Full-waveform LiDAR echo decomposition based on wavelet decomposition and particle swarm optimization

To obtain echo components, a full-waveform echo, $s(n)$ can be modelled by

$$\begin{eqnarray}&&s(n)=\underset{i=1}{\overset{K}{{{\sum}^{}\,}}}\,{{f}_{i}}(n)\,+\,d~+\,\varepsilon (n)\end{eqnarray} \tag{ 1 }$$

where n is sample time, K is the number of echo components in $s(n)$, ${{f}_{i}}(n)$ is fitting model for the ith echo component, d is direct current (DC) offset in $s(n)$ and $\varepsilon (n)$ is residual error. ${{f}_{i}}(n)$ is determined by the pulse response of each full- waveform LiDAR system and scatter property of each reflection surface. Because the impulse response of our lab-build full-waveform LiDAR system is the most similar with Gaussian function, Gaussian functions are selected as fitting models. When the Gaussian functions are selected as the fitting models for echo components, $s(n)$ can be modelled by

$$\begin{eqnarray}&&s(n)=\underset{i=1}{\overset{K}{\sum}}\,{{A}_{i}}{{\text{e}}^{-\frac{{{\left(n-{{\mu}_{i}}\right)}^{2}}}{F_{i}^{2}/\left(4\ln 2\right)}}}\text{+}d\text{+}\varepsilon (n)\end{eqnarray} \tag{ 2 }$$

where ${{A}_{i}}$, ${{\mu}_{i}}$ and ${{F}_{i}}$ are the amplitude, mean and full width at half maximum (FWHM) of the ith Gaussian function, which represent the amplitude, location and pulse width of the ith echo component, respectively.

Since a full-waveform echo is contaminated by background light, noises of photoelectric detector, preamplifier and A/D converter, wavelet-decomposition-based filter is utilized for removing the noise effect. The flow diagram of the wavelet-decomposition-based filter is shown in figure 1.

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As shown in figure 1, at Step 1, s(n) is decomposed by Symlets 6 wavelet from level 1 to N. Since the shape of scaling function of Symlets 6 wavelet is the most similar with fitting models of the echo components, the Symlets 6 wavelet is selected as wavelet function to decompose full-waveform echoes so that the characteristics of echo components can be best preserved and the most of noise in echo components can be easily removed. At the jth level, s(n) is decomposed using [21]

$$\begin{eqnarray}&&\left\{\begin{array}{*{35}{l}} a_{j}^{k}=\langle s(n),\quad {{2}^{-\frac{j}{2}}}\phi \left({{2}^{-j}}n-k\right)\rangle \\ d_{j}^{k}=\langle s(n),\quad {{2}^{-\frac{j}{2}}}\psi \left({{2}^{-j}}n-k\right)\rangle \end{array}\right. \end{eqnarray} \tag{ 3 }$$

where $a_{j}^{k}$ is the kth approximate coefficient at the jth level, $\langle.,~.\rangle$ denotes inner product operator, $\,{{2}^{-\frac{j}{2}}}\phi \left({{2}^{-j}}n-k\right)$ is the kth basis in approximate space at the jth level which is obtained from the dilation and translation of $\phi (n)$. $d_{j}^{k}$ is the kth detail coefficient at the jth level, ${{2}^{-\frac{j}{2}}}\psi \left({{2}^{-j}}n-k\right)$ is the kth basis in detail space at the jth level which is obtained from dilation and translation of $\psi (n)$. $\phi (n)$ and $\psi (n)$ are scaling and wavelet functions of Symlet 6 wavelet, respectively.

At Step 2, thresholding of detail coefficients from level 1 to N are achieved. Thresholding of detail coefficients at jth level are achieved by [22]

$$\begin{eqnarray}\hat{d}_{j}^{k}=\left\{\begin{array}{*{35}{l}} \text{sign}\left(d_{j}^{k}\right)\left(|d_{j}^{k}|-{{T}_{j}}\right), & |d_{j}^{k}|\geqslant {{T}_{j}} \\ 0 & |d_{j}^{k}|<{{T}_{j}} \end{array}\right. \end{eqnarray} \tag{ 4 }$$

where $\hat{d}_{j}^{k}$ is the kth detail coefficient after thresholding, $\text{sign}\left(\centerdot \right)$ is sign operator, ${{T}_{j}}$ is heursure threshold [23] and determined by

$$\begin{eqnarray}&&{{T}_{j}}=\left\{\begin{array}{*{35}{l}} T_{j}^{f},\quad {{u}_{j}}<{{v}_{j}} \\ T_{j}^{r},\quad {{u}_{j}}\geqslant {{v}_{j}} \end{array}\right. \end{eqnarray} \tag{ 5 }$$

where $T_{j}^{f}$ are calculated by

$$\begin{eqnarray}&&T_{j}^{f}=\sqrt{2\ln {{N}_{j}}}\end{eqnarray} \tag{ 6 }$$

where ${{N}_{j}}$ is the number of the detail coefficients at the jth level. $T_{j}^{r}$ is calculated by the following three steps:

• 1.  
  All $d_{j}^{k}$s are sorted in an ascending order depending on their square $g_{j}^{m}$

  $$\begin{eqnarray}&&g_{j}^{m}={{\left(d_{j}^{k}\right)}^{2}}\end{eqnarray} \tag{ 7 }$$

  where m is the order of $d_{j}^{k}$. Then, a vector composed of all $g_{j}^{m}$ is denoted as ${{\mathbf{G}}_{j}}$

  $$\begin{eqnarray}&&{{\mathbf{G}}_{j}}=\left[g_{j}^{1},g_{j}^{2},\cdots,g_{j}^{m},\cdots,g_{j}^{{{N}_{j}}-1},g_{j}^{{{N}_{j}}}\right]\end{eqnarray} \tag{ 8 }$$
• 2.  
  A risk vector R_j is constructed by

  $$\begin{eqnarray}&&{{\mathbf{R}}_{j}}=\left[r_{j}^{1},r_{j}^{2},\cdots,r_{j}^{m},\cdots,r_{j}^{{{N}_{j}}-1},r_{j}^{{{N}_{j}}}\right]\end{eqnarray} \tag{ 9 }$$

  where $r_{j}^{m}$ is calculated by

  $$\begin{eqnarray}&&r_{j}^{m}=\left[{{N}_{j}}-2m-\left({{N}_{j}}-m\right)g_{j}^{m}+\underset{i=1}{\overset{m}{\sum}}\,g_{j}^{i}\right]/{{N}_{j}}\end{eqnarray} \tag{ 10 }$$
• 3.  
  The element with the minimum value in R_j is found. The index of the minimum element in R_j is denoted as i_min. Then, $T_{j}^{r}$ is calculated by

  $$\begin{eqnarray}&&T_{j}^{r}=\sqrt{g_{j}^{{{i}_{\min}}}}\end{eqnarray} \tag{ 11 }$$

  u_j and v_j are calculated by

  $$\begin{eqnarray}&&{{\mu}_{j}}=\frac{\underset{i=1}{\overset{{{N}_{j}}}{\sum}}\,{{\left(d_{j}^{i}\right)}^{2}}-{{N}_{j}}}{{{N}_{j}}}\end{eqnarray} \tag{ 12 }$$

  $$\begin{eqnarray}&&{{\nu}_{j}}=\sqrt{\frac{1}{{{N}_{j}}}{{\left(\frac{\ln {{N}_{j}}}{\ln 2}\right)}^{3}}}\end{eqnarray} \tag{ 13 }$$

  As shown in figure 1, at Step 3, the full-waveform echo after filtering is denoted as c_N(n) and constructed by

  $$\begin{eqnarray}&&{{c}_{N}}(n)=\underset{j=1}{\overset{N}{\sum}}\,\underset{k=-\infty}{\overset{+\infty}{\sum}}\,\left[\hat{d}_{j}^{k}\times {{2}^{-\frac{j}{2}}}\psi \left({{2}^{-j}}n-k\right)\right]\\ &&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\underset{k=-\infty}{\overset{+\infty}{\sum}}\,a_{N}^{k}\times {{2}^{-\frac{N}{2}}}\phi \left({{2}^{-N}}n-k\right)\end{eqnarray} \tag{ 14 }$$

In stop criterion, th is set as 1.01. After filtering, the noise is removed from s(n). Thus, the residual between c_N(n) and s(n) can be regarded as the primary noise. The mean and standard deviation of the noise in the full-waveform echo are calculated using the residual, and denoted as ${{\mu}_{n}}$ and ${{\sigma}_{n}}$, respectively. A raw and the filtered full-waveform echoes are shown in figure 2.

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2.3.1. Detection of peak and inflection points.

After filtering noise, peak and inflection points of ${{c}_{N}}(n)$ are used to detect the echo components. The peak and inflection points are detected depending on the first and second order differences of ${{c}_{N}}(n)$, respectively. If the first order difference at a sample changes from positive to negative, the sample in ${{c}_{N}}(n)$ is regarded as a peak point of an echo component. A zero-crossing point of the second order difference of ${{c}_{N}}(n)$ is regarded as an inflection point of an echo component.

Because the noise in s(n) cannot be completely removed, the false peak and inflection points caused by noise may exist in the detected peak and inflection points. Thus, to remove the noise-caused peak and inflection points, a noise threshold, $\text{t}{{\text{h}}_{\text{n}}}$ is set as

$$\begin{eqnarray}&&t{{h}_{\text{n}}}={{\mu}_{\text{n}}}+3{{\sigma}_{\text{n}}}\end{eqnarray} \tag{ 15 }$$

where ${{\mu}_{\text{n}}}$ and ${{\sigma}_{\text{n}}}$ are the mean and standard deviation of the noise in the full-waveform echo, respectively. Then, a peak point or an inflection point whose amplitude is less than $t{{h}_{\text{n}}}$ is regarded as a false peak point or a false inflection point, and hence it should be removed.

2.3.2. Initial estimation of the location of a separated echo component.

A separated echo component is defined as a non-overlapping echo component in a full-waveform and can be detected by the peak point. Figure 3 shows a simulated full-waveform echo with four echo components, which are denoted as E₁, E₂, E₃ and E₄, respectively. It can be seen that E₁, E₂ and E₄ are separated echo components. While the gap between E₃ and E₄ is so small that the peak point of E₃ cannot be distinguished. Therefore, E₃ is regarded as an overlapping echo component. The steps to estimate the lower and upper limits of the location of a separated echo component are as follows:

• (1)  
  For a peak point ${{P}_{i}}\left({{t}_{i}},~{{a}_{i}}\right)$, where ${{t}_{i}}$ and ${{a}_{i}}$ are the time coordinate and amplitude of the ith peak point, the monotone intervals of the waveform at the left and right sides of ${{P}_{i}}$ are noted as $I_{i}^{\text{L}}$ and $I_{i}^{\text{R}}$, respectively. Then, the inflection points in $I_{i}^{\text{L}}$ and $I_{i}^{\text{R}}$ are detected, respectively. If there is no inflection point in both $I_{i}^{\text{L}}$ and $I_{i}^{\text{R}}$, ${{P}_{i}}$ does not belong to any echo component and is removed. Then, the detection of echo component at ${{P}_{i}}$ stops. Otherwise, ${{P}_{i}}$ may belong to an echo component and the detection of echo component at ${{P}_{i}}$ continues. Figure 4 shows the enlarged view of area 1 in figure 3. It can be seen that $I_{1}^{\text{L}}$ and $I_{1}^{\text{R}}$ are the monotone intervals of the waveform at the left and right sides of ${{P}_{1}}$, respectively. ${{B}_{1}}$  −  ${{B}_{12}}$ are the inflection points located in $I_{1}^{\text{L}}$. ${{B}_{13}}$  −  ${{B}_{19}}$ are the inflection points located in $I_{i}^{\text{R}}$.
• (2)  
  The mean of the time coordinates of the inflection points in $I_{i}^{\text{L}}$ and $I_{i}^{\text{R}}$ are denoted as $\mu _{i}^{\text{L}}$ and $\mu _{i}^{\text{R}}$, respectively. If there is at least one inflection point in $I_{i}^{\text{L}}$, $\mu _{i}^{\text{L}}$ is calculated. Otherwise, $\mu _{i}^{\text{L}}$ is set as $2{{t}_{i}}-\mu _{i}^{\text{R}}$. The calculation of $\mu _{i}^{\text{R}}$ is similar to that of $\mu _{i}^{\text{L}}$. As shown in figure 4, $\mu _{1}^{\text{L}}$ and $\mu _{1}^{\text{R}}$ are 212.45 ns and 289.14 ns, respectively.
• (3)  
  The absolute value of the time difference between $\mu _{i}^{\text{L}}$ or $\mu _{i}^{\text{R}}$ and ${{t}_{i}}$ is denoted as $d_{i}^{k}$ and calculated by

  $$\begin{eqnarray}&&d_{i}^{k}=\,|\mu _{i}^{k}-{{t}_{i}}|\end{eqnarray} \tag{ 16 }$$

  where k is L or R. As shown in figure 4, $d_{1}^{\text{L}}$ and $d_{1}^{\text{R}}$ are 37.55 ns and 39.14 ns, respectively.
• (4)  
  The lower and upper limits of the location of the ith echo component are denoted as $L_{i}^{\text{L}}$ and $L_{i}^{\text{U}}$, respectively. They are set as

  $$\begin{eqnarray}&&\left\{\begin{array}{*{35}{l}} \text{if}\,d_{i}^{\text{L}}<d_{i}^{\text{R}}:\quad L_{i}^{\text{L}}=u_{i}^{\text{L}},\quad L_{i}^{\text{U}}=2\times {{t}_{i}}-u_{i}^{\text{L}} \\ \text{else}:\quad L_{i}^{\text{L}}=2\times {{t}_{i}}-u_{i}^{\text{R}},\quad L_{i}^{\text{U}}=u_{i}^{\text{R}} \end{array}\right. \end{eqnarray} \tag{ 17 }$$

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2.3.3. Initial estimation of the location of an overlapping echo component.

The overlapping echo components are detected by using the inflection points which are located in the same monotone interval of the full-waveform echo and not located in $\left[L_{i}^{\text{L}},~L_{i}^{\text{U}}\right]$, where $L_{i}^{\text{L}}$ and $L_{i}^{\text{U}}$ are the lower and upper limits of the location of the ith echo component detected at ${{P}_{i}}$, respectively. As shown in figure 5, inflection points ${{D}_{1}}$, ${{D}_{2}}$ are located in $I_{3}^{\text{L}}$ but not in $\left[L_{3}^{\text{L}},~L_{3}^{\text{U}}\right]$. Thus, a new echo component is deemed. The lower limit of location of the new echo component is set as the mean of the time coordinates of ${{D}_{1}}$ and ${{D}_{2}}$. The upper limit of the location of the new echo component is set as the $L_{3}^{\text{L}}$.

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The location intervals of echo components detected from the simulated echo are shown in table 1. It can be seen that the 1th, 2th, 3th and 8th location intervals contain echo components given in the simulated full-waveform echo, thus the echo components in these location intervals are considered as true echo components. However, there are still some location intervals which do not include any echo components and the echo components in these location intervals are referred as false echo components. As shown in figures 4 and 5, location intervals which contain echo components are shown using blue dash lines. Meanwhile, location intervals which do not contain echo components are shown using red dash lines.

Table 1. The location intervals detected from the simulated full-waveform echo.

NO.	Lower limit	Upper limit	Contain echo component
1	212	288	Yes
2	482	496	Yes
3	423	437	Yes
4	194	212	No
5	288	308	No
6	420	423	No
7	437	439	No
8	466	482	Yes

After detection of echo components, the number of echo components detected from ${{c}_{N}}(n)$ is denoted as M. If M is equal to zero, there is no echo components detected from ${{c}_{N}}(n)$; Otherwise, PSO is used to select the true echo components and optimize the parameters of the selected echo components. Steps to select echo components and optimize the parameters of echo components are as follows:

• (1)  
  The number of echo components in s(n) is set as 1 (K  =  1).
• (2)  
  The amplitudes, locations and FWHMs of the K echo components are simultaneously and iteratively renewed using the common PSO algorithm [24]. The variables subjected to be optimized by PSO method are shown in table 2. The parameters of the common PSO algorithm are shown in table 3. In the common PSO algorithm, configurations of the lower and upper limits of amplitude, location and FWHM of the ith echo component and DC offset of s(n) are shown in table 4. In PSO algorithm, residual sum of squares (RSS) between the full-waveform echo reconstructed using optimized variables and the raw full-waveform echo is selected as cost function.
• (3)  
  After renewing, if the amplitudes of echo components are equal to th_n, the echo components are regarded as false echo components. The number of false echo components is denoted as F. If the false echo components are un-detected, go to next step. Else, the false echo components and location intervals where false echo components are located are removed. Meanwhile, $K~=~K~-~F$, $M~=~M~-~F$ and go to step (2).
• (4)  
  Residual error $\varepsilon (n)$ is calculated depending on equation (2). Then, standard deviation of $\varepsilon (n)$ is calculated and denoted as ${{\sigma}_{d}}$.
• (5)  
  The selection and optimization of echo components stop if one of the following three conditions is satisfied. The first condition is ${{\sigma}_{d}}<{{\sigma}_{n}}$. The second condition is that K is equal to G. G is a given maximum number of echo components in one full-waveform echo. The third condition is that K is equal to M. Otherwise, $K~=~K+1$. Then, go to Step (2).

Table 4. Configurations of the lower and upper limits of amplitude, location and FWHM of ith echo component and DC offset of s(n).

	Amplitude	Location	FWHM	D
Lower limit	$\text{t}{{\text{h}}_{n}}$	$L_{i}^{\text{L}}$	${{F}_{t}}~$b	0
Upper limit	${{M}_{c}}$a	$L_{i}^{\text{U}}$	${{l}_{r}}~$c	${{M}_{c}}$

^aMaximum count of A/D convertor. ^bFWHM of the transmitted pulse ^cThe half of sample length of s(n).

The echo components in each full-waveform echo were simulated using Gaussian functions. The noise in the full-waveform echo was regarded as Gaussian white. The steps to carry out the simulation were as follows:

Firstly, the number of the simulated full-waveform echoes is 1000. The sampling frequency and time length of each simulated full-waveform echo is 1 GHz and 499 ns, respectively. Therefore, each simulated full-waveform echo is of 500 samples. The FWHM of the emitted laser pulse was set as 10 ns. The maximum number of the echo components in each simulated full-waveform echo was set as 6.

Secondly, each simulated full-waveform echo, $m(n)$, was generated by

$$\begin{eqnarray}&&m(t)=\underset{i=1}{\overset{{{N}_{c}}}{\sum}}\,{{a}_{i}}{{\text{e}}^{-\frac{{{\left(t-{{\tau}_{i}}\right)}^{2}}}{\omega _{i}^{2}/\left(4\ln 2\right)}}}+n(t)\quad 0\leqslant t\leqslant 499\end{eqnarray} \tag{ 22 }$$

where t is sample time, ${{N}_{c}}$ is the number of the echo components in $m(t)$ and set as a random integer between 1 and 6, ${{a}_{i}}$ is the amplitude of the ith echo component and set as a random value between 30 and 255, ${{\tau}_{i}}$ is the location of the ith echo component and set as a random integer between 0 and 499 ns, ${{\omega}_{i}}$ is the FWHM of the ith echo component and set as a random value between 10 and 100 ns, $n(t)$ is Gaussian white noise and generated according to the given SNR and the power of $m(t)$. The SNR of each simulated full-waveform echo was set as a random value between 20 and 60 dB. Figure 6 shows a simulated full-waveform echo with three echo components. The amplitudes of the three echo components are 41, 82 and 83, respectively. The locations of the three echo components are 142 ns, 240 ns and 356 ns, respectively. The FWHMs of the three echo components are 84.04 ns, 72.27 ns and 73.72 ns, respectively. The SNR of the full-waveform echo is 40 dB.

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Thirdly, for each simulated full-waveform echo, the standard deviation of noise in the simulated full-waveform echo was estimated using front-based, rear-based and wavelet-decomposition-based methods, respectively. In front-based and rear-based methods, the number of samples used to estimate noise level was 50. Then, the simulated full-waveform echo was filtered using Gaussian smoothing filter, Winer filter and wavelet-decomposition-based filter. Last, the simulated full-waveform echo was decomposed using WD-PSO and GS-LM methods, respectively.

For each filter, mean and standard deviation of SNRG of the 1000 simulated full-waveform echoes after filtering were calculated and denoted as ${{u}_{\text{g}}}$ and ${{\sigma}_{\text{g}}}$, respectively. ${{u}_{\text{g}}}\text{s}$ and ${{\sigma}_{\text{g}}}\text{s}$ of the 1000 simulated full-waveform echoes filtered using the three filters were listed in table 5, respectively. It can be seen that the full-waveform echoes which are filtered using wavelet-decomposition-based filter is of the largest mean and the smallest standard deviation of SNRG, implying that the wavelet-decomposition-based filter is of the best improvement of SNR and the noise in the full-waveform echoes can be significantly decreased.

Table 5. Means and standard deviations of SNRGs of different filters.

Filters	${{u}_{g}}$ (dB)	${{\sigma}_{g}}$ (dB)
Gaussian smoothing filter	−6.783	11.31
Wiener filter	2.094	2.183
Wavelet-decomposition-based filter	8.079	1.306

For each estimation method for noise level, the mean and standard deviation of ${{e}_{\sigma}}$ of the 1000 simulated full-waveform echoes were calculated and denoted as ${{u}_{{{e}_{\sigma}}}}$ and ${{\sigma}_{{{e}_{\sigma}}}}$. ${{u}_{{{e}_{\sigma}}}}\text{s}$ and ${{\sigma}_{{{e}_{\sigma}}}}\text{s}$ of the three noise level estimation methods were calculated and listed in table 6. It can be seen that estimation error of standard deviation of noise obtained using wavelet-decomposition-based method is of the smallest mean and standard deviation, implying that the wavelet-decomposition-based method is of the highest estimation accuracy of noise level.

Table 6. Means and standard deviations of EENs of different methods.

Methods	${{u}_{{{e}_{\sigma}}}}$	${{\sigma}_{{{e}_{\sigma}}}}$
Before-based	4.344	8.548
Rear-based	4.530	9.098
Wavelet-decomposition-based	−0.040	0.053

DSR of GS-LM and WD-PSO were calculated and listed in table 7. It can be seen that the proposed method is of 79.3% DSR, which is larger than it of GS-LM, implying that (1) noise in the full-waveform echoes can be effectively removed using wavelet-decomposition-based filter; (2) the echo components can be accurately detected by estimating the upper and lower of the locations of the echo components; (3) PSO-based optimization method can effectively remove false echo components; (4) PSO method is of better validation than it of L–M method, which is used in GS-LM.

Table 7. DSRs, means and standard deviations of EEAs, EELs and EEFs of the two decomposition methods.

Methods	${{S}_{\text{r}}}$ (%)	$\mu _{\text{e}}^{\text{A}}$	$\sigma _{\text{e}}^{\text{A}}$	$\mu _{\text{e}}^{\text{L}}$	$\sigma _{\text{e}}^{\text{L}}$	$\mu _{\text{e}}^{\text{F}}$	$\sigma _{\text{e}}^{\text{F}}$
GS-LM	70.8	−2.085	4.147	−0.484	0.177	0.551	0.429
WD-PSO	79.3	−0.562	1.070	−0.009	0.248	0.126	0.386

The mean and standard deviation of ${{E}^{\text{A}}}$, ${{E}^{\text{L}}}$ and ${{E}^{\text{F}}}$ were denoted as $\mu _{\text{e}}^{\text{A}}$, $\sigma _{\text{e}}^{\text{A}}$, $\mu _{\text{e}}^{\text{L}}$, $\sigma _{\text{e}}^{\text{L}}$, $\mu _{\text{e}}^{\text{F}}$ and $\sigma _{\text{e}}^{\text{F}}$, respectively. $\mu _{\text{e}}^{\text{A}}$, $\sigma _{\text{e}}^{\text{A}}$, $\mu _{\text{e}}^{\text{L}}$, $\sigma _{\text{e}}^{\text{L}}$, $\mu _{\text{e}}^{\text{F}}$ and $\sigma _{\text{e}}^{\text{F}}$, of the echo components decomposed from the full-waveform echoes using GS-LM and WD-PSO were calculated and also listed in table 7. It can be seen that the proposed method is of smaller $\mu _{\text{e}}^{\text{A}}$, $\sigma _{\text{e}}^{\text{A}}$, $\mu _{\text{e}}^{\text{L}}$, $\sigma _{\text{e}}^{\text{L}}$, $\mu _{\text{e}}^{\text{F}}$ and $\sigma _{\text{e}}^{\text{F}}$, implying that (1) estimation accuracy of the parameters of echo components obtained using WD-PSO are higher than those obtained by GS-LM. (2) noise in full-waveform echo can be effectively removed using wavelet-decomposition-based filter. (3) attenuation of amplitudes of the echo components caused by wavelet-decomposition-based filter is smaller than it caused by Gaussian smoothing filter, which is used in GS-LM.

To verify the performance of the proposed method for the true full-waveform echo, a lab-build full-waveform LiDAR system was set-up, as shown in figure 7. A Nd:YAG laser emits laser pulses with FWHM of 10 ns, wavelength of 1064 nm and repetition frequency of 5 kHz. The emitted laser pulses are split into three beams by beam splitters 1 and 2. The three beams are referred as beam 1, 2 and 3, respectively.

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Beam 1 is reflected towards to objects by reflector 1. If there is an object at the transmission path of beam 1, a part of beam 1 is scattered by the object. The scattered laser beam is collected by an assembled lens. The collected laser beam is converted to electrical signal by a photoelectric detector, which is referred as detector 1. The electrical signal is acquired by channel 0 of a high-speed digitizer produced by National Instruments (NI). The acquired electrical signal is the full-waveform echo scattered from the object at the transmission path of beam 1. The NI digitizer is of three input channels, which are channel 0, channel 1 and trigger channel referred as CH0, CH1 and Trig, respectively. The CH0 and CH1 are of 1 GHz sampling frequency and 8-bit resolution, respectively.

Beam 2 is used to produce a trigger signal. The trigger signal is connected to Trig of the NI digitizer and used to trigger the NI digitizer start acquiring the data in CH0 and CH1. Beam 3 is reflected, attenuated and detected by reflector 2, neutral filter and detector 2, respectively. The output of the detector 2 is voltage signal of the attenuated beam 3. The voltage signal is acquired by CH1 of the NI digitizer as the waveform of the transmitted laser pulse.

In order to obtain full-waveform echoes composed of different echo components, three objects were set on the transmission path of beam 1. Figure 8 shows the experimental scene. In experiment, the distance between object 1 and the lab-build full-waveform LiDAR system approximately was 35 m. Eight types of full-waveform echoes were obtained by adjusting the relative locations of the three objects. For each type of full-waveform echo, 1000 full-waveform echoes were acquired. Figure 9 shows eight types of full-waveform echo. Seen from the first to the forth types of full-waveform, the distance between object1 and object 2 gradually decreases. However, the distance between object 2 and 3 is almost constant. The four types of full-waveform echoes were used to evaluate the detection performances of different decomposition methods for two objects with different distance intervals. Seen from the fifth to eighth types of full-waveform echoes, both distances between object 1 and 2 and distance between object 2 and 3 gradually decrease. The four types of full-waveform echoes were used to evaluate the detection performance of different decomposition methods for three objects with different distance intervals.

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After obtaining the full-waveform echoes, all full-waveform echoes were decomposed using GS-LM and WD-PSO. Meanwhile, to obtain the expected parameters of echo components, all full-waveform echoes were decomposed using MATLAB curve fitting tool.