CN104614718B - Method for decomposing laser radar waveform data based on particle swarm optimization - Google Patents

Abstract

The invention provides a three-dimensional laser echo decomposition algorithm based on the combination of particle swarm algorithm and Levenberg-Marquardt Algorithm (LM algorithm), including smoothing and denoising, peak detection, waveform decomposition and fitting. Use the set threshold and peak detection to determine the number of waveforms with good signal-to-noise ratio after smoothing and denoising, obtain the approximate value of the intensity parameter and the approximate value of the width parameter of a single waveform through the particle swarm optimization algorithm, and use it as the initial value of the LM algorithm to improve the decomposition Accuracy, reducing the error impact caused by the initial value.

Description

Decomposition method of lidar waveform data based on particle swarm optimization

technical field

The invention belongs to the technical field of laser data processing, and specifically refers to a method for decomposing laser radar waveform data based on a particle swarm algorithm.

Background technique

The laser echo waveform contains a large amount of surface information inside the laser spot, and the fine features of the surface target can be extracted by analyzing the echo waveform. Therefore, finding an effective and accurate echo decomposition algorithm is a topic worthy of research.

Background noise will cause random changes in waveform amplitude, and too many burrs may cause detection errors, so smoothing filtering has a greater impact on parameter fitting. However, some filtering algorithms will lead to amplitude distortion and excessive smoothing, resulting in loss of details or loss of peak points, so it is necessary to compare and choose a better algorithm.

In the existing algorithms, the accuracy of the LM algorithm depends on the initial value. If the initial value has a large deviation, it is difficult to obtain an accurate fitting effect. In the previous algorithms, the pulse width parameters were often determined by the detection of the zero-crossing inflection point, but the actual laser model is not a standard Gaussian model and is not symmetrical. Therefore, the echo parameters obtained by this method are Not accurate as an initial value.

To sum up, the current laser echo decomposition algorithm is not ideal in the case of many parameters and superimposed broadening, so the vondrak smoothing algorithm with better effect is more suitable for echo decomposition, and the improved particle swarm optimization algorithm is used to obtain its Using parameter values as the initial values of the LM algorithm is a way to optimize its results.

Particle swarm optimization algorithm, also known as particle swarm optimization algorithm (Particle Swarm Optimization), abbreviated as PSO, is a new evolutionary algorithm (Evolutionary Algorithm-EA) developed in recent years. The PSO algorithm is a kind of evolutionary algorithm. Similar to the simulated annealing algorithm, it also starts from a random solution and finds the optimal solution through iteration. It also evaluates the quality of the solution through fitness, but it is simpler than the rules of the genetic algorithm. It does not have the "Crossover" and "Mutation" operations of the genetic algorithm, and it searches for the global optimum by following the optimal value currently searched. This algorithm has attracted the attention of academic circles for its advantages of easy implementation, high precision, and fast convergence, and has demonstrated its superiority in solving practical problems. Particle swarm algorithm is a parallel algorithm.

PSO is initialized as a group of random particles (random solutions). The optimal solution is then found through iteration. In each iteration, the particle updates itself by tracking two "extrema". The first is the optimal solution found by the particle itself, which is called the individual extremum pBest. Another extremum is the optimal solution currently found by the entire population, and this extremum is the global extremum gBest. In addition, instead of the whole population, only a part of it can be used as the neighbor of the particle, then the extremum among all the neighbors is the local extremum.

The LM algorithm, the full name of the Levenberg-Marquard algorithm, can be used to solve nonlinear least squares problems, and is mostly used in curve fitting and other occasions.

Contents of the invention

The purpose of the present invention is to develop a complete set of laser echo data decomposition methods based on the three-dimensional laser imaging system. In the case of more parameters and superposition and broadening, the vondrak smoothing algorithm with better effect is adopted, and the particle swarm algorithm is improved. Obtain its parameter value as the initial value of the LM algorithm to optimize its result.

The technical scheme of the present invention is: the method for the laser radar waveform data decomposition based on particle swarm algorithm, concrete steps are as follows:

(1) Obtain full waveform echo data of lidar;

(2) Denoise the waveform;

(3) smoothing the waveform;

(4) Detect the peak point of the echo through peak detection, set the peak point threshold according to the background noise level, remove the redundant peak point caused by noise, and determine the position and number of the echo peak point;

(5) Use the improved particle swarm optimization algorithm to iterate to obtain the approximate value of a single waveform parameter;

(6) The approximate value of the parameters obtained in step (5) is used as the initial value of the LM iterative algorithm, and the final waveform parameters are obtained by the least square method; the fitting effect is calculated according to the degree of fit formula, and the degree of fit formula is as follows:

Among them: obs _i is the target waveform to be fitted, that is, the actual waveform data after steps (1)-(3), y _i is the fitting result, N is the number of actual echo sampling points, when the R value is closer to 1 is the higher the fit.

Further, in step (2), wavelet algorithm is used to denoise the waveform.

Further, step (5) uses the improved particle swarm optimization algorithm to iterate to obtain the approximate value of a single waveform parameter; the specific process is as follows:

a. Determine the parameters, randomly initialize the position and velocity of the particle group, and record the individual extreme value and the group extreme value;

b. Calculate the fitness value of each particle;

c. Compare the fitness value of each particle with the individual extremum, and if it is better, update the individual extremum of the particle;

d. Compare the fitness value of each particle with the group extremum, and if it is better, update the particle swarm group extremum;

e. Update the position and flight speed of each particle;

f. Set the number of iterations, and stop the calculation when the number of iterations is reached;

The parameters are delay, intensity and pulse width parameters of a single waveform, and the adaptation value is determined by the function calculation, where obs _i is the target waveform to be fitted, that is, the actual waveform data after steps (1)-(3), y _i is the waveform reconstructed by the current parameter particles, N is the number of actual echo sampling points, When the R value is closer to 1, the effect is better;

The update for parameter particles is:

The individual extremum refers to the optimal solution found by the particle itself, that is, the The population extremum refers to the optimal solution found globally, that is, the is the parameter particle, is the flight speed, ω is the inertia factor, c ₁ and c ₂ are acceleration constants.

Further, adopt vondrak algorithm to carry out smooth processing to waveform in step (3), concrete process is as follows:

For the measured waveform data sequence ( _xi , y _i ), _xi is the sampling time, y _i is the data sampling value, is the smoothed value, express , P _i is the weight of the measurement value, F is the degree of approximation, S is the smoothness, and 1/λ ² becomes the smoothing coefficient; the smoothing function used in the vondrak smoothing method is expressed in polynomial form, and the specific method is to adjacent four sets of data Expressed by a cubic Lagrangian polynomial, every four smoothing values constitute a Lagrangian polynomial, which is used to represent the middle two smoothing values, and the basic equations of the vondrak smoothing method are:

(i=1,2,...,n)

There are n equations in total, among which:

A _-3i = a _i-3 d _i-3 ; A _-2i = a _i-2 c _i-2 + b _i-3 d _i-3 ; A _-1i = a _i-1 b _i-1 + b _{i -2} c _i-2 +c _i-3 d _i-3 ;

A _1i ＝a _i b _i +b _i-1 c _i-1 +c _i-2 d _i-2

A _2i ＝a _i c _i +b _i-1 d _i-1 ; A _3i ＝a _i d _i ; ε=1/λ ² ; B _i ＝εP _i ; A _i ＝0(j+i≤0 or j +i≥n+1)

in:

The smoothed data can be obtained by solving this system of linear equations.

Further, in step (4), the noise caused by peak detection is removed by obtaining the mean value and variance of a section of background noise.

The beneficial effects of the present invention are: in the case of many parameters and superimposition and broadening, the vondrak smoothing algorithm with better effect is adopted, and the parameter value obtained by improving the particle swarm algorithm is used as the initial value of the LM algorithm to optimize the result. The LM algorithm used in traditional fitting has different initial value settings, and the final fitting accuracy is also different. Several groups of initial values are randomly set, and the final average fitting degree is less than 0.98. However, through the present invention, the particle swarm algorithm is used to obtain The value is used as the initial value, and the final fitting degree can reach 0.989.

Description of drawings

The present invention will be further described below in conjunction with accompanying drawing and example.

Fig. 1 is a technical flow chart of the present invention;

Figure 2 is the laser radar full waveform echo data;

Fig. 3 is the waveform data after wavelet denoising;

Figure 4 is the result of vondrak smoothing on the denoised waveform, the enlarged part of the waveform is inside the box, and the loss of surface details;

Figure 5 is the result of five-point three-time smoothing of the denoised waveform, and the enlarged part of the waveform is in the box, compared with vondrak;

Figure 6 is the peak detection result, the background noise is selected in the box, and the threshold value is set according to the background noise level;

Figure 7 is the final result of the decomposition.

detailed description

The technical implementation of the present invention will be described in detail below in conjunction with the accompanying drawings.

As shown in Figure 1, the method of decomposing lidar waveform data based on particle swarm optimization algorithm, the specific steps are as follows:

(1) According to the actual laser system used, select the corresponding function to obtain the full waveform data of the actual echo of the laser radar, as shown in Figure 2, the abscissa is the sampling interval of the waveform data (unit: ns), and the ordinate is the amplitude value .

(2) Process the noise of waveform data by wavelet algorithm. During the flight process and reflection process of the signal, waveform noise will be generated due to multiple factors such as atmospheric and system noise. Wavelet denoising is used to process the waveform data, and the noise in the filtered waveform data has been significantly suppressed. The result is shown in Figure 3 shown.

(3) For the waveform after denoising, there are still many burrs, which have a serious impact on the decomposition of waveform data. Simple smoothing will lead to loss of waveform data or unsatisfactory smoothing effect, so it is necessary to select a smoothing algorithm with relatively better effect. The invention improves the signal-to-noise ratio through vondrak algorithm smoothing; the method is firstly applied in astronomy, and the purpose is to reduce the influence of errors introduced by astronomical observation instruments and environmental factors on astronomical observation data. The Vondrak data smoothing method is not only suitable for measuring data with equal spacing and equal precision, but also can be used for smoothing data with unequal spacing and unequal precision, so it has a wide range of applications.

The specific steps of vondrak algorithm smoothing are as follows: the basic assumption is

Q＝F+λ ² S＝Minimum

in

For the measured waveform data sequence ( _xi , y _i ), _xi is the sampling time, y _i is the data sampling value, is the smoothed value, express The third derivative of , P _i is the weight of the measured value, F is the degree of approximation, S is the degree of smoothness, and 1/λ ² becomes the smoothing coefficient. The smoothing function used in the vondrak smoothing method is expressed in the form of a polynomial, and the specific method is to compare the adjacent four sets of data Expressed by a cubic Lagrangian polynomial, every four smoothing values constitute a Lagrangian polynomial, which is used to represent the middle two smoothing values.

The basic system of equations (in matrix form) for the vondrak smoothing method is:

(i=1,2,...,n)

There are n equations in total, among which:

A _-3i = a _i-3 d _i-3 ; A _-2i = a _i-2 c _i-2 + b _i-3 d _i-3 ; A _-1i = a _i-1 b _i-1 + b _{i -2} c _i-2 +c _i-3 d _i-3 ;

A _1i ＝a _i b _i +b _i-1 c _i-1 +c _i-2 d _i-2

A _2i ＝a _i c _i +b _i-1 d _i-1 ; A _3i ＝a _i d _i ; ε=1/λ ² ; B _i ＝εP _i ; A _i ＝0(j+i≤0 or j +i≥n+1)

in

The smoothed data can be obtained by solving the linear equation system. In the present invention, ε= ¹ /λ2 is selected as 450. The result after vondrak smoothing is shown in Figure 4, and the result of the previous five-point three-time smoothing is shown in Figure 5. From the comparison, it can be seen that after smoothing, the burrs can be effectively filtered out, and at the same time, it can be well guaranteed signal integrity.

(4) The peak point is detected by peak detection, the peak point threshold is set according to the background noise level, and the redundant peak point due to noise is removed; for the peak value, the first order derivative = 0 is searched, but due to the influence of noise, the searched point It is bound to be much more than the actual one, so it is necessary to set a threshold level according to the intensity of the background noise to remove the noise. As shown in Figure 6, the noise caused by peak detection is removed by obtaining the mean and variance of a segment of background noise.

(5) Obtain the approximate value of each parameter of a single waveform by improving the particle swarm optimization algorithm, and use it as the initial value for further fitting;

Nonlinear fitting is the core of waveform decomposition. However, since the accuracy of the LM (Levenberg-Marquardt algorithm) is highly dependent on the initial value, and the actual laser waveform is not a symmetrical Gaussian model, if the half-width parameter is obtained through inflection point detection, it will affect the As a result, certain errors occur. The particle swarm optimization (PSO) algorithm is derived from the simulation of the predation behavior of birds or fish, and has strong global search capabilities. The process is as follows:

a. Determine the parameters, randomly initialize the position and velocity of the particle group, and record the individual extreme value and the group extreme value.

b. Calculate the fitness value of each particle

c. Compare the fitness value of each particle with the individual extreme value, if better, update the individual extreme value of the particle

d. Compare the fitness value of each particle with the group extremum, if better, update the particle swarm group extremum

e. Update the position and flight speed of each particle

f. Set the number of iterations, stop the calculation when it reaches

In the present invention, the parameters are the delay, intensity and pulse width parameters of a single waveform, and the adaptation value is determined by the function calculation, where obs _i is the target waveform to be fitted, that is, the actual waveform data after steps (1)-(3), y _i is the waveform reconstructed by the current parameter particles, N is the number of actual echo sampling points, The effect is better when the R value is closer to 1. The update for parameter particles is:

The individual extremum refers to the optimal solution found by the particle itself, that is, the The group extremum refers to the optimal solution found globally, that is, the is the parameter particle, For flight speed, the maximum value is set to 0.5, and the minimum value is set to -0.5. The inertia factor w is 0.729 by comparison, the convergence is better, c ₁ and c ₂ are acceleration constants, in this paper it is 1.454, and the number of iterations is set to 10000.

(6) The optimal value obtained by the particle swarm optimization algorithm is used as the initial value of the LM algorithm, the number of cycles is set, and the fitting effect is according to the fitting degree formula calculation, where obs _i is the target waveform to be fitted, that is, the actual waveform data after step (1)-(3) processing, y _i is the fitting result, N is the number of actual echo sampling points, when the R value is closer to The higher the fitting degree at 1, the better the effect. After decomposition, the decomposition result is finally obtained. The final result is shown in Figure 7. For each decomposed wavelet, its delay is the elevation information of the corresponding pixel point, and its amplitude is the intensity information of the corresponding pixel point.

The LM algorithm used in traditional fitting has different initial value settings, and the final fitting accuracy is also different. Several groups of initial values are randomly set, and the final average fitting degree is less than 0.98. However, through the present invention, the particle swarm algorithm is used to obtain The value is used as the initial value, and the final fitting degree can reach 0.989.

It should be understood that the application of the present invention is not limited to the above-mentioned examples, and those skilled in the art can make improvements or transformations according to the above-mentioned descriptions, and all these changes and transformations should belong to the protection scope of the appended claims of the present invention.

Claims (4)

1. the method based on the laser radar waveform data decomposition of particle swarm algorithm, it is characterized in that: concrete steps are as follows: (1) Obtain full waveform echo data of lidar; (2) Denoise the waveform; (3) smoothing the waveform; (4) Detect the peak point of the echo through peak detection, set the peak point threshold according to the background noise level, remove the redundant peak point caused by noise, and determine the position and number of the echo peak point; (5) Use the improved particle swarm optimization algorithm to iterate to obtain the approximate value of a single waveform parameter; the specific process is as follows: a. Determine the parameters, randomly initialize the position and velocity of the particle group, and record the individual extreme value and the group extreme value; b. Calculate the fitness value of each particle; c. Compare the fitness value of each particle with the individual extremum, and if it is better, update the individual extremum of the particle; d. Compare the fitness value of each particle with the group extremum, and if it is better, update the particle swarm group extremum; e. Update the position and flight speed of each particle; f. Set the number of iterations, and stop the calculation when the number of iterations is reached; The parameters are delay, intensity and pulse width parameters of a single waveform, and the adaptation value is determined by the function calculation, where obs _i is the target waveform to be fitted, that is, the actual waveform data after steps (1)-(3), y _i is the waveform reconstructed by the current parameter particles, N is the number of actual echo sampling points, When the R value is closer to 1, the effect is better; The update for parameter particles is: ${\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; *</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; *</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; *</span>\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; no</span>}\mathrm{<span\; class="notranslate">\; d</span>}<span\; class="notranslate">\; *</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span>$ ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; k</span>}}$ The individual extremum refers to the optimal solution found by the particle itself, that is, the The population extremum refers to the optimal solution found globally, that is, the is the parameter particle, is the flight speed, ω is the inertia factor, c ₁ and c ₂ are acceleration constants; (6) The approximate value of the parameters obtained in step (5) is used as the initial value of the LM iterative algorithm, and the final waveform parameters are obtained by the least square method; the fitting effect is calculated according to the degree of fit formula, and the degree of fit formula is as follows: $\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; obs</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; /</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; N</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; obs</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$ Among them: obs _i is the target waveform to be fitted, that is, the actual waveform data after steps (1)-(3), y _i is the fitting result, N is the number of actual echo sampling points, when the R value is closer to 1 is the higher the fit. 2. The method for decomposing laser radar waveform data based on particle swarm optimization algorithm according to claim 1, characterized in that: in step (2), wavelet algorithm is used to denoise the waveform. 3. the method for the laser radar waveform data decomposition based on particle swarm optimization algorithm according to claim 1, is characterized in that: adopt vondrak algorithm to carry out smooth processing to waveform in step (3), concrete process is as follows: For the measured waveform data sequence ( _xi , y _i ), _xi is the sampling time, y _i is the data sampling value, is the smoothed value, express , P _i is the weight of the measurement value, F is the degree of approximation, S is the smoothness, and 1/λ ² becomes the smoothing coefficient; the smoothing function used in the vondrak smoothing method is expressed in polynomial form, and the specific method is to adjacent four sets of data Expressed by a cubic Lagrangian polynomial, every four smoothing values constitute a Lagrangian polynomial, which is used to represent the middle two smoothing values, and the basic equations of the vondrak smoothing method are: $\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}}{\overset{\mathrm{<span\; class="notranslate">\; 3</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\stackrel{<span\; class="notranslate">\; -</span>}{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\stackrel{<span\; class="notranslate">\; -</span>}{{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; ...</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$ There are n equations in total, among which: A _-3i = a _i-3 d _i-3 ; A _-2i = a _i-2 c _i-2 + b _i-3 d _i-3 ; A _-1i = a _i-1 b _i-1 + b _{i -2} c _i-2 +c _i-3 d _i-3 ; ${\mathrm{<span\; class="notranslate">\; A</span>}}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;P</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}$ A _2i ＝a _i c _i +b _i-1 d _i-1 ; A _3i ＝a _i d _i ; ε=1/λ ² ; B _i ＝εP _i ; A _i ＝0(j+i≤0 or j +i≥n+1) in: ${\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 6</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 6</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>$ ${\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 6</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 6</span>}\sqrt{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}}}{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>$ The smoothed data can be obtained by solving this system of linear equations. 4. the method for the laser radar waveform data decomposition based on particle swarm optimization algorithm according to claim 1, is characterized in that: in step (4), remove the noise that peak detection brings by obtaining the mean value and the variance of a section of background noise.