the received backscattered signals are susceptible to interference from electronic thermal noise and background optical noise. In real-world environments, these noise sources are typically complex, and the actual noise energy can be conceptualized as the aggregate of several independent random noises, each with distinct probability distributions. According to the central limit theorem, as the number of noise sources increases, the normalized sum of all noises tends to approximate a Poisson distribution to a certain degree. Consequently, in the context of various noise interferences, the noise can be treated as conforming to a Poisson distribution.34,35 This premise underpins the formulation of the equation for the echo signal received by the LiDAR receiving system:
(2)
where x(t) is the true echo signal and n(t) is the Poisson noise caused by electronic thermal noise and background optical noise. The purpose of the adaptive continuous threshold wavelet denoising method proposed in this paper is to reduce the noise n(t) and obtain the true echo signal x(t) with good performance. Figure 2 illustrates both the ideal, noise-free LiDAR echo signal and the actual LiDAR echo signal, highlighting the substantial noise fluctuations at extended distances in the latter.
FIG. 2.
(a) Noise-free LiDAR echo signal. (b) Measured LiDAR echo signal.
View largeDownload slide

(a) Noise-free LiDAR echo signal. (b) Measured LiDAR echo signal.

FIG. 2.
(a) Noise-free LiDAR echo signal. (b) Measured LiDAR echo signal.
View largeDownload slide

(a) Noise-free LiDAR echo signal. (b) Measured LiDAR echo signal.

Close modal

The wavelet transform method,36 building upon the Fourier transform, is known for its strong time–frequency characteristics. It uses the Mallat algorithm to decompose the signal in layers to generate wavelet coefficients of different levels, including high-frequency (noise and detail) and low-frequency (main feature) components. Thresholding, either soft or hard, is then applied to these coefficients to filter out noise.37 The final step involves reconstructing the signal with the refined coefficients, resulting in a denoised output. The flowchart of wavelet transform denoising is presented in Fig. 3.

FIG. 3.
Flowchart of wavelet transform denoising.
View largeDownload slide

Flowchart of wavelet transform denoising.

FIG. 3.
Flowchart of wavelet transform denoising.
View largeDownload slide

Flowchart of wavelet transform denoising.

Close modal

In contrast to the wavelet decomposition and reconstruction of the signal, the selection of the threshold value and function assumes greater significance. The prevailing threshold in current practice is the Donoho uniform threshold, where the threshold for each layer is set to a consistent value. The formula for this threshold is expressed as λ=σln⁡N⁠, where σ denotes the standard deviation of the noise, and N represents the signal length.

In the actual denoising process, the standard deviation of the signal noise is unknown. Therefore, the standard deviation of the noise is determined by the estimation method only when the threshold is selected: σj,k=median(cj,k)/0.6754⁠,where median(cj,k) sets the noise to be the median of the selected detail coefficients.

The main choice with regard to the threshold function is between a hard threshold function and a soft threshold function, which are defined as
(3)
and
(4)
respectively, where wj,k are the original wavelet coefficients, λ is the threshold of the corresponding layer, ŵj,k are the estimated wavelet coefficients after thresholding, and sgn(·) is the sign function, taking values of ±1. The curves of the hard and soft threshold function curves are shown in Fig. 4.
FIG. 4.
Hard and soft threshold functions.
View largeDownload slide

Hard and soft threshold functions.

FIG. 4.
Hard and soft threshold functions.
View largeDownload slide

Hard and soft threshold functions.

Close modal

Hard and soft thresholding methods, as depicted in Fig. 4, differ in how they handle wavelet coefficients relative to the threshold value. Hard thresholding zeros out coefficients below the threshold while keeping those above unchanged, preserving signal edges but causing discontinuity and root-mean-square error, and thus affecting signal smoothness. Soft thresholding zeros out coefficients below the threshold but reduces those above toward zero. This produces a smoother continuous signal, but introduces a constant bias, resulting in blurred edges and loss of some signal details.

1. Adaptive thresholding

The accuracy of traditional wavelet thresholding denoising methods depends critically on correct selection of the wavelet threshold. An inappropriately high threshold might remove essential parts of the signal as noise, while too low a threshold can leave excessive noise, reducing the denoising effect and SNR. Thus, precise threshold selection is vital for effective noise elimination and maintaining the original signal’s integrity. Uniformly applying the same threshold across all decomposition layers in wavelet decomposition is not advisable, since noise coefficients decrease with increasing number of layers, and this approach could misjudge noisy signals and compromise denoising outcomes.

The aforementioned Donoho uniform threshold fails to satisfy these requirements. To ensure that the chosen threshold aligns more closely with the variability of the noise coefficient, this study introduces a novel threshold that varies according to the number of decomposition layers:
(5)
where σ is the noise variance, N is the signal length, j is the number of decomposition layers, and as the number of decomposition layers increases, λj becomes smaller, thus realizing flexibility of different thresholds for different layers. When the number of decomposition layers j = 1, λ1=σ2⁡ln⁡N⁠, and the result is consistent with the uniform threshold, avoiding the problem of inconsistent thresholds in the first layer; when the scale j > 1, the threshold becomes smaller as the number of decomposition layers increases, which satisfies the property that the noise decreases with the number of decomposition layers under the wavelet transform.

2. Improved threshold function

The effectiveness of wavelet transform denoising hinges significantly on the threshold function, which influences both the continuity and accuracy of the reconstructed signal. Traditional hard and soft thresholding functions, despite their effectiveness, have limitations. Thus, there is a need for a new class of threshold functions that combine the best of both the hard and soft methods. Such a function should not only correct the constant deviation in postprocessed wavelet coefficients, but also ensure continuity, potentially enhancing the overall denoising effect. We propose the following improved function:
(6)
where wj,k is the kth original wavelet coefficient of the jth layer, λ is the threshold of the corresponding layer, and ŵj,k is the estimated wavelet coefficient of the kth threshold processing of the jth layer. This function has the following characteristics:
  1. Continuity. When wj,k → λ+,
    (7)
    When wj,k → −λ−,
    (8)
    Thus, it is clear that the improved function is continuous at wj,k = ±λ, overcoming the problem of signal oscillations caused by discontinuities in the hard threshold function.
  2. Asymptotic behavior. When wj,k → +∞,
    (9)
    When wj,k → −∞,
    (10)
    Thus, the asymptote of the improved function is wj,k=ŵj,k⁠. As wj,k continues to increase, the deviation between the function and ŵj,k becomes smaller and smaller.
  3. Deviation. When wj,k → +∞,
    (11)
    When wj,k → −∞,
    (12)
    Thus, ŵj,k→wj,k when wj,k → ±∞, which further shows that the deviation of the improved threshold function will become smaller and smaller, thus overcoming the shortcomings of the soft threshold function.
In this paper, the SNR and root mean square error (RMSE) are utilized as metrics to assess the effectiveness of the proposed denoising algorithm. A higher SNR and a lower RMSE are indicative of a more successful denoising performance. The SNR measures the level of the desired signal relative to the background noise, while the RMSE quantifies the difference between values predicted by a model and the values actually observed. These metrics are critical for evaluating the accuracy and efficiency of denoising algorithms in signal processing. The SNR and RMSE are defined as follows:
(13)
(14)
where Soriginal is the original signal, Sre is the reconstructed signal, and N is the signal length.

The flowchart of the proposed method based on the above analysis is shown in Fig. 5. The overall process consists of four key stages: initial acquisition and preprocessing of raw noisy signals, where preprocessing primarily involves denoising background noise; determination of adaptive thresholds; wavelet coefficient processing using an improved thresholding function; and, finally, signal reconstruction.

FIG. 5.
Flowchart of ACT-WD algorithm.
View largeDownload slide

Flowchart of ACT-WD algorithm.

FIG. 5.
Flowchart of ACT-WD algorithm.
View largeDownload slide

Flowchart of ACT-WD algorithm.

Close modal

Bumps and Blocks signals are critical in evaluating denoising algorithms because of their complexity and variety, which mirror real-world signal challenges. Bumps signals, with their sharp variations, test an algorithm’s ability to maintain intricate details. Blocks signals, with their flat areas and abrupt shifts, assess an algorithm’s edge preservation capabilities. These signals provide a standard benchmark for comparing different denoising algorithms, allowing for a thorough evaluation of their performance, including how well they handle smoothness, edge retention, and noise rejection.

To assess the performance of the improved wavelet transform denoising algorithm, four types of original signals were used: Bumps, Blocks, simulated LiDAR echo, and measured LiDAR echo signals, each augmented with Poisson noise. The proposed ACT-WD method was compared with hard thresholding, soft thresholding, and Improved Wavelet Denoising Algorithm with Parameter Self-Tuning (IWDA-PST) methods through simulation experiments. The efficiency and reliability of the improved algorithm were evaluated using the SNR and RMSE metrics, providing a comprehensive analysis of its denoising capabilities.

Random Poisson noise was added to the Bumps and Blocks signals, as well as to ideal and aerosol-laden LiDAR echo signals, using MATLAB platforms. The wavelet threshold denoising process, analyzed through multiple coupling experiments, set to six layers and employing the sym8 wavelet base, was implemented to accomplish the denoising task.

To assess the efficacy of the ACT-WD method, denoising simulations were conducted on the synthetic Bumps signal, which had an input SNR (SNR_in) of 10 dB and a signal length N = 1024. This analysis compared the ACT-WD method against the hard thresholding method, the soft thresholding method, and the IWDA-PST, providing a comprehensive evaluation of its performance in enhancing signal clarity and reducing noise.

Figure 6 compares the performance of various denoising methods on the Bumps signal. The red and blue lines in Fig. 6(a) denote the original and noise signals, respectively. Figures 6(b)–6(e) show the denoised signals obtained using hard and soft thresholding, IWDA-PST, and ACT-WD, respectively. From a comparison of Figs. 6(b) and 6(e), it can be seen that although the hard thresholding noise reduction process diminishes noise to some extent, it proves unsatisfactory in the areas indicated by the black, blue, and purple circles, which represent intricate signal details in the original noisy signal. Conversely, ACT-WD demonstrates greater efficacy in processing signals within these regions, preserving detailed signals as evidenced in the post-noise reduction plot in Fig. 6(e). Hence, it can be inferred that the denoising efficacy of hard thresholding is markedly inferior to that of ACT-WD. On comparing Figs. 6(c) and 6(e), it can be seen that akin to hard threshold denoising, the signal after the soft threshold noise reduction process is excessively smooth, which is particularly conspicuous in the regions demarcated by the black, blue, and purple circles. Such overly smooth processing is undesirable in signal processing, since it risks the loss of numerous details in the original signal. From Figs. 6(d) and 6(e), it is evident that the signal denoised using the ACT-WD method closely resembles the original signal, effectively mitigating high-frequency noise signals in the original noisy signal and exhibiting superior noise suppression (as indicated by the orange, green, and brown circles). Conversely, while the IWDA-PST method displays some efficacy in denoising, ACT-WD proves more adept in high-frequency noise processing, reducing high-frequency noise while retaining more details of the original signal.

FIG. 6.
Experimental results of denoising of bumps simulation signals: (a) original signal and noise signal; (b) hard threshold; (c) soft threshold; (d) IWDA-PST; (e) ACT-WD.
View largeDownload slide

Experimental results of denoising of bumps simulation signals: (a) original signal and noise signal; (b) hard threshold; (c) soft threshold; (d) IWDA-PST; (e) ACT-WD.

FIG. 6.
Experimental results of denoising of bumps simulation signals: (a) original signal and noise signal; (b) hard threshold; (c) soft threshold; (d) IWDA-PST; (e) ACT-WD.
View largeDownload slide

Experimental results of denoising of bumps simulation signals: (a) original signal and noise signal; (b) hard threshold; (c) soft threshold; (d) IWDA-PST; (e) ACT-WD.

Close modal

Table I presents the SNR_out and RMSE values for the Bumps signal with SNR_in = 10 dB. The results demonstrate that the ACT-WD method achieves the highest SNR_out (18.7197 dB) and the lowest RMSE (0.2085), signifying its superior noise reduction capabilities for the input signal compared with the other three methods. Both Fig. 6 and Table I confirm that the ACT-WD method excels over the other methods in denoising and maintaining signal integrity.

TABLE I.

SNR and RMSE of bumps signal denoising.

MethodSNRRMSE
Hard thresholding 15.6401 0.2973 
Soft thresholding 12.2550 0.4390 
IWDA-PST 17.2973 0.2456 
ACT-WD 18.7197 0.2085 
MethodSNRRMSE
Hard thresholding 15.6401 0.2973 
Soft thresholding 12.2550 0.4390 
IWDA-PST 17.2973 0.2456 
ACT-WD 18.7197 0.2085 
View Large

Similar to the Bumps signal, denoising simulation was carried out using the same methods for the synthesized Blocks signal, characterized by an SNR_in of 15 dB and a signal length N = 1024. Denoising of this signal was compared for the hard thresholding, soft thresholding, IWDA-PST, and ACT-WD methods.

Figure 7 presents the denoising effect of the different methods on the Blocks signal. The red and blue lines in Fig. 7(a) denote the original and noise signals, respectively. Figures 7(b)–7(e) show the denoised signals obtained using hard and soft thresholding, IWDA-PST, and ACT-WD, respectively. Comparison of Figs. 7(b) and 7(e) reveals that while hard thresholding effectively reduces noise in the signal stabilization region, it performs inadequately in the signal hopping region, where notable noise residuals persist (the regions marked by black, blue, and green circles), indicating limited efficacy of hard thresholding in noise removal within signal mutation regions. On comparing Figs. 7(c) and 7(e), it can be seen that the signal processed with soft threshold noise reduction appears excessively smoothed, this being particularly evident in the areas indicated by black, blue, and green circles. This over-smoothing results in loss of numerous details from the original signal, consequently distorting the noise reduction signal significantly. From a comparison between Figs. 7(d) and 7(e), it is evident that the ACT-WD method not only preserves the structure of the original signal in the processed signal, but also effectively reduces noise, especially in the regions indicated by blue, purple, orange, and green circles, which contain critical signal mutations or feature points. Conversely, although the IWDA-PST method also reduces noise to some extent, its efficacy is not as pronounced as that of the ACT-WD method in the critical circle-marked areas. These findings underscore the significant advantages of the ACT-WD method in detail preservation and noise suppression.

FIG. 7.
Experimental results of denoising of blocks simulation signals: (a) original signal and noise signal; (b) hard thresholding; (c) soft thresholding; (d) IWDA-PST; (e) ACT-WD.
View largeDownload slide

Experimental results of denoising of blocks simulation signals: (a) original signal and noise signal; (b) hard thresholding; (c) soft thresholding; (d) IWDA-PST; (e) ACT-WD.

FIG. 7.
Experimental results of denoising of blocks simulation signals: (a) original signal and noise signal; (b) hard thresholding; (c) soft thresholding; (d) IWDA-PST; (e) ACT-WD.
View largeDownload slide

Experimental results of denoising of blocks simulation signals: (a) original signal and noise signal; (b) hard thresholding; (c) soft thresholding; (d) IWDA-PST; (e) ACT-WD.

Close modal

Table II lists SNR_out and RMSE for the four methods. The results demonstrate that the ACT-WD method achieves the highest SNR_out (19.3269 dB) and the lowest RMSE (0.3209), indicating its superior noise reduction capabilities. In conjunction with Fig. 7, these results clearly show that the ACT-WD method’s denoising performance surpasses that of the other three methods.

TABLE II.

SNR and RMSE of Blocks signal denoising.

MethodSNRRMSE
Hard thresholding 18.1590 0.3671 
Soft thresholding 14.2872 0.5733 
IWDA-PST 18.6933 0.3452 
ACT-WD 19.3269 0.3209 
MethodSNRRMSE
Hard thresholding 18.1590 0.3671 
Soft thresholding 14.2872 0.5733 
IWDA-PST 18.6933 0.3452 
ACT-WD 19.3269 0.3209 
View Large

To further verify the denoising effect of the ACT-WD method on LiDAR echo signals, a Mie-scattered LiDAR echo smooth signal was simulated in advance. There were 500 sampling points in the echo signal, the spacing between each point was 0.015 km, and the detection distance was 7.5 km. Subsequently, noise with an SNR_in of 20 dB was added, followed by denoising using the hard and soft threshold methods, IWDA-PST, and ACT-WD.

Figure 8 compares the denoising results for the simulated LiDAR echo signal. Figures 8(a) and 8(b) show the original signal and the band noise signal after noise addition, respectively. Figures 8(c) show the denoised signals obtained using hard and soft thresholding, IWDA-PST, and denoised ACT-WD, respectively. It can be seen from Figs. 8(c) and 8(f) that while the overall disparity between signals denoised using the hard thresholding and ACT-WD methods may not be stark, the ACT-WD-denoised signals exhibit smoother characteristics in the region delineated by the black circle and demonstrate higher accuracy based on the RMSE metric compared with the hard thresholding method. Comparison between Figs. 8(d) and 8(f) suggests that signals processed by the soft thresholding method closely resemble those from the hard thresholding method, with minimal overall discrepancy. However, in the region demarcated by the black circle, ACT-WD not only adeptly preserves original signal details, but also mitigates excessive smoothing, thereby reducing loss of intricate signal information. As shown in Figs. 8(e) and 8(f), in contrast to the IWDA-PST method, ACT-WD significantly reduces high-frequency noise at the 5.5 km mark indicated by the blue circle, mitigating the impact of sudden noise on the original signal and achieving signal recovery closer to its original state. In summary, post-denoising of the ideal LiDAR signal, ACT-WD not only effectively eliminates noise, but also preserves the primary structure and details of the signal, rendering it highly valuable for practical applications.

FIG. 8.
Experimental results of denoising of simulated LiDAR echo signal: (a) original signal; (b) signal with noise; (c) hard thresholding; (d) soft thresholding; (e) IWDA-PST; (f) ACT-WD.
View largeDownload slide

Experimental results of denoising of simulated LiDAR echo signal: (a) original signal; (b) signal with noise; (c) hard thresholding; (d) soft thresholding; (e) IWDA-PST; (f) ACT-WD.

FIG. 8.
Experimental results of denoising of simulated LiDAR echo signal: (a) original signal; (b) signal with noise; (c) hard thresholding; (d) soft thresholding; (e) IWDA-PST; (f) ACT-WD.
View largeDownload slide

Experimental results of denoising of simulated LiDAR echo signal: (a) original signal; (b) signal with noise; (c) hard thresholding; (d) soft thresholding; (e) IWDA-PST; (f) ACT-WD.

Close modal

Table III shows a comparison of the SNR and RMSE for the four denoising methods. Post-denoising, the SNR achieved using the ACT-WD method is 36.3427 dB. This signifies an enhancement of 8.0065 dB over the soft thresholding method and an improvement of 1.8954 dB compared with IWDA-PST, indicating the superior denoising efficacy of ACT-WD. Additionally, the RMSE value after employing the ACT-WD method is 1.1576, representing a substantial reduction of 91.3% relative to the soft thresholding method and 82.6% relative to IWDA-PST.

TABLE III.

SNR and RMSE of ideal echo signal denoising.

MethodSNRRMSE
Hard thresholding 33.0385 1.8649 
Soft thresholding 28.4262 3.1715 
IWDA-PST 34.4473 1.5857 
ACT-WD 36.3427 0.2748 
MethodSNRRMSE
Hard thresholding 33.0385 1.8649 
Soft thresholding 28.4262 3.1715 
IWDA-PST 34.4473 1.5857 
ACT-WD 36.3427 0.2748 
View Large

To validate the efficacy of the proposed method on measured LiDAR echo signals, which are inherently complex owing to various interferences, experimental verification was undertaken. The study utilized the GYLZ-P01-T single-photon LiDAR from Shandong Guoyao Quantum Technology Co., Ltd., featuring a 1550 nm laser wavelength, in conjunction with the P-SPD1550 single-photon detector from Fudong Quantum Technology Co., Ltd. for horizontal aerosol detection. The LiDAR data were detected by the LiDAR located at the Yangtze River Delta Research Institute of Beijing Institute of Technology, Jiaxing, Zhejiang Province, and the LiDAR sampling period was from October 13 to 24, 2023. The resulting echo signal, characterized by multiple peaks and notable fluctuations at extended distances, is illustrated in Fig. 9. These fluctuations stem predominantly from the intensity of the LiDAR echo signal being inversely proportional to the square of the distance. The subtle variations in aerosols during detection are manifested in the peaks, while the fluctuations at longer distances are primarily attributed to Poisson noise within the system.

FIG. 9.
Experimental results of denoising of measured LiDAR echo signal: (a) hard thresholding; (b) soft thresholding; (c) IWDA-PST; (d) ACT-WD.
View largeDownload slide

Experimental results of denoising of measured LiDAR echo signal: (a) hard thresholding; (b) soft thresholding; (c) IWDA-PST; (d) ACT-WD.

FIG. 9.
Experimental results of denoising of measured LiDAR echo signal: (a) hard thresholding; (b) soft thresholding; (c) IWDA-PST; (d) ACT-WD.
View largeDownload slide

Experimental results of denoising of measured LiDAR echo signal: (a) hard thresholding; (b) soft thresholding; (c) IWDA-PST; (d) ACT-WD.

Close modal

Figure 9 compares the results of different denoising methods for a measured LiDAR echo signal. The blue curve in the background of each plot is the detected signal with noise and the red curve is the denoised signal. Figures 9(a)–9(d) show the denoised signal obtained using hard and soft thresholding, IWDA-PST, and ACT-WD, respectively. On comparing Figs. 9(a) and 9(d), it can be seen that although the hard thresholding method effectively suppresses noise in the 0–5 km range, its overall efficacy is unsatisfactory. In particular, significant noise residue persists at the location indicated by the purple circle in the long-range region, adversely impacting signal quality. Comparison between Figs. 9(b) and 9(d) reveals that the soft thresholding method exhibits relatively smooth processing in long-range signal processing. However, in peak areas at close range, such as those indicated by the black circle at 1.65 km and the blue circle at 3 km, soft thresholding leads to serious loss of signal details, resulting in the loss of numerous useful signals and subsequently affecting the accuracy of the analysis.

In Figs. 9(c) and 9(d), the denoised signals from IWDA-PST and ACT-WD exhibit a high degree of consistency in overall trend, accurately tracking changes in the real signal. However, although the IWDA-PST method achieves noise removal in most areas, significant noise remains uncleared in the regions indicated by the orange and green circles, detrimentally impacting precision and accuracy of subsequent data processing results. By contrast, the ACT-WD method not only better aligns with the real signal trend overall, but also effectively removes high-frequency noise, thereby preserving signal integrity to the greatest extent possible. In summary, during the denoising process of real LiDAR detection signals, the ACT-WD method exhibits excellent noise reduction capability, a critical advantage that is particularly significant in processing LiDAR detection signals, ultimately enhancing data accuracy and reliability.

The data presented in Table IV reveal that the SNR_out for the ACT-WD method post-denoising stands at 37.0523 dB. This represents an improvement of 4.3653 dB over the hard thresholding method and 1.5689 dB over the IWDA-PST, indicating the superior denoising effect of the ACT-WD method. Furthermore, the RMSE post-denoising with the ACT-WD method is 1.1576, marking a significant reduction of 39.5% compared with the hard thresholding method and 16.5% compared with IWDA-PST. This performance underscores the ACT-WD method’s significant advantages in processing measured LiDAR echo signals, particularly over long measurement distances, effectively enhancing signal quality, which is pivotal for the functionality of LiDAR systems in practical applications. Overall, these experimental findings confirm the robust denoising capability of the ACT-WD method, particularly in handling real signals laden with Poisson noise, thus validating its efficacy as an innovative denoising approach.

TABLE IV.

SNR and RMSE for denoising measured LiDAR echo signals.

MethodSNRRMSE
Hard thresholding 32.6870 1.9135 
Soft thresholding 34.3713 1.5762 
IWDA-PST 35.4834 1.3868 
ACT-WD 37.0523 1.1576 
MethodSNRRMSE
Hard thresholding 32.6870 1.9135 
Soft thresholding 34.3713 1.5762 
IWDA-PST 35.4834 1.3868 
ACT-WD 37.0523 1.1576 
View Large

This paper has addressed the challenges of denoising single-photon LiDAR echo signals, critically examining traditional wavelet transform denoising techniques, with a particular focus on Donoho thresholds and hard and soft threshold functions. It has introduced an innovative adaptive continuous threshold wavelet denoising method, distinguished by its adaptive threshold that dynamically adjusts in response to the decomposition level and a threshold function specifically designed to minimize constant error.

The ACT-WD method, evaluated through three simulated signals and one measured LiDAR echo signal against three other denoising methods, demonstrates superior performance. In comparison with both the hard and soft thresholding methods, ACT-WD achieves a higher SNR and a lower RMSE, particularly excelling in high-frequency noise removal. Unlike the soft thresholding method, ACT-WD avoids excessive signal smoothness during transitions, preventing serious signal distortion. In the presence of a robust noise background, ACT-WD stands out when compared with IWDA-PST. Not only does it efficiently eliminate noise, but. more importantly, it achieves this while preserving intricate signal details, a critical advantage in LiDAR signal processing, where the extraction of weak signals is a common requirement. This capability holds substantial significance in enhancing the detection range, detection rate, and measurement accuracy of single-photon LiDAR systems.

This research was funded by the National Key R&D Program of China (Grant No. 2022YFC3300705) and the National Natural Science Foundation of China (Grant Nos. 62203056, 12202048, and 62201056).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

1.
Pöschl
U
.
Atmospheric aerosols: Composition, transformation, climate and health effects
.
Angew Chem Int Ed
2005
;
44
:
7520
–
7540
.
https://doi.org/10.1002/chin.200607299
Google Scholar
Crossref
Search ADS
 
2.
Tao
W
,
Hongyan
Z
,
Haihang
C
.
Numerical simulation of the effect of gravitational direction on particle deposition in pulmonary acinus under the condition of breath holding
.
Nanotechnol Precis Eng
2018
;
1
:
66
–
72
.
https://doi.org/10.13494/j.npe.20170003
Google Scholar
 
3.
McFiggans
G
.
Marine aerosols and iodine emissions
.
Nature
2005
;
433
:
E13
.
https://doi.org/10.1038/nature03372
Google Scholar
Crossref
Search ADS
PubMed
 
4.
Li
Y
,
Fu
M
,
Pang
W
,
Chang
Y
,
Duan
X
.
A combined virtual impactor and field-effect transistor microsystem for particulate matter separation and detection
.
Nanotechnol Precis Eng
2021
;
4
:
21
–
29
.
https://doi.org/10.1063/10.0003447
Google Scholar
Crossref
Search ADS
 
5.
Hoher
P
,
Wirtensohn
S
,
Baur
T
,
Reuter
J
,
Govaers
F
,
Koch
W
.
Extended target tracking with a lidar sensor using random matrices and a virtual measurement model
.
IEEE Trans Signal Process
2022
;
70
:
228
–
239
.
https://doi.org/10.1109/tsp.2021.3138006
Google Scholar
Crossref
Search ADS
 
6.
Li
Z
,
Liu
B
,
Wang
H
,
Yi
H
,
Chen
Z
.
Advancement on target ranging and tracking by single-point photon counting lidar
.
Opt Express
2022
;
30
:
29907
–
29922
.
https://doi.org/10.1364/OE.464070
Google Scholar
Crossref
Search ADS
PubMed
 
7.
Omasa
K
,
Hosoi
F
,
Konishi
A
.
3D lidar imaging for detecting and understanding plant responses and canopy structure
.
J Exp Bot
2007
;
58
:
881
–
898
.
https://doi.org/10.1093/jxb/erl142
Google Scholar
Crossref
Search ADS
PubMed
 
8.
Yakovlev
S
,
Sadovnikov
S
,
Kharchenko
O
,
Kravtsova
N
.
Remote sensing of atmospheric methane with IR OPO lidar system
.
Atmosphere
2020
;
11
:
70
–
82
.
https://doi.org/10.3390/atmos11010070
Google Scholar
Crossref
Search ADS
 
9.
Sun
X
,
Kolbeck
PT
,
Abshire
JB
,
Kawa
SR
,
Mao
J
.
Attenuated atmospheric backscatter profiles measured by the CO2 Sounder lidar in the 2017 ASCENDS/ABoVE airborne campaign
.
Earth Syst Sci Data
2022
;
14
:
3821
–
3833
.
https://doi.org/10.5194/essd-14-3821-2022
Google Scholar
Crossref
Search ADS
 
10.
Fei
R
,
Kong
Z
,
Wang
X
,
Zhang
B
,
Gong
Z
,
Liu
K
,
Hua
D
,
Mei
L
.
Retrieval of the aerosol extinction coefficient from scanning Scheimpflug lidar measurements for atmospheric pollution monitoring
.
Atmos Environ
2023
;
309
:
e119945
.
https://doi.org/10.1016/j.atmosenv.2023.119945
Google Scholar
Crossref
Search ADS
 
11.
Liu
H
,
Qin
C
,
Papangelakis
G
,
Iu
M
,
Helmy
A
.
Compact all-fiber quantum-inspired LiDAR with over 100 dB noise rejection and single photon sensitivity
.
Nat Commun
2023
;
14
:
5344
.
https://doi.org/10.1038/s41467-023-40914-6
Google Scholar
Crossref
Search ADS
PubMed
 
12.
Cao
N
,
Zhu
C
,
Kai
Y
,
Yan
P
.
A method of background noise reduction in lidar data
.
Appl Phys B
2013
;
113
:
115
–
123
.
https://doi.org/10.1007/s00340-013-5447-9
Google Scholar
Crossref
Search ADS
 
13.
Mei
L
,
Zhang
L
,
Kong
Z
,
Li
H
.
Noise modeling, evaluation and reduction for the atmospheric lidar technique employing an image sensor
.
Opt Commun
2018
;
426
:
463
–
470
.
https://doi.org/10.1016/j.optcom.2018.05.072
Google Scholar
Crossref
Search ADS
 
14.
Mao
J.
Noise reduction for lidar returns using local threshold wavelet analysis
.
Opt Quantum Electron
2012
;
43
:
59
–
68
.
https://doi.org/10.1007/s11082-011-9503-6
Google Scholar
Crossref
Search ADS
 
15.
Cheng
X
,
Mao
J
,
Li
J
,
Zhao
H
,
Zhou
C
,
Gong
X
,
Rao
Z
.
An EEMD-SVD-LWT algorithm for denoising a lidar signal
.
Measurement
2021
;
168
:
e108405
.
https://doi.org/10.1016/j.measurement.2020.108405
Google Scholar
Crossref
Search ADS
 
16.
Chang
J
,
Zhu
L
,
Li
H
,
Xu
F
,
Liu
B
,
Yang
Z
.
Noise reduction in Lidar signal using correlation-based EMD combined with soft thresholding and roughness penalty
.
Opt Commun
2018
;
407
:
290
–
295
.
https://doi.org/10.1016/j.optcom.2017.09.063
Google Scholar
Crossref
Search ADS
 
17.
Hua
T
,
Dai
K
,
Zhang
X
,
Yao
Z
,
Wang
H
,
Xie
K
,
Feng
T
,
Zhang
H
.
Optimal VMD-based signal denoising for laser radar via Hausdorff distance and wavelet transform
.
IEEE Access
2019
;
7
:
167997
–
168010
.
https://doi.org/10.1109/access.2019.2949063
Google Scholar
Crossref
Search ADS
 
18.
Hu
M
,
Mao
J
,
Li
J
,
Wang
Q
,
Zhang
Y
.
A novel lidar signal denoising method based on convolutional autoencoding deep learning neural network
.
Atmosphere
2021
;
12
:
1403
–
1420
.
https://doi.org/10.3390/atmos12111403
Google Scholar
Crossref
Search ADS
 
19.
Wu
S
,
Liu
Z
,
Liu
B
.
Enhancement of lidar backscatters signal-to-noise ratio using empirical mode decomposition method
.
Opt Commun
2006
;
267
:
137
–
144
.
https://doi.org/10.1016/j.optcom.2006.05.069
Google Scholar
Crossref
Search ADS
 
20.
Sarvani
M
,
Raghunath
K
,
Rao
SVB
.
Lidar signal denoising methods- application to NARL Rayleigh lidar
.
J Opt
2015
;
44
:
164
–
171
.
https://doi.org/10.1007/s12596-015-0247-8
Google Scholar
Crossref
Search ADS
 
21.
Tang
B
,
Dong
S
,
Song
T
.
Method for eliminating mode mixing of empirical mode decomposition based on the revised blind source separation
.
Signal Process
2012
;
92
:
248
–
258
.
https://doi.org/10.1016/j.sigpro.2011.07.013
Google Scholar
Crossref
Search ADS
 
22.
Dragomiretskiy
K
,
Zosso
D
.
Variational mode decomposition
.
IEEE Trans Signal Process
2014
;
62
:
531
–
544
.
https://doi.org/10.1109/TSP.2013.2288675
Google Scholar
Crossref
Search ADS
 
23.
Akyol
EA
,
Erzin
E
,
Tekalp
AM
.
Robust speech recognition using adaptively denoised wavelet coefficients
.
Proceedings of the IEEE 12th Signal Processing and Communications Applications Conference
.
IEEE
.
2004
. pp.
407
–
409
.
Google Scholar
Crossref
Search ADS
 
24.
Bhutada
GG
,
Anand
RS
,
Saxena
SC
.
Edge preserved image enhancement using adaptive fusion of images denoised by wavelet and curvelet transform
.
Digit Signal Process
2011
;
21
:
118
–
130
.
https://doi.org/10.1016/j.dsp.2010.09.002
Google Scholar
Crossref
Search ADS
 
25.
To
AC
,
Moore
JR
,
Glaser
SD
.
Wavelet denoising techniques with applications to experimental geophysical data
.
Signal Process
2009
;
89
:
144
–
160
.
https://doi.org/10.1016/j.sigpro.2008.07.023
Google Scholar
Crossref
Search ADS
 
26.
Nasri
M
,
Nezamabadi-pour
H
.
Image denoising in the wavelet domain using a new adaptive thresholding function
.
Neurocomputing
2009
;
72
:
1012
–
1025
.
https://doi.org/10.1016/j.neucom.2008.04.016
Google Scholar
Crossref
Search ADS
 
27.
Fang
H
,
Huang
D
.
Noise reduction in lidar signal based on discrete wavelet transform
.
Opt. Commun.
2004
;
233
:
67
–
76
.
https://doi.org/10.1016/j.optcom.2004.01.017
Google Scholar
Crossref
Search ADS
 
28.
Mao
J
,
Hua
D
,
Wang
Y
,
Wang
L
.
Noise reduction in lidar signal based on wavelet packet analysis
.
Chin J Lasers
2011
;
38
:
226
–
233
.
https://doi.org/10.3788/cjl201138.0209001
Google Scholar
Crossref
Search ADS
 
29.
Qin
X
,
Mao
J
.
Noise reduction for lidar returns using self-adaptive wavelet neural network
.
Opt Rev
2017
;
24
:
416
–
427
.
https://doi.org/10.1007/s10043-017-0337-8
Google Scholar
Crossref
Search ADS
 
30.
Liu
B
,
Feng
J
,
Song
S
,
Ye
H
.
Research on an improved wavelet denoising algorithm with parameter self-tuning
.
Control Eng China
2020
;
27
:
444
–
450
.
https://doi.org/10.14107/j.cnki.kzgc.170770
Google Scholar
 
31.
Li
M
,
Wang
Z
,
Luo
J
,
Liu
Y
,
Cai
S
.
Wavelet denoising of vehicle platform vibration signal based on threshold neural network
.
Shock Vib
;
2017
:
1
–
12
.
https://doi.org/10.1155/2017/7962828
Crossref
Search ADS
 
32.
Guo
W
,
Xu
P
,
Xu
P
.
Investigation of material removal characteristics of Si (100) wafer during linear field atmospheric-pressure plasma etching
.
Nanotechnol Precis Eng
2020
;
3
:
244
–
249
.
https://doi.org/10.1016/j.npe.2020.09.003
Google Scholar
Crossref
Search ADS
 
33.
Zhao
X
,
Xia
H
,
Zhao
J
,
Zhou
F
.
Adaptive wavelet threshold denoising for bathymetric laser full-waveforms with weak bottom returns
.
IEEE Geosci Remote Sens Lett
2022
;
19
:
1
–
5
.
https://doi.org/10.1109/LGRS.2022.3141057
Google Scholar
Crossref
Search ADS
 
34.
Xu
X
,
Luo
M
,
Tan
Z
,
Pei
R
.
Echo signal extraction method of laser radar based on improved singular value decomposition and wavelet threshold denoising
.
Infrared Phys Technol
2018
;
92
:
327
–
335
.
https://doi.org/10.1016/j.infrared.2018.06.028
Google Scholar
Crossref
Search ADS
 
35.
Guo
D
,
Wu
Y
,
Shitz
SS
,
S.
Verdú
.
Estimation in Gaussian noise: Properties of the minimum mean-square error
.
IEEE Trans Inf Theory
2011
;
57
:
2371
–
2385
.
https://doi.org/10.1109/TIT.2011.2111010
Google Scholar
Crossref
Search ADS
 
36.
Mallat
SG
.
A theory for multiresolution signal decomposition: The wavelet representation
.
IEEE Trans Pattern Anal Mach Intell
1989
;
11
:
674
–
693
.
https://doi.org/10.1109/34.192463
Google Scholar
Crossref
Search ADS
 
37.
Donoho
DL
.
De-noising by soft-thresholding
.
IEEE transactions on information theory
1995
;
41
:
613
–
627
.
https://doi.org/10.1109/18.382009
Google Scholar
Crossref
Search ADS
 

Dezhi Zheng received his B.E. and Ph.D. degrees from Beijing University of Aeronautics and Astronautics, Beijing, China, in 2000 and 2006, respectively. He is currently a Professor in the School of Information and Electronics, Beijing Institute of Technology. His research includes sensor sensitive mechanisms, weak EEG signal perception, and scientific detection of complex environments.

Tianchi Qu received his Bachelor’s degree from Jilin University, Jilin, China, in 2021. He is pursuing a Master’s degree in Electronic Information at Beijing Institute of Technology. His current research focuses on complex environment detection and aerosol LiDAR detection

Chun Hu received the Ph.D. degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 2000. He is currently a Professor in the School of Information and Electronics, Beijing Institute of Technology. His research includes advanced sensing and intelligent instrumentation, meteorological information detection, and intelligent signal processing.

Zhongxiang Li received the Ph.D. degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 2022. He is currently a Postdoctoral Fellow in the School of Frontier Interdisciplinary Sciences, Beijing Institute of Technology. His research includes resonant sensors and extremely weak signal perception technology.

Shijia Lu received her Master’s degree from Shenzhen University, Guangdong, China, in 2023. She is working as an engineering technician at the Yangtze Delta Region Academy of the Beijing Institute of Technology, Jiaxing, China. Her research focuses on photodetectors and quantum optics.

© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).