# Point Cloud Intensity Correction for 2D LiDAR Mobile Laser Scanning

Authors: Xu Liu; Qiujie Li; Youlin Xu; Xuefeng Wei
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

The acquisition of point cloud data by mobile laser scanning (MLS) includes not only the information about the 3D geometry of the object but also the intensity from the scanned object. However, due to the influence of various factors, there is a large deviation between the intensity and the spectral reflection characteristics of the scanned object. Intensity correction should be carried out before this method is applied to object recognition. A new point cloud intensity correction method for 2D MLS that was developed by combining theoretical derivation with empirical correction is proposed in this paper. First, based on the LiDAR formula, the main factors influencing MLS intensity are investigated, and a distance piecewise polynomial and an incident angle cosine polynomial are adopted to obtain the intensity correction model of UTM-30LX 2D LiDAR on a diffuse reflector plate. Second, according to the scan pattern, a 2D scan grid is constructed to organize the MLS intensity, and a new method of spherical neighborhood search fitting plane is proposed to accurately calculate the cosine of the incident angle. Finally, the obtained intensity correction model is utilized to correct the MLS intensity of a wall. Two groups of verification experiments are carried out on single sites and multiple sites to test the effect of the intensity correction model. Overall, the improvements in intensity consistency range from 70% to 92.7% after correction within the tested ranges of distance and incident angles [0.52 m-5.34 m, 0°-74°]. The results indicate that the proposed intensity correction model yields highly accurate fitting and can effectively remove the deviation in MLS intensity caused by the distance and incident angle.


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## Body

## 1. Introduction

Light detection and ranging (LiDAR), a type of noncontact active remote sensing sensor, can quickly scan high-resolution and high-precision 3D point cloud data on the target surface. LiDAR has been successfully applied in fields, such as global climate change research, smart cities, forest resource surveys, environmental monitoring, and basic mapping [1, 2]. LiDAR can not only obtain the 3D geometric information of the target surface, but it can also record the intensity information. The intensity corresponds to the coordinate information one-to-one without registration and has the feature of pixel-level fusion. It represents the reflection spectral characteristics of the object target to the laser and can be used as an important feature of target classification [3–5]. However, due to the influence of various factors, such as scanner characteristics, atmospheric transmission characteristics, target surface parameters, and data acquisition parameters, there is a large deviation between the intensity and the spectral reflection characteristics of the object target. The phenomena of the same object target with different spectra and the same spectrum of different object targets are apt to occur. Therefore, it is necessary to eliminate the influence of various factors through correction [6–8].Intensity correction methods can be divided into theoretical correction and empirical correction [9]. The theoretical correction method focuses on the analysis of the relationship between multiple factors causing the intensity change, and the regression model of each influencing factor and intensity is established according to the LiDAR ranging principle. Song et al. [10] systematically discussed and analyzed the radiation characteristics, as well as the key influencing factors of intensity data from the perspective of the LiDAR radiation transmission mechanism. They also eliminated these factors through a theoretical correction model. Bolkas [11] studied the intensity correction method of the incident angle and evaluated the intensity correction effect of four kinds of light reflection models under different colors and glosses. Cheng et al. [12] proposed a method to eliminate the distance effect of laser intensity by using a piecewise polynomial model, which effectively eliminated the intensity deviation caused by distance. Fang et al. [13], based on the principle of laser imaging, deduced an intensity theoretical model that considered the defocus effect of the receiving optical system and applied the model to the intensity correction of fresco. The theoretical correction model is applicable to a wide range of scenarios, but the process of the established model is more complex. The parameters in the theoretical correction model are always set as the measured value under the ideal state, which produces large errors in the actual intensity correction. For example, the measurement of the atmospheric attenuation coefficient and transmission coefficient of a LiDAR optical system usually has fluctuation errors. The reflection characteristics of the scanned object’s surface are very complex, and it is difficult to regard it as an ideal diffuse reflection target.The empirical correction method does not rely on any theoretical model. However, it only establishes the intensity correction model in the form of elementary functions (such as polynomial, cosine, or exponential functions). It estimates the function parameters from the actual measured data, which is suitable for situations where the structure and parameters of LiDAR are poorly known. Li and Cheng [14] established two data-driven models to correct the effects of the distance and incident angle on intensity and then applied the corrected intensity to the damage detection of historic buildings. Coren and Sterzai [15] adopted the empirical correction method to establish an exponential function model between the intensity and atmospheric attenuation coefficient, which reduced the fluctuation of the intensity data of asphalt pavement. Vain et al. [16] and Korpela et al. [17] used the empirical correction method to compensate for the intensity change caused by the LiDAR internal automatic gain control system. For the intensity data of specific scenes, the empirical correction method has a higher accuracy than the theoretical correction method, but it is only applicable to this specific application scene; otherwise, it needs to be remodeled.LiDAR measurements are often classified into three types: terrestrial laser scanning (TLS), mobile laser scanning (MLS), and airborne laser scanning (ALS). TLS can directly obtain high-precision 3D point cloud data from the surrounding natural environment by setting up fixed sites. MLS, on the other hand, can quickly scan 3D point cloud data on both sides of the road or on one side by carrying LiDAR on mobile devices, such as cars [18]. Compared with TLS, MLS is flexible and expands the scanning range of LiDAR. MLS also has mature applications in digital city 3D modeling, urban environment monitoring, and urban resource surveys [19, 20]. ALS can obtain the surface spatial information within a large area in a short time with high working efficiency by carrying LiDAR onto the aircraft to realize scanning. However, compared with MLS, point cloud data obtained via ALS measurements are less accurate. Moreover, MLS has an advantage over ALS in obtaining vertical point cloud data. Yang et al. [21] showed that, compared with ALS, MLS has considerable advantages in street tree identification at the scale of a single tree, as well as a strong data acquisition ability to penetrate the inner canopy and trunk.LiDAR sensors are usually divided into two kinds: 3D and 2D (also known as multithread and single-thread sensors). The 3D LiDAR can accurately obtain 3D geometric information of the surrounding environment, and it is mainly used in the field of unmanned driving, which requires high accuracy. However, 3D LiDAR is usually expensive. In contrast, 2D LiDAR can obtain high-precision 3D information about the environment throughout the process of moving through fan-shaped scanning frame by frame, which is advantageous because of the fast scanning speed, the small size, low power consumption, and low manufacturing cost. In addition, 2D LiDAR’s acquisition of point cloud data has low redundancy and simple data fusion. Thus, it can be directly indexed according to the frame number and in-frame number of the measured points, avoiding accuracy loss caused by grid partitioning [22]. 2D LiDAR using the MLS measurement method is widely used in the field of tree parameter extraction [23, 24] and urban block ground object classification [25, 26]. The point cloud data collection process by 2D LiDAR is shown in Figure 1. Xu et al. [27] and Nan et al. [28] designed a real-time automatic target spray system based on MLS 2D LiDAR detection and compared the performance with that of infrared, ultrasonic sensors, and 3D LiDAR, proving that 2D LiDAR has great advantages in the identification accuracy and rapid measurement of distance.Figure 1
Data acquisition process of the MLS system based on a single 2D LiDAR. (a) A single frame of the 2D LiDAR. (b) Index structure for the 2D LiDAR data.
(a)(b)To the author’s knowledge, most previous intensity correction studies have focused on 3D LiDAR sensors for ALS and TLS, whereas 2D LiDAR sensors for MLS are still relatively rare. Based on the advantages of MLS over ALS and TLS analyzed above in application value, the important contribution of this work proposes a high-precision intensity correction method suitable for 2D MLS. In recent years, there have been some studies on intensity correction for TLS similar to this paper. Tan and Cheng [29] studied TLS intensity correction based on the laser ranging formula. The relationship between intensity and new variables was established by combining the cosine of the incident angle and the square of the distance. In the intensity correction for different areas of the white wall, the intensity variance-to-mean ratio ε was between 0.6 and 0.78, indicating that the improvements in intensity consistency ranged from 22% to 40%. Subsequently, Tan and Cheng [30] investigated the effects of incidence angle and distance on intensity data and corrected the intensity data of Faro Focus3D 120. For four reference targets with reflectances of 20%, 40%, 60%, and 80%, the linear interpolation method was used to fit the relationship between incidence angle versus intensity and distance versus intensity. A total of 20 small regions with a size of approximately 15cm×15cm in the white lime wall surface were randomly sampled to verify the intensity correction effect. The intensity variance-to-mean ratio ε was 0.26, 0.14, 0.19, and 0.21. The results showed that the intensity consistency was improved by 74%~86% by using the intensity correction model established by the above four reference targets. For MLS intensity correction research, Teo and Yu [31] proposed a distance intensity correction workflow for MLS road point clouds using a data-driven approach. The relationship between distance and intensity was fitted by a piecewise polynomial. By comparing the differences between the scanners and the stripe intensities before and after correction, the improvements in intensity consistency ranged from 47% to 56%.In contrast from the general data-driven intensity correction model, which directly uses various models to establish the approximate relationship between intensity and its influencing factors, lacking the necessary theoretical basis, a new point cloud intensity correction method for 2D MLS is proposed by combining theoretical derivation with empirical correction. In addition, according to the scan pattern, a 2D scan grid is constructed to organize the MLS intensity of the wall, and a new method of a spherical neighborhood search fitting plane is proposed to accurately calculate the cosine of the incident angle. In particular, the relationship between the neighborhood radius and the distance is discussed in the process of plane fitting. The accuracy of incident angle measurement is improved by selecting an appropriate neighborhood radius. Two groups of verification experiments are carried out on single sites and multiple sites to test the effect of the intensity correction model. A single-site experiment shows that theε value of the five same area regions within the range of the distance and incident angle [0.52 m-1.55 m, 0°-74°] is approximately 0.3, indicating that the consistency of intensity has been improved by 70% after correction. Multisite experiments concluded that the ε values of sites A, B, C, and D were 0.073, 0.079, 0.233, and 0.280, respectively, within the range of distance and incident angle [1.52 m-5.34 m, 0°-62°]. This means that the improvements in intensity consistency range from 72% to 92.7% after correction. Overall, this approach is superior to the latest study mentioned above.The proposed method of the intensity correction model is introduced in Section2. The process of obtaining the intensity correction model is described in Section 3. To demonstrate the practical effectiveness of the intensity correction model, two different scenes of wall intensity correction are presented and analyzed in Section 4. Section 5 concludes with the findings.


## 2. Methods

### 2.1. MLS 2D LiDAR Intensity Correction Process

Figure2 is the flow chart of intensity correction for the MLS 2D LiDAR point cloud. In this paper, the intensity correction process consists of two steps. The first step is for model establishment, and the second step is for model validation. Model establishment is when the intensity correction model is obtained. First, the influencing factors of MLS intensity (including target reflectivity, distance, and incident angle) are analyzed for the acquisition of the intensity correction model. Second, the intensity multiplicative model is established, and the intensity correction formula for the distance and incident angle is deduced. Finally, the parameters of the intensity correction model are estimated, including the order and coefficient of the polynomial. Model validation uses the intensity correction model to correct the intensity data of the wall point cloud. For the wall point cloud intensity data scanned by MLS 2D LiDAR, the distance intensity and incident angle intensity are extracted. Then, the obtained intensity correction model is used to correct the intensity data of the wall point cloud.Figure 2
Flow chart of intensity correction for the MLS 2D LiDAR point cloud.


### 2.2. Influencing Factors of MLS Intensity

Assuming that the measured target surface is a Lambert surface (the surface with ideal diffuse reflection characteristics), according to the laser ranging formula [32], the echo power received by the laser detector is
(1)Pr=PtDr2ηatmηsys4∙λcosθR2,where the received laser power Pr is a function of the transmitting laser power Pt, the aperture of the receiving detector is denoted by Dr, the atmospheric one-way extinction coefficient is ηatm, the transmission coefficient of the optical system is ηsys, the target reflectivity is λ, the distance R is between the measuring point and the receiving end of the LiDAR laser (referred to as the distance), and the cosine of the laser incident angle is denoted by cosθ. For the MLS system with a short distance, it can be considered that Pt, Dr, ηatm, and ηsys are constants. Tan and Cheng [29] showed that intensity is the laser echo energy represented digitally. Additionally, there is a certain linear relationship between the received laser power and intensity. According to the above analysis, the main influencing factors of MLS intensity are target reflectivity λ, distance R, and incident angle θ.


### 2.3. The Multiplicative Model of Intensity by MLS

The objective of intensity correction in this paper is to remove the influence of the distance and incident angle on MLS intensity. Then, the intensity value is only related to the reflectivity of the target. Since the received laser power is nonlinearly processed internally by LiDAR, it is not possible to directly use Formula (1) for theoretical correction. In addition, considering that the effects of the reflectivity, distance, and incident angle are independent in theory, they can each be solved separately. The multiplicative model of intensity can be established as
(2)Iλ,R,θ=fλλfRRfθcosθ.Iλ,R,θ is the original intensity, and fλλ, fRR, and fθcosθ represent functions of the independent influence of the target reflectivity, distance, and incident angle as Iλ,R,θ. To eliminate the influence of the distance and incident angle on intensity, Iλ,R,θ at any distance and incident angle should be transformed to the corrected intensity Ic at reference distance R0 and reference incident angle θ0 for a Lambert body with the same target reflectivity.
(3)Ic=fλλfRR0fθcosθ0=fRR0fθcosθ0fRRfθcosθ∙Iλ,R,θ.Formula (3) shows that the transformation relationship between Ic and Iλ,R,θ is established by fRR and fθcosθ.After satisfying the conditions of a target with reflectivity,λ is scanned at different distances R and at a constant incident angleθ0; then, the intensity after the distance correction Irc can be written as
(4)Irc=fRR0fθcosθ0fRRfθcosθ0∙Iλ,R,θ=fRR0fRR∙Iλ,R,θ.Under the above premise conditions,Rmin≤R≤Rmax (Rmin and Rmax are the minimum value and maximum value of the distance measuring range, respectively). According to the Weierstrass theorem [33], a continuous function on a closed interval can be uniformly approximated by a polynomial series. According to the definition, fRR describes the relationship between distance R and Iλ,R,θ in the distance-intensity data. The intensity regression value Iλ,R,θ0 under reflectivity λ and reference incident angle θ0 can be obtained by polynomial regression fitting. Tan and Cheng [30] showed that due to the short-distance effect of the LiDAR optical system, the intensity increases with an increasing distance when the distance is relatively close and decreases with an increasing distance when the distance is relatively far. According to the variable of distance R, piecewise polynomial modeling can be adopted here to illustrate the relationship between R and Iλ,R,θ0 as
(5)fRR=Iλ,R,θ0=∑k=0KakRk,R≤Rt,∑l=0Lbl1Rl,R>Rt,where ak and bl are the coefficients of the distance polynomial, K and L are the order of the distance polynomial, and Rt is the distance cutoff point. Combined with Formula (4), Irc can be expressed as follows:
(6)Irc=fRR0fRR∙Iλ,R,θ=Iλ,R,θ∑k=0KakR0k∑k=0KakRk,R≤Rt,Iλ,R,θ∑l=0Lbl1/R0l∑l=0Lbl1/Rl,R>Rt.Similarly, satisfying the conditions of a reference target with reflectivity,λ is scanned at different incident angles θ and at a constant distance R0, deduced from Formula (3). The intensity after the incident angle correction Iθc can be written as
(7)Iθc=fRR0fθcosθ0fRR0fθcosθ∙Iλ,R,θ=fθcosθ0fθcosθ∙Iλ,R,θ.In the above incident angle-intensity data,0≤cosθ≤1. According to the Weierstrass theorem mentioned above, the intensity regression value Iλ,R0,θ under the reference reflectivity λ0 and the reference incident angle θ0 can also be obtained by polynomial regression fitting. A cosine polynomial is used for modeling fθcosθ, as shown in the following formula:
(8)fθcosθ=Iλ,R0,θ=∑m=0Mcmcosθm,where cm is the coefficient of the incident angle cosine polynomial and M is the order of the incident angle cosine polynomial. By substituting the obtained fθcosθ into Formula (7), Iθc is established as follows:
(9)Iθc=fθcosθ0fθcosθ∙Iλ,R,θ=Iλ,R,θ∑m=0Mcmcosθ0m∑m=0Mcmcosθm.


### 2.4. Parameter Estimation of the MLS Intensity Correction Model

In this paper, the elbow rule method and the least square method are used to determine the polynomial order and the fitting polynomial coefficients. For example, in the polynomial in the short-distance segment (R≤Rt) of Formula (5), the values of K and ak need to be determined in a way that enables the error sum of squares Sa0,a1,⋯aK between Iλ,R,θ0 and Iλ,R,θ to be minimized. Sa0,a1,⋯,aK is the cost function. The root mean square error (RMSE) of the intensity sample is defined as
(10)RMSE=Sa0,a1,⋯aKN=∑n=1NIλ,R,θ0n−Iλ,R,θn2N,where N is the number of Iλ,R,θand Iλ,R,θ0 and n is the serial number of the intensity sample. The smaller the RMSE value is, the lower the fitting error is. In general, the higher the polynomial order (the larger the K value), the smaller the RMSE value. However, it should be noted that if the fitting order K value is selected too high, it is easy to overfit when the distance correction model is applied to the actual scene for intensity correction and the correction effect is poor. If the K value is too low, the distance intensity correction cannot be carried out effectively. To avoid overfitting, according to the change curve of different polynomial order RMSE values, the elbow rule in machine learning is used to determine that the elbow position is the best polynomial order.To obtain the polynomial coefficientak under Sa0,a1,⋯aK the minimum value, the polynomial in the short-distance segment (R≤Rt) of Formula (5) is brought into Formula (10). Then, the partial derivative of each polynomial coefficient ak is transformed into the problem of finding the extremum; thus,
(11)∂Sa0,a1,⋯aK∂ak=2∑n=1N∑k=0KakRnk−Iλ,R,θnRk=0.It can be further obtained that(12)∑k=0KakRnk=Iλ,R,θn.The linear formulas can be obtained as follows:(13)R10R11⋯R1kR20R21⋯R2k⋮⋮⋱⋮Rn0Rn1⋯Rnka0a1⋮ak=Iλ,R,θ1Iλ,R,θ2⋮Iλ,R,θn.In this way, the polynomial coefficientsa0,a1,⋯,ak of fRR can be solved by the Gaussian elimination method. Similarly, bringing the polynomial in the long-distance segment (R>Rt) of Formula (5) and the cosine polynomial of Formula (8) into Formula (10), based on the elbow method and the least square method, the corresponding polynomial coefficient parameters b0,b1,⋯,bl andc0,c1,⋯cm are obtained.


### 2.5. Evaluation Indices of the MLS Intensity Correction Model

To evaluate the effect of distance and incident angle correction models on MLS intensity correction, the coefficient of variationCV is used to represent the degree of discretization before and after intensity correction. The coefficient of variation CV is defined by
(14)CV=STDMean×100%,where STD is the standard deviation of intensity andMean is the mean value of intensity. CV is positively correlated with the dispersion of intensity data. The smaller its value, the smaller the data dispersion. In addition, the evaluation index variance-to-mean ratio ε of the intensity correction model is defined as follows:
(15)ε=CVcorCVori=STD/MeancorSTD/Meanori.CVori and CVcor represent the variation coefficient of the original intensity and the variation coefficient of the corrected intensity, which can be calculated by Formula (14). When ε is less than 1, it means that the variability of the intensity value after correction is less than that before correction, and the intensity correction model is effective. The smaller the value of ε is, the better the intensity consistency of the model after correction.


## 2.1. MLS 2D LiDAR Intensity Correction Process

Figure2 is the flow chart of intensity correction for the MLS 2D LiDAR point cloud. In this paper, the intensity correction process consists of two steps. The first step is for model establishment, and the second step is for model validation. Model establishment is when the intensity correction model is obtained. First, the influencing factors of MLS intensity (including target reflectivity, distance, and incident angle) are analyzed for the acquisition of the intensity correction model. Second, the intensity multiplicative model is established, and the intensity correction formula for the distance and incident angle is deduced. Finally, the parameters of the intensity correction model are estimated, including the order and coefficient of the polynomial. Model validation uses the intensity correction model to correct the intensity data of the wall point cloud. For the wall point cloud intensity data scanned by MLS 2D LiDAR, the distance intensity and incident angle intensity are extracted. Then, the obtained intensity correction model is used to correct the intensity data of the wall point cloud.Figure 2
Flow chart of intensity correction for the MLS 2D LiDAR point cloud.


## 2.2. Influencing Factors of MLS Intensity

Assuming that the measured target surface is a Lambert surface (the surface with ideal diffuse reflection characteristics), according to the laser ranging formula [32], the echo power received by the laser detector is
(1)Pr=PtDr2ηatmηsys4∙λcosθR2,where the received laser power Pr is a function of the transmitting laser power Pt, the aperture of the receiving detector is denoted by Dr, the atmospheric one-way extinction coefficient is ηatm, the transmission coefficient of the optical system is ηsys, the target reflectivity is λ, the distance R is between the measuring point and the receiving end of the LiDAR laser (referred to as the distance), and the cosine of the laser incident angle is denoted by cosθ. For the MLS system with a short distance, it can be considered that Pt, Dr, ηatm, and ηsys are constants. Tan and Cheng [29] showed that intensity is the laser echo energy represented digitally. Additionally, there is a certain linear relationship between the received laser power and intensity. According to the above analysis, the main influencing factors of MLS intensity are target reflectivity λ, distance R, and incident angle θ.


## 2.3. The Multiplicative Model of Intensity by MLS

The objective of intensity correction in this paper is to remove the influence of the distance and incident angle on MLS intensity. Then, the intensity value is only related to the reflectivity of the target. Since the received laser power is nonlinearly processed internally by LiDAR, it is not possible to directly use Formula (1) for theoretical correction. In addition, considering that the effects of the reflectivity, distance, and incident angle are independent in theory, they can each be solved separately. The multiplicative model of intensity can be established as
(2)Iλ,R,θ=fλλfRRfθcosθ.Iλ,R,θ is the original intensity, and fλλ, fRR, and fθcosθ represent functions of the independent influence of the target reflectivity, distance, and incident angle as Iλ,R,θ. To eliminate the influence of the distance and incident angle on intensity, Iλ,R,θ at any distance and incident angle should be transformed to the corrected intensity Ic at reference distance R0 and reference incident angle θ0 for a Lambert body with the same target reflectivity.
(3)Ic=fλλfRR0fθcosθ0=fRR0fθcosθ0fRRfθcosθ∙Iλ,R,θ.Formula (3) shows that the transformation relationship between Ic and Iλ,R,θ is established by fRR and fθcosθ.After satisfying the conditions of a target with reflectivity,λ is scanned at different distances R and at a constant incident angleθ0; then, the intensity after the distance correction Irc can be written as
(4)Irc=fRR0fθcosθ0fRRfθcosθ0∙Iλ,R,θ=fRR0fRR∙Iλ,R,θ.Under the above premise conditions,Rmin≤R≤Rmax (Rmin and Rmax are the minimum value and maximum value of the distance measuring range, respectively). According to the Weierstrass theorem [33], a continuous function on a closed interval can be uniformly approximated by a polynomial series. According to the definition, fRR describes the relationship between distance R and Iλ,R,θ in the distance-intensity data. The intensity regression value Iλ,R,θ0 under reflectivity λ and reference incident angle θ0 can be obtained by polynomial regression fitting. Tan and Cheng [30] showed that due to the short-distance effect of the LiDAR optical system, the intensity increases with an increasing distance when the distance is relatively close and decreases with an increasing distance when the distance is relatively far. According to the variable of distance R, piecewise polynomial modeling can be adopted here to illustrate the relationship between R and Iλ,R,θ0 as
(5)fRR=Iλ,R,θ0=∑k=0KakRk,R≤Rt,∑l=0Lbl1Rl,R>Rt,where ak and bl are the coefficients of the distance polynomial, K and L are the order of the distance polynomial, and Rt is the distance cutoff point. Combined with Formula (4), Irc can be expressed as follows:
(6)Irc=fRR0fRR∙Iλ,R,θ=Iλ,R,θ∑k=0KakR0k∑k=0KakRk,R≤Rt,Iλ,R,θ∑l=0Lbl1/R0l∑l=0Lbl1/Rl,R>Rt.Similarly, satisfying the conditions of a reference target with reflectivity,λ is scanned at different incident angles θ and at a constant distance R0, deduced from Formula (3). The intensity after the incident angle correction Iθc can be written as
(7)Iθc=fRR0fθcosθ0fRR0fθcosθ∙Iλ,R,θ=fθcosθ0fθcosθ∙Iλ,R,θ.In the above incident angle-intensity data,0≤cosθ≤1. According to the Weierstrass theorem mentioned above, the intensity regression value Iλ,R0,θ under the reference reflectivity λ0 and the reference incident angle θ0 can also be obtained by polynomial regression fitting. A cosine polynomial is used for modeling fθcosθ, as shown in the following formula:
(8)fθcosθ=Iλ,R0,θ=∑m=0Mcmcosθm,where cm is the coefficient of the incident angle cosine polynomial and M is the order of the incident angle cosine polynomial. By substituting the obtained fθcosθ into Formula (7), Iθc is established as follows:
(9)Iθc=fθcosθ0fθcosθ∙Iλ,R,θ=Iλ,R,θ∑m=0Mcmcosθ0m∑m=0Mcmcosθm.


## 2.4. Parameter Estimation of the MLS Intensity Correction Model

In this paper, the elbow rule method and the least square method are used to determine the polynomial order and the fitting polynomial coefficients. For example, in the polynomial in the short-distance segment (R≤Rt) of Formula (5), the values of K and ak need to be determined in a way that enables the error sum of squares Sa0,a1,⋯aK between Iλ,R,θ0 and Iλ,R,θ to be minimized. Sa0,a1,⋯,aK is the cost function. The root mean square error (RMSE) of the intensity sample is defined as
(10)RMSE=Sa0,a1,⋯aKN=∑n=1NIλ,R,θ0n−Iλ,R,θn2N,where N is the number of Iλ,R,θand Iλ,R,θ0 and n is the serial number of the intensity sample. The smaller the RMSE value is, the lower the fitting error is. In general, the higher the polynomial order (the larger the K value), the smaller the RMSE value. However, it should be noted that if the fitting order K value is selected too high, it is easy to overfit when the distance correction model is applied to the actual scene for intensity correction and the correction effect is poor. If the K value is too low, the distance intensity correction cannot be carried out effectively. To avoid overfitting, according to the change curve of different polynomial order RMSE values, the elbow rule in machine learning is used to determine that the elbow position is the best polynomial order.To obtain the polynomial coefficientak under Sa0,a1,⋯aK the minimum value, the polynomial in the short-distance segment (R≤Rt) of Formula (5) is brought into Formula (10). Then, the partial derivative of each polynomial coefficient ak is transformed into the problem of finding the extremum; thus,
(11)∂Sa0,a1,⋯aK∂ak=2∑n=1N∑k=0KakRnk−Iλ,R,θnRk=0.It can be further obtained that(12)∑k=0KakRnk=Iλ,R,θn.The linear formulas can be obtained as follows:(13)R10R11⋯R1kR20R21⋯R2k⋮⋮⋱⋮Rn0Rn1⋯Rnka0a1⋮ak=Iλ,R,θ1Iλ,R,θ2⋮Iλ,R,θn.In this way, the polynomial coefficientsa0,a1,⋯,ak of fRR can be solved by the Gaussian elimination method. Similarly, bringing the polynomial in the long-distance segment (R>Rt) of Formula (5) and the cosine polynomial of Formula (8) into Formula (10), based on the elbow method and the least square method, the corresponding polynomial coefficient parameters b0,b1,⋯,bl andc0,c1,⋯cm are obtained.


## 2.5. Evaluation Indices of the MLS Intensity Correction Model

To evaluate the effect of distance and incident angle correction models on MLS intensity correction, the coefficient of variationCV is used to represent the degree of discretization before and after intensity correction. The coefficient of variation CV is defined by
(14)CV=STDMean×100%,where STD is the standard deviation of intensity andMean is the mean value of intensity. CV is positively correlated with the dispersion of intensity data. The smaller its value, the smaller the data dispersion. In addition, the evaluation index variance-to-mean ratio ε of the intensity correction model is defined as follows:
(15)ε=CVcorCVori=STD/MeancorSTD/Meanori.CVori and CVcor represent the variation coefficient of the original intensity and the variation coefficient of the corrected intensity, which can be calculated by Formula (14). When ε is less than 1, it means that the variability of the intensity value after correction is less than that before correction, and the intensity correction model is effective. The smaller the value of ε is, the better the intensity consistency of the model after correction.


## 3. Experiment and Data Acquisition

### 3.1. 2D LiDAR and Diffuse Reflector Plate

UTM-30LX 2D LiDAR [34] utilizes time-of-flight technology to measure the distance. The maximum detection distance of the 2D LiDAR is 30 m. The accuracy is ±30 mm within the range of 0.1-10 m and ±50 mm within the range of 10-30 m. The 2D LiDAR laser beam has a wavelength of 905 nm, an angular resolution of 0.25°, a scanning period of 25 ms, and a scanning range of 0° to 270°, as shown in Figure 3. A total of 1081 target distances and intensity values in different directions are included in one frame of scan data and stored in a file as 4-byte and 2-byte unsigned integer data.Figure 3
UTM-30LX scan area.The reference target is a standard diffuse reflectance plate with a size of50cm×50cm and a reflectance of 50% (that is, the reflectance of the reference target is λ0=0.5), as shown in Figure 4. The diffuse reflector plate is composed of highly diffuse reflector materials. When irradiated by a laser beam with a wavelength of 905 nm, the diffuse reflector plate can be regarded as a standard Lambert body.Figure 4
Standard diffuse reflector.The UTM-30LX 2D LiDAR was placed vertically against the diffuse reflectance plate in an indoor environment. Since the laser performance in different emission directions is the same, only the single laser beam in the middle was used to obtain the distance-intensity measurement value. The order and coefficient of polynomialsfRR and fθcosθ were obtained by designing two groups of experiments of distance and incident angle correction model acquisition, respectively.


### 3.2. Distance Correction Model Acquisition Experiment

The reference incident angle was set toθ0=0°, and the distance-intensity data Ri,Iλ0,Ri,θ0 were obtained by setting different distance sites to scan the diffuse reflectance plate. Since the intensity measurement error was large when the distance was long, the intensity data of the diffuse reflectance plate were collected within the range of 0.1 m-14.4 m by taking half of the maximum detection distance. Nonuniform sampling was adopted, and a total of 39 sites were set. Dense sampling with an interval of 0.1 m was carried out in the range of 0.1 m-1.8 m, and 18 sites were set; sparse sampling with an interval of 0.6 m was carried out in the range of 1.8 m-14.4 m, and 21 sites were set. The experimental scenario is shown in Figure 5.Figure 5
Intensity measurement at different sites in incident angle experiments: (a) intensity measurement at different sites of 0.1 m-14.4 m under the reference incident angle of 0°; (b) the picture of the scanning scene.
(a)(b)The 2D LiDAR was fixed on the ground and placed horizontally opposite the diffuse reflector plate. To ensure that the distance-intensity dataRi,Iλ0,Ri,θ0 were obtained at the reference angle θ0=0°, the laser beam at the intermediate point of the LiDAR at each site (frame 541) was perpendicular to the diffuse reflector plate. To improve the measurement accuracy of distance-intensity data, 10 frames were collected continuously at each site, and then, the average intensity value was calculated. Setting the reference incident angle θ0=0°, within the range of 0.1 m-14.4 m, the relationship between different distances and the corresponding intensity is shown in Figure 6.Figure 6
The relationship between the distance and intensity of different sites in the range of 0.1 m-14.4 m under the reference angle of 0°.In the range of 0.1-0.7 m, the intensity increases rapidly with an increasing distance value, and after 0.7 m, the intensity decreases slowly with an increasing distance value. Therefore, the polynomial function can be fitted by Formula (5) with 0.7 m as the distance cutoff point.


### 3.3. Incident Angle Correction Model Acquisition Experiment

The reference distance was set toR0=1.2m, and the intensities of different incident angles within the range of 0°-80° were collected. To ensure that the incident angle-intensity data θi,Iλ0,R0,θi were measured at R0=1.2m, the laser beam at frame 541 of LiDAR was positioned directly against the central axis of the diffuse reflector plate. The position of LiDAR was fixed, and the diffuse reflector plate central axis was taken to change the position within the range of 0°-80° at intervals of 10°. The experiment is shown in Figure 7.Figure 7
Intensity measurement at different sites in distance experiments: (a) intensity measurement at different sites of 0°-80° at a reference distance of 1.2 m; (b) the picture of the scanning scene.
(a)(b)Ten frames were collected by continuous scanning at each angle site (frame 541), the incident angle-intensity data of the laser beam position were obtained, and the average intensity value was calculated. The experimental result is shown in Figure8.Figure 8
The relationship between the incident angle and intensity at different sites from 0° to 80° at a reference distance of 1.2 m.


### 3.4. Determining the Order and Coefficient of the Polynomial Model

The least square method described in Section2.4 is used to determine the order and coefficient of the polynomial model of distance and incident angle. To avoid overfitting the intensity data, according to the elbow rule in machine learning, the best order of the polynomial is determined to be the elbow position. To determine the ideal polynomial order, the distance correction model acquisition experiment in Section 3.2 and the incident angle correction model acquisition experiment in Section 3.3 were repeated 5 times to expand the dataset. Table 1 lists RMSE for different polynomial order K, L, and M values. The process of determining the optimal values of K, L, and M based on RMSE is shown in Figure 9. It is proven that K=4 and L=3 are the ideal distance piecewise polynomial orders in the short-distance and long-distance segments, respectively, and M=1 is the ideal incident angle cosine polynomial order.Table 1
RMSE forK, L, and M values at different polynomial orders.
OrderRMSEKLM1182.0138.57.3275.637.67.3338.617.65.6417.615.44.2517.510.04.268.79.83.4Figure 9
The process of determining the optimal values ofK, L, and M based on RMSE. (a) The corresponding RMSE value change curve as the distance polynomial order K and L increase. (b) The corresponding RMSE value change curve as the incidence angle cosine polynomial order M increases.
(a)(b)As shown in Figure9(a), the RMSE decreases substantially with increasing K and L values in the distance correction experiment, and K≥4 and L≥3 tend to be flat. According to the principle of the elbow rule in machine learning, the order of the distance polynomial model is set to K=4 and L=3. The blue and orange circles in Figure 9(a) represent the optimal values of K and L, respectively. In Figure 9(b), compared with K and L, the different RMSE values and fluctuation ranges corresponding to the order M of the incident angle cosine polynomial are relatively small. To avoid overfitting, the order of the incident angle polynomial model is set to M=1. The gray circle in Figure 9(b) represents the optimal value of M.In the short-distance segment (R≤Rt), the distance-intensity data Ri,Iλ0,Ri,θ0 are substituted into Formula (13) for calculation. Then, a0=3.9332×103, a1=−2.39 ×104, a2=1.2268×105, a3=−2.1138×105, and a4=1.2328×105. Similarly, in the long-distance segment (R>Rt),b0=−99.7915, b1=1.2582×104, b2=−1.5033×104, and b3=6.0276×103. The incident angle polynomial coefficients c0=2.8033×103 and c1=607.177 can also be obtained by using the incident angle-intensity data θi,Iλ0,R0,θi. The specific expressions of the distance and incident angle cosine polynomial correction functions are shown in Table 2.Table 2
Specific expression of the distance and incident angle cosine polynomial correction function.
fRR=1.2328×105R4−2.1138×105R3+1.2268×105R2−2.39×104R+3.9332×103,R≤0.7m.fRR=6.0276×1031/R3−1.5033×1041/R2+1.2582×1041/R−99.7915,R>0.7m.fθcosθ=607.177cosθ+2.8033×103,0°≤θ≤80°.The function expression in Table2 is only applicable to the point cloud intensity data obtained by UTM-30LX 2D LiDAR scanning the same target reflectivity Lambert body and can only fit the intensity data within the range of 0.1 m-14.4 m and incident angle value of 0°-80°. The fitting results of the piecewise distance and incident angle cosine polynomial are shown in Figure 10.Figure 10
Results of polynomial fitting: (a) the result of piecewise fitting of the distance polynomial; (b) the result of cosine polynomial fitting of the incident angle.
(a)
fRR (K=4, L=3)(b)
fθcosθ (M=1)R0=1.2 m and θ0=0° are taken and substituted by the distance-intensity data Ri,Iλ0,Ri,θ0 and the incident angle-intensity data θi,Iλ0,R0,θi mentioned above into Formulas (6) and (9) for intensity correction. The intensity after correction at different sites is shown in Figure 11. The intensity distribution of each site is approximately 3400, although there are some model errors.Figure 11
Corrected intensity at different sites in distance and incident angle experiments. (a) The intensity value of different distances from 0 to 15 m was corrected by the distance polynomial model under the reference incident angle of 0°. (b) The intensity value of different incident angles from 0° to 80° was corrected by the incident angle cosine polynomial model under the reference distance of 1.2 m.
(a)(b)


## 3.1. 2D LiDAR and Diffuse Reflector Plate

UTM-30LX 2D LiDAR [34] utilizes time-of-flight technology to measure the distance. The maximum detection distance of the 2D LiDAR is 30 m. The accuracy is ±30 mm within the range of 0.1-10 m and ±50 mm within the range of 10-30 m. The 2D LiDAR laser beam has a wavelength of 905 nm, an angular resolution of 0.25°, a scanning period of 25 ms, and a scanning range of 0° to 270°, as shown in Figure 3. A total of 1081 target distances and intensity values in different directions are included in one frame of scan data and stored in a file as 4-byte and 2-byte unsigned integer data.Figure 3
UTM-30LX scan area.The reference target is a standard diffuse reflectance plate with a size of50cm×50cm and a reflectance of 50% (that is, the reflectance of the reference target is λ0=0.5), as shown in Figure 4. The diffuse reflector plate is composed of highly diffuse reflector materials. When irradiated by a laser beam with a wavelength of 905 nm, the diffuse reflector plate can be regarded as a standard Lambert body.Figure 4
Standard diffuse reflector.The UTM-30LX 2D LiDAR was placed vertically against the diffuse reflectance plate in an indoor environment. Since the laser performance in different emission directions is the same, only the single laser beam in the middle was used to obtain the distance-intensity measurement value. The order and coefficient of polynomialsfRR and fθcosθ were obtained by designing two groups of experiments of distance and incident angle correction model acquisition, respectively.


## 3.2. Distance Correction Model Acquisition Experiment

The reference incident angle was set toθ0=0°, and the distance-intensity data Ri,Iλ0,Ri,θ0 were obtained by setting different distance sites to scan the diffuse reflectance plate. Since the intensity measurement error was large when the distance was long, the intensity data of the diffuse reflectance plate were collected within the range of 0.1 m-14.4 m by taking half of the maximum detection distance. Nonuniform sampling was adopted, and a total of 39 sites were set. Dense sampling with an interval of 0.1 m was carried out in the range of 0.1 m-1.8 m, and 18 sites were set; sparse sampling with an interval of 0.6 m was carried out in the range of 1.8 m-14.4 m, and 21 sites were set. The experimental scenario is shown in Figure 5.Figure 5
Intensity measurement at different sites in incident angle experiments: (a) intensity measurement at different sites of 0.1 m-14.4 m under the reference incident angle of 0°; (b) the picture of the scanning scene.
(a)(b)The 2D LiDAR was fixed on the ground and placed horizontally opposite the diffuse reflector plate. To ensure that the distance-intensity dataRi,Iλ0,Ri,θ0 were obtained at the reference angle θ0=0°, the laser beam at the intermediate point of the LiDAR at each site (frame 541) was perpendicular to the diffuse reflector plate. To improve the measurement accuracy of distance-intensity data, 10 frames were collected continuously at each site, and then, the average intensity value was calculated. Setting the reference incident angle θ0=0°, within the range of 0.1 m-14.4 m, the relationship between different distances and the corresponding intensity is shown in Figure 6.Figure 6
The relationship between the distance and intensity of different sites in the range of 0.1 m-14.4 m under the reference angle of 0°.In the range of 0.1-0.7 m, the intensity increases rapidly with an increasing distance value, and after 0.7 m, the intensity decreases slowly with an increasing distance value. Therefore, the polynomial function can be fitted by Formula (5) with 0.7 m as the distance cutoff point.


## 3.3. Incident Angle Correction Model Acquisition Experiment

The reference distance was set toR0=1.2m, and the intensities of different incident angles within the range of 0°-80° were collected. To ensure that the incident angle-intensity data θi,Iλ0,R0,θi were measured at R0=1.2m, the laser beam at frame 541 of LiDAR was positioned directly against the central axis of the diffuse reflector plate. The position of LiDAR was fixed, and the diffuse reflector plate central axis was taken to change the position within the range of 0°-80° at intervals of 10°. The experiment is shown in Figure 7.Figure 7
Intensity measurement at different sites in distance experiments: (a) intensity measurement at different sites of 0°-80° at a reference distance of 1.2 m; (b) the picture of the scanning scene.
(a)(b)Ten frames were collected by continuous scanning at each angle site (frame 541), the incident angle-intensity data of the laser beam position were obtained, and the average intensity value was calculated. The experimental result is shown in Figure8.Figure 8
The relationship between the incident angle and intensity at different sites from 0° to 80° at a reference distance of 1.2 m.


## 3.4. Determining the Order and Coefficient of the Polynomial Model

The least square method described in Section2.4 is used to determine the order and coefficient of the polynomial model of distance and incident angle. To avoid overfitting the intensity data, according to the elbow rule in machine learning, the best order of the polynomial is determined to be the elbow position. To determine the ideal polynomial order, the distance correction model acquisition experiment in Section 3.2 and the incident angle correction model acquisition experiment in Section 3.3 were repeated 5 times to expand the dataset. Table 1 lists RMSE for different polynomial order K, L, and M values. The process of determining the optimal values of K, L, and M based on RMSE is shown in Figure 9. It is proven that K=4 and L=3 are the ideal distance piecewise polynomial orders in the short-distance and long-distance segments, respectively, and M=1 is the ideal incident angle cosine polynomial order.Table 1
RMSE forK, L, and M values at different polynomial orders.
OrderRMSEKLM1182.0138.57.3275.637.67.3338.617.65.6417.615.44.2517.510.04.268.79.83.4Figure 9
The process of determining the optimal values ofK, L, and M based on RMSE. (a) The corresponding RMSE value change curve as the distance polynomial order K and L increase. (b) The corresponding RMSE value change curve as the incidence angle cosine polynomial order M increases.
(a)(b)As shown in Figure9(a), the RMSE decreases substantially with increasing K and L values in the distance correction experiment, and K≥4 and L≥3 tend to be flat. According to the principle of the elbow rule in machine learning, the order of the distance polynomial model is set to K=4 and L=3. The blue and orange circles in Figure 9(a) represent the optimal values of K and L, respectively. In Figure 9(b), compared with K and L, the different RMSE values and fluctuation ranges corresponding to the order M of the incident angle cosine polynomial are relatively small. To avoid overfitting, the order of the incident angle polynomial model is set to M=1. The gray circle in Figure 9(b) represents the optimal value of M.In the short-distance segment (R≤Rt), the distance-intensity data Ri,Iλ0,Ri,θ0 are substituted into Formula (13) for calculation. Then, a0=3.9332×103, a1=−2.39 ×104, a2=1.2268×105, a3=−2.1138×105, and a4=1.2328×105. Similarly, in the long-distance segment (R>Rt),b0=−99.7915, b1=1.2582×104, b2=−1.5033×104, and b3=6.0276×103. The incident angle polynomial coefficients c0=2.8033×103 and c1=607.177 can also be obtained by using the incident angle-intensity data θi,Iλ0,R0,θi. The specific expressions of the distance and incident angle cosine polynomial correction functions are shown in Table 2.Table 2
Specific expression of the distance and incident angle cosine polynomial correction function.
fRR=1.2328×105R4−2.1138×105R3+1.2268×105R2−2.39×104R+3.9332×103,R≤0.7m.fRR=6.0276×1031/R3−1.5033×1041/R2+1.2582×1041/R−99.7915,R>0.7m.fθcosθ=607.177cosθ+2.8033×103,0°≤θ≤80°.The function expression in Table2 is only applicable to the point cloud intensity data obtained by UTM-30LX 2D LiDAR scanning the same target reflectivity Lambert body and can only fit the intensity data within the range of 0.1 m-14.4 m and incident angle value of 0°-80°. The fitting results of the piecewise distance and incident angle cosine polynomial are shown in Figure 10.Figure 10
Results of polynomial fitting: (a) the result of piecewise fitting of the distance polynomial; (b) the result of cosine polynomial fitting of the incident angle.
(a)
fRR (K=4, L=3)(b)
fθcosθ (M=1)R0=1.2 m and θ0=0° are taken and substituted by the distance-intensity data Ri,Iλ0,Ri,θ0 and the incident angle-intensity data θi,Iλ0,R0,θi mentioned above into Formulas (6) and (9) for intensity correction. The intensity after correction at different sites is shown in Figure 11. The intensity distribution of each site is approximately 3400, although there are some model errors.Figure 11
Corrected intensity at different sites in distance and incident angle experiments. (a) The intensity value of different distances from 0 to 15 m was corrected by the distance polynomial model under the reference incident angle of 0°. (b) The intensity value of different incident angles from 0° to 80° was corrected by the incident angle cosine polynomial model under the reference distance of 1.2 m.
(a)(b)


## 4. Results and Analysis

### 4.1. MLS 2D LiDAR Point Cloud Data Acquisition System and Coordinate System

#### 4.1.1. MLS 2D LiDAR Point Cloud Data Acquisition System

To verify the validity of the intensity correction model for the distance and incident angle, a flat wall was selected as the experimental object, which was roughly considered a Lambert surface. To visually observe the intensity point cloud, the intensity value was converted into RGB in CloudCompare, a software point cloud development tool. The 3D point cloud RGB intensity map was also converted into a 2D pseudocolor map for display. The actual scene diagram of the experimental scanning wall is shown in Figure12. The MLS 2D LiDAR measurement system emitted laser pulses in all directions through internal rotating optical components to form a 2D fan-shaped scanning surface. The moving platform carried the laser pulses along the direction perpendicular to the scanning surface to realize the 3D measurement of the target surface. A moving slide mounted UTM-30LX 2D LiDAR was used to move in a straight line at a constant speed of 0.01 m/s in a direction parallel to the wall. The fan-shaped scanning surface was placed directly against the wall. Inside the red rectangular box was the wall study area, with a height of 1.5 m from top to bottom and a width of 1.2 m from left to right. The vertical distance from the mobile slide to the wall was 0.5 m, and the range of distance and incident angle of the wall study area was [0.52 m-1.55 m] and [0°-74°], respectively.Figure 12
The scene of scanning the wall surface: (a) the picture of the scanning scene; (b) 2D pseudocolor intensity map of the scanning scene.
(a)(b)


#### 4.1.2. MLS 2D LiDAR Coordinate System

The starting point position of LiDAR is taken as the origin, and the right-handed coordinate system of the point cloud of the MLS 2D LiDAR measurement system is established, as shown in Figure13.Figure 13
Coordinate system of point clouds acquired by MLS 2D LiDAR.Thex-axis is the moving direction of LiDAR, the y-axis is the scanning direction of LiDAR, the z-axis is vertical to the ground, αi,j is the scanning angle of the ith measuring point in the jth frame, and R is the distance between the LiDAR laser receiver and the measuring point. The calculation of the coordinate system is as follows:
(16)xi,j=v∆t∙j,yi,j=−R∙cosαi,j,zi,j=R∙sinαi,j,where i is the in-frame number, j is the frame number of the measured points, andxi,j, yi,j, and zi,j represent the X, Y, and Z coordinates of the ith measuring point in the jth frame. v is the speed of the moving slide platform in the direction of the motion, and ∆t is the scan cycle of the 2D LiDAR.The geometric relationship between the moving slide with 2D LiDAR and the wall is shown in Figure14. The coordinate of LiDAR center point O at different scanning moments was set as xi,j,0,0; the 3D coordinate of the ith measurement point in frame jth was pi,j=(xi,j,yi,j,zi,j), incident laser vector li,j=0,yi,j,zi,j, and the normal vector of scanning point ni,j=n1,n2,n3.Figure 14
Moving slide and wall geometrical relationship diagram.


### 4.2. ObtainingR and cosθ

Generally, the density of point cloud data obtained by 3D LiDAR is relatively high. Usually, to reduce data redundancy, a support vector machine (SVM) is used to integrate different feature weights of the point cloud into the classifier for training [35–38]. In addition, to improve the efficiency of neighborhood searching, the k-D tree algorithm is usually used for downsampling 3D LiDAR point clouds [39]. Compared with 3D LiDAR point clouds, the data structure of 2D LiDAR point clouds is relatively simple. It only needs to be indexed by the grid in the established coordinate system, and the k-nearest neighbor algorithm [40] can be used to establish the point cloud neighborhood set. Referring to our previous research [41], a 3D spherical neighborhood Si,j is defined as a set of the nearest adjacent points within a sphere centered at the measurement point pi,j with a radius of δ. According to the idea of the least square method, the plane is fitted in the neighborhood set Si,j, and the normal vector ni,j corresponding to each measuring point pi,j is obtained. Then, cosθi,j of the LiDAR laser receiver to each measuring point can be calculated according to Formula (17). Obviously, Ri,j can be obtained directly.
(17)cosθi,j=li,j∙ni,jli,jni,j.Sinceli,j does not change, the accuracy of cosθi,j depends on ni,j. In general, the smaller the value of δ, the smaller the fitting plane, and the more accurate the value of ni,j. However, if the value of δ is too small, the point cloud data in the two adjacent frames cannot be included in the neighborhood; then, the plane cannot be fitted either. In the MLS 2D LiDAR measurement system, the value of δ is closely related to distance Ri,j and the resolution of αi,j. The resolution of the moving direction and the scanning direction of LiDAR is defined as ∆x and ∆s, respectively:
(18)∆x=v∆t,∆s=Ri,jsin∆α.∆α is the 2D LiDAR angular resolution, and its value is constant. The larger the Ri,j is, the greater the value of ∆s. Therefore, if the value of δ is less than ∆s, in the LiDAR scanning direction, all of the in-frame data points adjacent to the measurement point pi,j cannot be included in Si,j. Then, ni,j cannot be obtained by plane fitting. According to Formula (18), the necessary conditions for the measurement point pi,j to have neighborhood points in the LiDAR scanning direction can be deduced:
(19)δ>Ri,jmaxsin∆α,Ri,j≤Ri,jmax.As for UTM-30LX, the value of∆α is 0.25°. In the wall intensity correction experiment, the maximum distance of LiDAR from the wall is not more than 5.5 m. According to Formula (19), Ri,jmaxsin∆α is 0.024. Considering that the range resolution of the LiDAR sensor within 10 m is ±30 mm, the value of δ is set to 0.03 m.


### 4.3. Correction of the Wall Point Cloud Intensity Data

According to the calculation method of the incident angle and distance, the incident angleθ and the distance value R of each measurement point in the wall point cloud intensity data were calculated. The study of Tan and Cheng [42] shows that since the impact of the incident angle and distance on intensity is independent, the sequence of the incident angle correction and distance correction does not affect the correction results. According to the established intensity correction model, the reference incident angle and reference distance are set to be θ0=0° and R0=1.2 m, respectively. The incident angle and distance are corrected for the wall point cloud intensity data. Figure 15 shows the intensity 2D pseudocolor map and intensity distribution histogram of the red rectangular wall area described in Figure 12 before and after intensity correction.Figure 15
The intensity 2D pseudocolor map and intensity distribution histogram of the red rectangular wall area described in Figure12 before and after intensity correction: (a) original intensity pseudocolor chart; (b) intensity pseudocolor chart after the incident angle correction; (c) intensity pseudocolor chart of distance correction after the incident angle correction; (d) histogram of original intensity distribution; (e) histogram of intensity distribution after incident angle correction; (f) intensity distribution histogram of distance correction after the incident angle correction.
(a)(b)(c)(d)(e)(f)As seen from the RGB diagram of the original intensity in Figure15(a), the intensity value at different distances and incident angles is considerably different before correction under the same target reflectivity. The closer the distance or the smaller the incident angle, the higher the intensity value will be. In contrast, the farther the distance or the larger the incident angle, the lower the intensity value will be. The RGB of the intensity after the correction of the incident angle in Figure 15(b) shows that the uniformity of the wall point cloud intensity data has been improved. The RGB diagram of the distance correction intensity after the incident angle correction is shown in Figure 15(c). The influence of the distance and incident angle on intensity is eliminated, and the corrected intensity value is consistent and similar. The histogram of the intensity distribution in Figure 15(d) shows that the original intensity data of the wall point cloud under the same target reflectivity have no obvious distribution pattern. From Figure 15(f), it is clear that after the correction of the incident angle and distance, the intensity value of the wall is concentrated and presents a Gaussian distribution.


### 4.4. Using an Evaluation Indexε to Verify the Intensity Correction Model

#### 4.4.1. Single-Site Multiregion Verification Experiment

In the wall point cloud intensity data obtained from the above experiment, 5 different areas of20cm×20cm were randomly selected, as shown in Figure 16. According to the established intensity correction model, the intensity data in different areas of S1-S5 were also corrected with reference distance R0=1.2 m and incident angle θ0=0°. In addition, the intensity value before and after correction was statistically analyzed. The intensity box chart is shown in Figure 17. The distribution of intensity maximum value (MAX), minimum value (MIN), mean, STD, CV, and the evaluation index ε before and after correction is shown in Table 3.Figure 16
Different areas of S1-S5 randomly selected on the wall.Figure 17
Intensity box diagram of different areas on the same wall before and after correction: (a) intensity box diagram of area S1-S5 before correction; (b) intensity box diagram of area S1-S5 after correction.
(a)(b)Table 3
The distribution of intensity MAX, MIN, Mean, STD, CV, and the evaluation indexε at S1-S5 different regions before and after correction.
IntensityRegion S1Region S2Region S3Region S4Region S5MAXOriginal26812668275028322896Corrected30423015306130063025MINOriginal25982346265727102775Corrected30192990303729813002MeanOriginal26122557271727852846Corrected30313003304929923013STDOriginal16.716.915.918.120.44Corrected5.84.35.65.38.3CVCVori0.00640.00660.00580.00650.0072CVcor0.00190.00140.00180.00180.0028ε0.2970.2120.3100.2770.389As shown in Figure17(a), the intensity value of different regions is substantially different and fluctuates in a large range within each region before intensity correction. From Figure 17(b), it is evident that the difference in the intensity value between different regions decreases, and the fluctuation range of the intensity within the region also becomes considerably smaller after the correction of the incident angle and distance. It is clear that the consistency of intensity between different areas and within different areas has been substantially improved by comparing the box graphs of intensity data from different areas of walls S1-S5 before and after intensity correction under the same reflectivity of the target.Table3 also shows a conclusion similar to that in Figure 17. Due to the small area of the S1-S5 region, the difference between the internal distance and incident angle of each region is not large. The STD values of the original intensity data are all below 21. In addition, due to the large difference in distance and incident angle value between different regions, their STD values are also relatively large, with a minimum value of 15.9 and a maximum value of 20.44. After intensity correction, the STD values of the intensity data in each region are all below 10, and the difference between them is small, indicating that the uniformity of the intensity distribution in each region has been substantially improved. The ε value fluctuates approximately 0.3 in S1-S5 different regions, indicating that the intensity consistency of the five same area regions with the range of distance and incident angle [0.52 m-1.55 m, 0°-74°] has been massively improved by 70% after applying the proposed correction method.Similar to the single-site multiregion verification experiment in this paper, Tan and Cheng [29] utilized Faro Focus3D TLS to obtain intensity point cloud data of ordinary white wall surfaces. In the range of distance and incident angle [6.70 m-14.76 m, 0°-80°], intensity correction experiments were carried out for the A-F region with the same area. The ε value corresponding to the intensity correction of the white wall obtained by the Tan method and the method proposed in this paper is shown in Table 4 below.Table 4
Comparison ofε values of the white wall intensity corrected by the Tan method and the method proposed in this paper.
MethodεRegion S1/ARegion S2/BRegion S3/CRegion S4/DRegion S5/ERegion S6/FThe proposed method0.2970.2120.3100.2770.389—Tan method0.6070.6560.5230.7871.001↑1.139↑Theε value of the Tan method has a large fluctuation range, approximately 0.6, while the ε value of the method proposed in this paper has a small fluctuation range, approximately 0.3. In the Tan experiment, especially in regions E and F, the ε value is greater than 1, indicating that the consistency of the corrected intensity is not as good as the original intensity, and there is an overfitting phenomenon. Moreover, the Tan method only conducts intensity correction research at distances where the intensity value is relatively stable, ignoring the LiDAR short-distance effect, and does not correct the intensity at distances less than 1 m. Overall, we can say that the proposed method in this paper is better than the Tan method and provides a higher accuracy of intensity correction.


#### 4.4.2. Multisite Scanning of the Whole Wall for the Verification Experiment

In the previous single-site experiment, the area of the S1-S5 region was small, and the incident angle and distance value of the point clouds in each region were not substantially different. The STD of the original intensity was also relatively small and could not effectively reflect the distribution characteristics of the original intensity data of the whole wall. A multisite experiment was devoted to the intensity correction of the same red rectangular area on the wall, as shown in Figure12, under multiple sites. Four sites were set up: A, B, C, and D. The vertical distances between the sliding platform and the wall were 1.5 m, 2.5 m, 3.5 m, and 4.5 m, respectively. The rectangular area of the wall was scanned to obtain the point cloud intensity data, as shown in Figure 18. According to the established intensity correction model, the intensity data at different sites were corrected at reference distance R0=1.2 m and incident angle θ0=0°. Table 5 lists the distance and incident angle range values of different sites A, B, C, and D. Figure 19 shows the intensity pseudocolor map of the red rectangular area on the wall described in Figure 12 before and after intensity correction at sites A, B, C and D. The histogram of the intensity distribution before and after intensity correction is shown in Figure 20.Figure 18
Mobile laser scan of 4 sites A, B, C, and D.Table 5
Distance and incident angle range values of different sites A, B, C, and D.
ABCDR (m)MAX3.203.744.535.34MIN1.522.533.524.53θ (°)MAX62483932MIN0000cosθMAX1111MIN0.46950.66910.77710.8480Figure 19
The intensity 2D pseudocolor map of the red rectangular area on the wall described in Figure12 before and after intensity correction at A, B, C, and D 4 sites, respectively. (a–d) Are the original intensity 2D pseudocolor map of sites A, B, C, and D. (e–h) Are the corresponding corrected intensity 2D pseudocolor map of sites A, B, C, and D, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 20
Histogram of intensity distribution before and after correction at A, B, C, and D different sites, respectively. (a–d) Are the original intensity distribution histogram of sites A, B, C, and D. (e–h) Are the corrected intensity distribution histograms of sites A, B, C, and D, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)It is evident from Figures19(a)–19(d) that as the distance increases, the density of point clouds on walls with the same area decreases, while the intensity value shows great differences within the whole wall of each site. It can be clearly seen from Figures 19(e)–19(h) that the variation of wall intensity of the four sites has been considerably reduced after distance and angle correction. By comparing the original intensity distribution in Figures 20(a)–20(d), it was found that quantitatively, the original intensity of each site decreases with an increasing distance and that the degree of dispersion of the intensity distribution is obviously different. As seen from Figures 20(e)–20(h), after intensity correction, the degree of discretization of the intensity value at sites A, B, C, and D is massively reduced, and the intensity value distribution is centralized and approximates a Gaussian distribution. Table 6 lists the distribution of MAX, MIN, Mean, STD, CV of the intensity, and the evaluation index ε at different sites A, B, C, and D before and after intensity correction.Table 6
The distribution of MAX, MIN, Mean, STD, CV of the intensity, and the evaluation indexε at different sites A, B, C, and D before and after intensity correction.
IntensityABCDMAXOriginal3518305227982610Corrected2748276528122762MINOriginal2563254124542334Corrected2632262426382554MeanOriginal3191285426532486Corrected2685265226732601STDOriginal254.40119.5071.7053.43Corrected15.508.7716.8215.86CVCVori0.07970.04190.02700.0215CVcor0.00580.00330.00630.0061ε0.0730.0790.2330.280The original intensity distributions of sites A, B, C, and D are [2563, 3518], [2541, 3052], [2454, 2798], and [2334, 2610], respectively. The STD values are 254.40, 119.50, 71.70, and 53.43. The intensity distribution of each site is successively [2632, 2748], [2624, 2765], [2638, 2812], and [2554, 2762] after correction. The STD value decreases to 15.50, 8.77, 16.82, and 15.86, which is better than the values before intensity correction.Meanwhile, the values ofCVori at A, B, C, and D are 0.0797, 0.0419, 0.0270, and 0.0215, respectively. The value of CVori decreases gradually with increasing distance between the sites and finally stabilizes at approximately 5 m. In combination with the distance and incident angle value in Table 5, it can be seen that the closer the site is to the wall, the larger the range of fluctuation of distance and incident angle in the same area will be, leading to the larger difference of original intensity value. Therefore, the values of CVori at sites A and B are slightly higher than those at sites C and D. After correction, the values of CVcor at the four sites are 0.0058, 0.0033, 0.0063, and 0.0061, respectively. It has been shown that the intensity variability after correction is smaller and the intensity correction effect is obvious. Among them, the CVcor values of sites A, C, and D are approximately 0.006, while the CVcor value of site B is only 0.0033, indicating that the corrected intensity consistency of site B is substantially better than that of other sites. The same conclusion can be obtained from Figure 19(f). The reason is that the original intensity distribution of site B is more concentrated than that of sites A, C, and D under the joint influence of distance and incident angle, as shown in Figure 20(b). Consequently, the intensity variability after correction at site B is lower than that at sites A, C, and D. Meanwhile, the ε values of sites A, B, C, and D are 0.073, 0.079, 0.233, and 0.280, respectively. This means that the intensity consistency of the four sites increased by 92.7%, 92.1%, 76.7%, and 72%, respectively. From Figure 19, we can intuitively see that the intensity correction effect of the short-distance sites A and B is better than that of the long-distance sites C and D.Similar to the research topic in this paper, Tan and Cheng [30] adopted a linear interpolation method to fit the relationship between incident angle versus intensity and distance versus intensity. In the range of distances and incident angles [1 m-29 m, 0°-80°], four reference targets with reflectance of 20%, 40%, 60%, and 80% were established for intensity correction models. Then, a total of 20 small regions with a size of approximately 15cm×15cm in the white lime wall surface were randomly sampled to verify the intensity correction effect over the whole range of distances and incident angles. After using the above four intensity correction models, the intensity variance-to-mean ratio ε was 0.26, 0.14, 0.19, and 0.21, meaning that the intensity consistency was improved by 74%, 86%, 81%, and 79%.Compared with Tan’s work, an intensity correction model based on a reference target with 50% reflectivity in the range of distance and incident angle [0.1-14.4 m, 0°-80°] is proposed. The intensity correction effect in this paper is better than Tan’s in eliminating the factors of the distance and incident angle. As mentioned above, the improvement rates of intensity consistency before and after correction for site A and site B are 92.7% and 92.1%, respectively. Obviously, regardless of what kind of reference target reflectance Tan’s intensity correction model is based on, the intensity consistency of the corrected white wall at sites A and B in this paper is higher than that of his research work. The reasons are as follows: first, to eliminate the distance factor, distance-intensity data with a distance value less than 1 m are considered in this paper. Therefore, compared with Tan’s method, the method presented in this paper performs better in short-distance intensity correction. Then, a new method of spherical neighborhood search fitting plane is proposed to accurately calculate the cosine of the incident angle. In particular, the relationship between neighborhood radius and distance is discussed in the process of plane fitting. The accuracy of incident angle measurement is improved by selecting an appropriate neighborhood radius. Finally, unlike Tan’s interpolation method, this paper adopts a piecewise distance polynomial and an incident angle cosine polynomial to fit distance-intensity and incident angle-intensity data. In conclusion, compared with Tan’s research, the intensity correction model established in this paper has a higher reliability and accuracy.


## 4.1. MLS 2D LiDAR Point Cloud Data Acquisition System and Coordinate System

### 4.1.1. MLS 2D LiDAR Point Cloud Data Acquisition System

To verify the validity of the intensity correction model for the distance and incident angle, a flat wall was selected as the experimental object, which was roughly considered a Lambert surface. To visually observe the intensity point cloud, the intensity value was converted into RGB in CloudCompare, a software point cloud development tool. The 3D point cloud RGB intensity map was also converted into a 2D pseudocolor map for display. The actual scene diagram of the experimental scanning wall is shown in Figure12. The MLS 2D LiDAR measurement system emitted laser pulses in all directions through internal rotating optical components to form a 2D fan-shaped scanning surface. The moving platform carried the laser pulses along the direction perpendicular to the scanning surface to realize the 3D measurement of the target surface. A moving slide mounted UTM-30LX 2D LiDAR was used to move in a straight line at a constant speed of 0.01 m/s in a direction parallel to the wall. The fan-shaped scanning surface was placed directly against the wall. Inside the red rectangular box was the wall study area, with a height of 1.5 m from top to bottom and a width of 1.2 m from left to right. The vertical distance from the mobile slide to the wall was 0.5 m, and the range of distance and incident angle of the wall study area was [0.52 m-1.55 m] and [0°-74°], respectively.Figure 12
The scene of scanning the wall surface: (a) the picture of the scanning scene; (b) 2D pseudocolor intensity map of the scanning scene.
(a)(b)


### 4.1.2. MLS 2D LiDAR Coordinate System

The starting point position of LiDAR is taken as the origin, and the right-handed coordinate system of the point cloud of the MLS 2D LiDAR measurement system is established, as shown in Figure13.Figure 13
Coordinate system of point clouds acquired by MLS 2D LiDAR.Thex-axis is the moving direction of LiDAR, the y-axis is the scanning direction of LiDAR, the z-axis is vertical to the ground, αi,j is the scanning angle of the ith measuring point in the jth frame, and R is the distance between the LiDAR laser receiver and the measuring point. The calculation of the coordinate system is as follows:
(16)xi,j=v∆t∙j,yi,j=−R∙cosαi,j,zi,j=R∙sinαi,j,where i is the in-frame number, j is the frame number of the measured points, andxi,j, yi,j, and zi,j represent the X, Y, and Z coordinates of the ith measuring point in the jth frame. v is the speed of the moving slide platform in the direction of the motion, and ∆t is the scan cycle of the 2D LiDAR.The geometric relationship between the moving slide with 2D LiDAR and the wall is shown in Figure14. The coordinate of LiDAR center point O at different scanning moments was set as xi,j,0,0; the 3D coordinate of the ith measurement point in frame jth was pi,j=(xi,j,yi,j,zi,j), incident laser vector li,j=0,yi,j,zi,j, and the normal vector of scanning point ni,j=n1,n2,n3.Figure 14
Moving slide and wall geometrical relationship diagram.


## 4.1.1. MLS 2D LiDAR Point Cloud Data Acquisition System

To verify the validity of the intensity correction model for the distance and incident angle, a flat wall was selected as the experimental object, which was roughly considered a Lambert surface. To visually observe the intensity point cloud, the intensity value was converted into RGB in CloudCompare, a software point cloud development tool. The 3D point cloud RGB intensity map was also converted into a 2D pseudocolor map for display. The actual scene diagram of the experimental scanning wall is shown in Figure12. The MLS 2D LiDAR measurement system emitted laser pulses in all directions through internal rotating optical components to form a 2D fan-shaped scanning surface. The moving platform carried the laser pulses along the direction perpendicular to the scanning surface to realize the 3D measurement of the target surface. A moving slide mounted UTM-30LX 2D LiDAR was used to move in a straight line at a constant speed of 0.01 m/s in a direction parallel to the wall. The fan-shaped scanning surface was placed directly against the wall. Inside the red rectangular box was the wall study area, with a height of 1.5 m from top to bottom and a width of 1.2 m from left to right. The vertical distance from the mobile slide to the wall was 0.5 m, and the range of distance and incident angle of the wall study area was [0.52 m-1.55 m] and [0°-74°], respectively.Figure 12
The scene of scanning the wall surface: (a) the picture of the scanning scene; (b) 2D pseudocolor intensity map of the scanning scene.
(a)(b)


## 4.1.2. MLS 2D LiDAR Coordinate System

The starting point position of LiDAR is taken as the origin, and the right-handed coordinate system of the point cloud of the MLS 2D LiDAR measurement system is established, as shown in Figure13.Figure 13
Coordinate system of point clouds acquired by MLS 2D LiDAR.Thex-axis is the moving direction of LiDAR, the y-axis is the scanning direction of LiDAR, the z-axis is vertical to the ground, αi,j is the scanning angle of the ith measuring point in the jth frame, and R is the distance between the LiDAR laser receiver and the measuring point. The calculation of the coordinate system is as follows:
(16)xi,j=v∆t∙j,yi,j=−R∙cosαi,j,zi,j=R∙sinαi,j,where i is the in-frame number, j is the frame number of the measured points, andxi,j, yi,j, and zi,j represent the X, Y, and Z coordinates of the ith measuring point in the jth frame. v is the speed of the moving slide platform in the direction of the motion, and ∆t is the scan cycle of the 2D LiDAR.The geometric relationship between the moving slide with 2D LiDAR and the wall is shown in Figure14. The coordinate of LiDAR center point O at different scanning moments was set as xi,j,0,0; the 3D coordinate of the ith measurement point in frame jth was pi,j=(xi,j,yi,j,zi,j), incident laser vector li,j=0,yi,j,zi,j, and the normal vector of scanning point ni,j=n1,n2,n3.Figure 14
Moving slide and wall geometrical relationship diagram.


## 4.2. ObtainingR and cosθ

Generally, the density of point cloud data obtained by 3D LiDAR is relatively high. Usually, to reduce data redundancy, a support vector machine (SVM) is used to integrate different feature weights of the point cloud into the classifier for training [35–38]. In addition, to improve the efficiency of neighborhood searching, the k-D tree algorithm is usually used for downsampling 3D LiDAR point clouds [39]. Compared with 3D LiDAR point clouds, the data structure of 2D LiDAR point clouds is relatively simple. It only needs to be indexed by the grid in the established coordinate system, and the k-nearest neighbor algorithm [40] can be used to establish the point cloud neighborhood set. Referring to our previous research [41], a 3D spherical neighborhood Si,j is defined as a set of the nearest adjacent points within a sphere centered at the measurement point pi,j with a radius of δ. According to the idea of the least square method, the plane is fitted in the neighborhood set Si,j, and the normal vector ni,j corresponding to each measuring point pi,j is obtained. Then, cosθi,j of the LiDAR laser receiver to each measuring point can be calculated according to Formula (17). Obviously, Ri,j can be obtained directly.
(17)cosθi,j=li,j∙ni,jli,jni,j.Sinceli,j does not change, the accuracy of cosθi,j depends on ni,j. In general, the smaller the value of δ, the smaller the fitting plane, and the more accurate the value of ni,j. However, if the value of δ is too small, the point cloud data in the two adjacent frames cannot be included in the neighborhood; then, the plane cannot be fitted either. In the MLS 2D LiDAR measurement system, the value of δ is closely related to distance Ri,j and the resolution of αi,j. The resolution of the moving direction and the scanning direction of LiDAR is defined as ∆x and ∆s, respectively:
(18)∆x=v∆t,∆s=Ri,jsin∆α.∆α is the 2D LiDAR angular resolution, and its value is constant. The larger the Ri,j is, the greater the value of ∆s. Therefore, if the value of δ is less than ∆s, in the LiDAR scanning direction, all of the in-frame data points adjacent to the measurement point pi,j cannot be included in Si,j. Then, ni,j cannot be obtained by plane fitting. According to Formula (18), the necessary conditions for the measurement point pi,j to have neighborhood points in the LiDAR scanning direction can be deduced:
(19)δ>Ri,jmaxsin∆α,Ri,j≤Ri,jmax.As for UTM-30LX, the value of∆α is 0.25°. In the wall intensity correction experiment, the maximum distance of LiDAR from the wall is not more than 5.5 m. According to Formula (19), Ri,jmaxsin∆α is 0.024. Considering that the range resolution of the LiDAR sensor within 10 m is ±30 mm, the value of δ is set to 0.03 m.


## 4.3. Correction of the Wall Point Cloud Intensity Data

According to the calculation method of the incident angle and distance, the incident angleθ and the distance value R of each measurement point in the wall point cloud intensity data were calculated. The study of Tan and Cheng [42] shows that since the impact of the incident angle and distance on intensity is independent, the sequence of the incident angle correction and distance correction does not affect the correction results. According to the established intensity correction model, the reference incident angle and reference distance are set to be θ0=0° and R0=1.2 m, respectively. The incident angle and distance are corrected for the wall point cloud intensity data. Figure 15 shows the intensity 2D pseudocolor map and intensity distribution histogram of the red rectangular wall area described in Figure 12 before and after intensity correction.Figure 15
The intensity 2D pseudocolor map and intensity distribution histogram of the red rectangular wall area described in Figure12 before and after intensity correction: (a) original intensity pseudocolor chart; (b) intensity pseudocolor chart after the incident angle correction; (c) intensity pseudocolor chart of distance correction after the incident angle correction; (d) histogram of original intensity distribution; (e) histogram of intensity distribution after incident angle correction; (f) intensity distribution histogram of distance correction after the incident angle correction.
(a)(b)(c)(d)(e)(f)As seen from the RGB diagram of the original intensity in Figure15(a), the intensity value at different distances and incident angles is considerably different before correction under the same target reflectivity. The closer the distance or the smaller the incident angle, the higher the intensity value will be. In contrast, the farther the distance or the larger the incident angle, the lower the intensity value will be. The RGB of the intensity after the correction of the incident angle in Figure 15(b) shows that the uniformity of the wall point cloud intensity data has been improved. The RGB diagram of the distance correction intensity after the incident angle correction is shown in Figure 15(c). The influence of the distance and incident angle on intensity is eliminated, and the corrected intensity value is consistent and similar. The histogram of the intensity distribution in Figure 15(d) shows that the original intensity data of the wall point cloud under the same target reflectivity have no obvious distribution pattern. From Figure 15(f), it is clear that after the correction of the incident angle and distance, the intensity value of the wall is concentrated and presents a Gaussian distribution.


## 4.4. Using an Evaluation Indexε to Verify the Intensity Correction Model

### 4.4.1. Single-Site Multiregion Verification Experiment

In the wall point cloud intensity data obtained from the above experiment, 5 different areas of20cm×20cm were randomly selected, as shown in Figure 16. According to the established intensity correction model, the intensity data in different areas of S1-S5 were also corrected with reference distance R0=1.2 m and incident angle θ0=0°. In addition, the intensity value before and after correction was statistically analyzed. The intensity box chart is shown in Figure 17. The distribution of intensity maximum value (MAX), minimum value (MIN), mean, STD, CV, and the evaluation index ε before and after correction is shown in Table 3.Figure 16
Different areas of S1-S5 randomly selected on the wall.Figure 17
Intensity box diagram of different areas on the same wall before and after correction: (a) intensity box diagram of area S1-S5 before correction; (b) intensity box diagram of area S1-S5 after correction.
(a)(b)Table 3
The distribution of intensity MAX, MIN, Mean, STD, CV, and the evaluation indexε at S1-S5 different regions before and after correction.
IntensityRegion S1Region S2Region S3Region S4Region S5MAXOriginal26812668275028322896Corrected30423015306130063025MINOriginal25982346265727102775Corrected30192990303729813002MeanOriginal26122557271727852846Corrected30313003304929923013STDOriginal16.716.915.918.120.44Corrected5.84.35.65.38.3CVCVori0.00640.00660.00580.00650.0072CVcor0.00190.00140.00180.00180.0028ε0.2970.2120.3100.2770.389As shown in Figure17(a), the intensity value of different regions is substantially different and fluctuates in a large range within each region before intensity correction. From Figure 17(b), it is evident that the difference in the intensity value between different regions decreases, and the fluctuation range of the intensity within the region also becomes considerably smaller after the correction of the incident angle and distance. It is clear that the consistency of intensity between different areas and within different areas has been substantially improved by comparing the box graphs of intensity data from different areas of walls S1-S5 before and after intensity correction under the same reflectivity of the target.Table3 also shows a conclusion similar to that in Figure 17. Due to the small area of the S1-S5 region, the difference between the internal distance and incident angle of each region is not large. The STD values of the original intensity data are all below 21. In addition, due to the large difference in distance and incident angle value between different regions, their STD values are also relatively large, with a minimum value of 15.9 and a maximum value of 20.44. After intensity correction, the STD values of the intensity data in each region are all below 10, and the difference between them is small, indicating that the uniformity of the intensity distribution in each region has been substantially improved. The ε value fluctuates approximately 0.3 in S1-S5 different regions, indicating that the intensity consistency of the five same area regions with the range of distance and incident angle [0.52 m-1.55 m, 0°-74°] has been massively improved by 70% after applying the proposed correction method.Similar to the single-site multiregion verification experiment in this paper, Tan and Cheng [29] utilized Faro Focus3D TLS to obtain intensity point cloud data of ordinary white wall surfaces. In the range of distance and incident angle [6.70 m-14.76 m, 0°-80°], intensity correction experiments were carried out for the A-F region with the same area. The ε value corresponding to the intensity correction of the white wall obtained by the Tan method and the method proposed in this paper is shown in Table 4 below.Table 4
Comparison ofε values of the white wall intensity corrected by the Tan method and the method proposed in this paper.
MethodεRegion S1/ARegion S2/BRegion S3/CRegion S4/DRegion S5/ERegion S6/FThe proposed method0.2970.2120.3100.2770.389—Tan method0.6070.6560.5230.7871.001↑1.139↑Theε value of the Tan method has a large fluctuation range, approximately 0.6, while the ε value of the method proposed in this paper has a small fluctuation range, approximately 0.3. In the Tan experiment, especially in regions E and F, the ε value is greater than 1, indicating that the consistency of the corrected intensity is not as good as the original intensity, and there is an overfitting phenomenon. Moreover, the Tan method only conducts intensity correction research at distances where the intensity value is relatively stable, ignoring the LiDAR short-distance effect, and does not correct the intensity at distances less than 1 m. Overall, we can say that the proposed method in this paper is better than the Tan method and provides a higher accuracy of intensity correction.


### 4.4.2. Multisite Scanning of the Whole Wall for the Verification Experiment

In the previous single-site experiment, the area of the S1-S5 region was small, and the incident angle and distance value of the point clouds in each region were not substantially different. The STD of the original intensity was also relatively small and could not effectively reflect the distribution characteristics of the original intensity data of the whole wall. A multisite experiment was devoted to the intensity correction of the same red rectangular area on the wall, as shown in Figure12, under multiple sites. Four sites were set up: A, B, C, and D. The vertical distances between the sliding platform and the wall were 1.5 m, 2.5 m, 3.5 m, and 4.5 m, respectively. The rectangular area of the wall was scanned to obtain the point cloud intensity data, as shown in Figure 18. According to the established intensity correction model, the intensity data at different sites were corrected at reference distance R0=1.2 m and incident angle θ0=0°. Table 5 lists the distance and incident angle range values of different sites A, B, C, and D. Figure 19 shows the intensity pseudocolor map of the red rectangular area on the wall described in Figure 12 before and after intensity correction at sites A, B, C and D. The histogram of the intensity distribution before and after intensity correction is shown in Figure 20.Figure 18
Mobile laser scan of 4 sites A, B, C, and D.Table 5
Distance and incident angle range values of different sites A, B, C, and D.
ABCDR (m)MAX3.203.744.535.34MIN1.522.533.524.53θ (°)MAX62483932MIN0000cosθMAX1111MIN0.46950.66910.77710.8480Figure 19
The intensity 2D pseudocolor map of the red rectangular area on the wall described in Figure12 before and after intensity correction at A, B, C, and D 4 sites, respectively. (a–d) Are the original intensity 2D pseudocolor map of sites A, B, C, and D. (e–h) Are the corresponding corrected intensity 2D pseudocolor map of sites A, B, C, and D, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 20
Histogram of intensity distribution before and after correction at A, B, C, and D different sites, respectively. (a–d) Are the original intensity distribution histogram of sites A, B, C, and D. (e–h) Are the corrected intensity distribution histograms of sites A, B, C, and D, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)It is evident from Figures19(a)–19(d) that as the distance increases, the density of point clouds on walls with the same area decreases, while the intensity value shows great differences within the whole wall of each site. It can be clearly seen from Figures 19(e)–19(h) that the variation of wall intensity of the four sites has been considerably reduced after distance and angle correction. By comparing the original intensity distribution in Figures 20(a)–20(d), it was found that quantitatively, the original intensity of each site decreases with an increasing distance and that the degree of dispersion of the intensity distribution is obviously different. As seen from Figures 20(e)–20(h), after intensity correction, the degree of discretization of the intensity value at sites A, B, C, and D is massively reduced, and the intensity value distribution is centralized and approximates a Gaussian distribution. Table 6 lists the distribution of MAX, MIN, Mean, STD, CV of the intensity, and the evaluation index ε at different sites A, B, C, and D before and after intensity correction.Table 6
The distribution of MAX, MIN, Mean, STD, CV of the intensity, and the evaluation indexε at different sites A, B, C, and D before and after intensity correction.
IntensityABCDMAXOriginal3518305227982610Corrected2748276528122762MINOriginal2563254124542334Corrected2632262426382554MeanOriginal3191285426532486Corrected2685265226732601STDOriginal254.40119.5071.7053.43Corrected15.508.7716.8215.86CVCVori0.07970.04190.02700.0215CVcor0.00580.00330.00630.0061ε0.0730.0790.2330.280The original intensity distributions of sites A, B, C, and D are [2563, 3518], [2541, 3052], [2454, 2798], and [2334, 2610], respectively. The STD values are 254.40, 119.50, 71.70, and 53.43. The intensity distribution of each site is successively [2632, 2748], [2624, 2765], [2638, 2812], and [2554, 2762] after correction. The STD value decreases to 15.50, 8.77, 16.82, and 15.86, which is better than the values before intensity correction.Meanwhile, the values ofCVori at A, B, C, and D are 0.0797, 0.0419, 0.0270, and 0.0215, respectively. The value of CVori decreases gradually with increasing distance between the sites and finally stabilizes at approximately 5 m. In combination with the distance and incident angle value in Table 5, it can be seen that the closer the site is to the wall, the larger the range of fluctuation of distance and incident angle in the same area will be, leading to the larger difference of original intensity value. Therefore, the values of CVori at sites A and B are slightly higher than those at sites C and D. After correction, the values of CVcor at the four sites are 0.0058, 0.0033, 0.0063, and 0.0061, respectively. It has been shown that the intensity variability after correction is smaller and the intensity correction effect is obvious. Among them, the CVcor values of sites A, C, and D are approximately 0.006, while the CVcor value of site B is only 0.0033, indicating that the corrected intensity consistency of site B is substantially better than that of other sites. The same conclusion can be obtained from Figure 19(f). The reason is that the original intensity distribution of site B is more concentrated than that of sites A, C, and D under the joint influence of distance and incident angle, as shown in Figure 20(b). Consequently, the intensity variability after correction at site B is lower than that at sites A, C, and D. Meanwhile, the ε values of sites A, B, C, and D are 0.073, 0.079, 0.233, and 0.280, respectively. This means that the intensity consistency of the four sites increased by 92.7%, 92.1%, 76.7%, and 72%, respectively. From Figure 19, we can intuitively see that the intensity correction effect of the short-distance sites A and B is better than that of the long-distance sites C and D.Similar to the research topic in this paper, Tan and Cheng [30] adopted a linear interpolation method to fit the relationship between incident angle versus intensity and distance versus intensity. In the range of distances and incident angles [1 m-29 m, 0°-80°], four reference targets with reflectance of 20%, 40%, 60%, and 80% were established for intensity correction models. Then, a total of 20 small regions with a size of approximately 15cm×15cm in the white lime wall surface were randomly sampled to verify the intensity correction effect over the whole range of distances and incident angles. After using the above four intensity correction models, the intensity variance-to-mean ratio ε was 0.26, 0.14, 0.19, and 0.21, meaning that the intensity consistency was improved by 74%, 86%, 81%, and 79%.Compared with Tan’s work, an intensity correction model based on a reference target with 50% reflectivity in the range of distance and incident angle [0.1-14.4 m, 0°-80°] is proposed. The intensity correction effect in this paper is better than Tan’s in eliminating the factors of the distance and incident angle. As mentioned above, the improvement rates of intensity consistency before and after correction for site A and site B are 92.7% and 92.1%, respectively. Obviously, regardless of what kind of reference target reflectance Tan’s intensity correction model is based on, the intensity consistency of the corrected white wall at sites A and B in this paper is higher than that of his research work. The reasons are as follows: first, to eliminate the distance factor, distance-intensity data with a distance value less than 1 m are considered in this paper. Therefore, compared with Tan’s method, the method presented in this paper performs better in short-distance intensity correction. Then, a new method of spherical neighborhood search fitting plane is proposed to accurately calculate the cosine of the incident angle. In particular, the relationship between neighborhood radius and distance is discussed in the process of plane fitting. The accuracy of incident angle measurement is improved by selecting an appropriate neighborhood radius. Finally, unlike Tan’s interpolation method, this paper adopts a piecewise distance polynomial and an incident angle cosine polynomial to fit distance-intensity and incident angle-intensity data. In conclusion, compared with Tan’s research, the intensity correction model established in this paper has a higher reliability and accuracy.


## 4.4.1. Single-Site Multiregion Verification Experiment

In the wall point cloud intensity data obtained from the above experiment, 5 different areas of20cm×20cm were randomly selected, as shown in Figure 16. According to the established intensity correction model, the intensity data in different areas of S1-S5 were also corrected with reference distance R0=1.2 m and incident angle θ0=0°. In addition, the intensity value before and after correction was statistically analyzed. The intensity box chart is shown in Figure 17. The distribution of intensity maximum value (MAX), minimum value (MIN), mean, STD, CV, and the evaluation index ε before and after correction is shown in Table 3.Figure 16
Different areas of S1-S5 randomly selected on the wall.Figure 17
Intensity box diagram of different areas on the same wall before and after correction: (a) intensity box diagram of area S1-S5 before correction; (b) intensity box diagram of area S1-S5 after correction.
(a)(b)Table 3
The distribution of intensity MAX, MIN, Mean, STD, CV, and the evaluation indexε at S1-S5 different regions before and after correction.
IntensityRegion S1Region S2Region S3Region S4Region S5MAXOriginal26812668275028322896Corrected30423015306130063025MINOriginal25982346265727102775Corrected30192990303729813002MeanOriginal26122557271727852846Corrected30313003304929923013STDOriginal16.716.915.918.120.44Corrected5.84.35.65.38.3CVCVori0.00640.00660.00580.00650.0072CVcor0.00190.00140.00180.00180.0028ε0.2970.2120.3100.2770.389As shown in Figure17(a), the intensity value of different regions is substantially different and fluctuates in a large range within each region before intensity correction. From Figure 17(b), it is evident that the difference in the intensity value between different regions decreases, and the fluctuation range of the intensity within the region also becomes considerably smaller after the correction of the incident angle and distance. It is clear that the consistency of intensity between different areas and within different areas has been substantially improved by comparing the box graphs of intensity data from different areas of walls S1-S5 before and after intensity correction under the same reflectivity of the target.Table3 also shows a conclusion similar to that in Figure 17. Due to the small area of the S1-S5 region, the difference between the internal distance and incident angle of each region is not large. The STD values of the original intensity data are all below 21. In addition, due to the large difference in distance and incident angle value between different regions, their STD values are also relatively large, with a minimum value of 15.9 and a maximum value of 20.44. After intensity correction, the STD values of the intensity data in each region are all below 10, and the difference between them is small, indicating that the uniformity of the intensity distribution in each region has been substantially improved. The ε value fluctuates approximately 0.3 in S1-S5 different regions, indicating that the intensity consistency of the five same area regions with the range of distance and incident angle [0.52 m-1.55 m, 0°-74°] has been massively improved by 70% after applying the proposed correction method.Similar to the single-site multiregion verification experiment in this paper, Tan and Cheng [29] utilized Faro Focus3D TLS to obtain intensity point cloud data of ordinary white wall surfaces. In the range of distance and incident angle [6.70 m-14.76 m, 0°-80°], intensity correction experiments were carried out for the A-F region with the same area. The ε value corresponding to the intensity correction of the white wall obtained by the Tan method and the method proposed in this paper is shown in Table 4 below.Table 4
Comparison ofε values of the white wall intensity corrected by the Tan method and the method proposed in this paper.
MethodεRegion S1/ARegion S2/BRegion S3/CRegion S4/DRegion S5/ERegion S6/FThe proposed method0.2970.2120.3100.2770.389—Tan method0.6070.6560.5230.7871.001↑1.139↑Theε value of the Tan method has a large fluctuation range, approximately 0.6, while the ε value of the method proposed in this paper has a small fluctuation range, approximately 0.3. In the Tan experiment, especially in regions E and F, the ε value is greater than 1, indicating that the consistency of the corrected intensity is not as good as the original intensity, and there is an overfitting phenomenon. Moreover, the Tan method only conducts intensity correction research at distances where the intensity value is relatively stable, ignoring the LiDAR short-distance effect, and does not correct the intensity at distances less than 1 m. Overall, we can say that the proposed method in this paper is better than the Tan method and provides a higher accuracy of intensity correction.


## 4.4.2. Multisite Scanning of the Whole Wall for the Verification Experiment

In the previous single-site experiment, the area of the S1-S5 region was small, and the incident angle and distance value of the point clouds in each region were not substantially different. The STD of the original intensity was also relatively small and could not effectively reflect the distribution characteristics of the original intensity data of the whole wall. A multisite experiment was devoted to the intensity correction of the same red rectangular area on the wall, as shown in Figure12, under multiple sites. Four sites were set up: A, B, C, and D. The vertical distances between the sliding platform and the wall were 1.5 m, 2.5 m, 3.5 m, and 4.5 m, respectively. The rectangular area of the wall was scanned to obtain the point cloud intensity data, as shown in Figure 18. According to the established intensity correction model, the intensity data at different sites were corrected at reference distance R0=1.2 m and incident angle θ0=0°. Table 5 lists the distance and incident angle range values of different sites A, B, C, and D. Figure 19 shows the intensity pseudocolor map of the red rectangular area on the wall described in Figure 12 before and after intensity correction at sites A, B, C and D. The histogram of the intensity distribution before and after intensity correction is shown in Figure 20.Figure 18
Mobile laser scan of 4 sites A, B, C, and D.Table 5
Distance and incident angle range values of different sites A, B, C, and D.
ABCDR (m)MAX3.203.744.535.34MIN1.522.533.524.53θ (°)MAX62483932MIN0000cosθMAX1111MIN0.46950.66910.77710.8480Figure 19
The intensity 2D pseudocolor map of the red rectangular area on the wall described in Figure12 before and after intensity correction at A, B, C, and D 4 sites, respectively. (a–d) Are the original intensity 2D pseudocolor map of sites A, B, C, and D. (e–h) Are the corresponding corrected intensity 2D pseudocolor map of sites A, B, C, and D, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 20
Histogram of intensity distribution before and after correction at A, B, C, and D different sites, respectively. (a–d) Are the original intensity distribution histogram of sites A, B, C, and D. (e–h) Are the corrected intensity distribution histograms of sites A, B, C, and D, respectively.
(a)(b)(c)(d)(e)(f)(g)(h)It is evident from Figures19(a)–19(d) that as the distance increases, the density of point clouds on walls with the same area decreases, while the intensity value shows great differences within the whole wall of each site. It can be clearly seen from Figures 19(e)–19(h) that the variation of wall intensity of the four sites has been considerably reduced after distance and angle correction. By comparing the original intensity distribution in Figures 20(a)–20(d), it was found that quantitatively, the original intensity of each site decreases with an increasing distance and that the degree of dispersion of the intensity distribution is obviously different. As seen from Figures 20(e)–20(h), after intensity correction, the degree of discretization of the intensity value at sites A, B, C, and D is massively reduced, and the intensity value distribution is centralized and approximates a Gaussian distribution. Table 6 lists the distribution of MAX, MIN, Mean, STD, CV of the intensity, and the evaluation index ε at different sites A, B, C, and D before and after intensity correction.Table 6
The distribution of MAX, MIN, Mean, STD, CV of the intensity, and the evaluation indexε at different sites A, B, C, and D before and after intensity correction.
IntensityABCDMAXOriginal3518305227982610Corrected2748276528122762MINOriginal2563254124542334Corrected2632262426382554MeanOriginal3191285426532486Corrected2685265226732601STDOriginal254.40119.5071.7053.43Corrected15.508.7716.8215.86CVCVori0.07970.04190.02700.0215CVcor0.00580.00330.00630.0061ε0.0730.0790.2330.280The original intensity distributions of sites A, B, C, and D are [2563, 3518], [2541, 3052], [2454, 2798], and [2334, 2610], respectively. The STD values are 254.40, 119.50, 71.70, and 53.43. The intensity distribution of each site is successively [2632, 2748], [2624, 2765], [2638, 2812], and [2554, 2762] after correction. The STD value decreases to 15.50, 8.77, 16.82, and 15.86, which is better than the values before intensity correction.Meanwhile, the values ofCVori at A, B, C, and D are 0.0797, 0.0419, 0.0270, and 0.0215, respectively. The value of CVori decreases gradually with increasing distance between the sites and finally stabilizes at approximately 5 m. In combination with the distance and incident angle value in Table 5, it can be seen that the closer the site is to the wall, the larger the range of fluctuation of distance and incident angle in the same area will be, leading to the larger difference of original intensity value. Therefore, the values of CVori at sites A and B are slightly higher than those at sites C and D. After correction, the values of CVcor at the four sites are 0.0058, 0.0033, 0.0063, and 0.0061, respectively. It has been shown that the intensity variability after correction is smaller and the intensity correction effect is obvious. Among them, the CVcor values of sites A, C, and D are approximately 0.006, while the CVcor value of site B is only 0.0033, indicating that the corrected intensity consistency of site B is substantially better than that of other sites. The same conclusion can be obtained from Figure 19(f). The reason is that the original intensity distribution of site B is more concentrated than that of sites A, C, and D under the joint influence of distance and incident angle, as shown in Figure 20(b). Consequently, the intensity variability after correction at site B is lower than that at sites A, C, and D. Meanwhile, the ε values of sites A, B, C, and D are 0.073, 0.079, 0.233, and 0.280, respectively. This means that the intensity consistency of the four sites increased by 92.7%, 92.1%, 76.7%, and 72%, respectively. From Figure 19, we can intuitively see that the intensity correction effect of the short-distance sites A and B is better than that of the long-distance sites C and D.Similar to the research topic in this paper, Tan and Cheng [30] adopted a linear interpolation method to fit the relationship between incident angle versus intensity and distance versus intensity. In the range of distances and incident angles [1 m-29 m, 0°-80°], four reference targets with reflectance of 20%, 40%, 60%, and 80% were established for intensity correction models. Then, a total of 20 small regions with a size of approximately 15cm×15cm in the white lime wall surface were randomly sampled to verify the intensity correction effect over the whole range of distances and incident angles. After using the above four intensity correction models, the intensity variance-to-mean ratio ε was 0.26, 0.14, 0.19, and 0.21, meaning that the intensity consistency was improved by 74%, 86%, 81%, and 79%.Compared with Tan’s work, an intensity correction model based on a reference target with 50% reflectivity in the range of distance and incident angle [0.1-14.4 m, 0°-80°] is proposed. The intensity correction effect in this paper is better than Tan’s in eliminating the factors of the distance and incident angle. As mentioned above, the improvement rates of intensity consistency before and after correction for site A and site B are 92.7% and 92.1%, respectively. Obviously, regardless of what kind of reference target reflectance Tan’s intensity correction model is based on, the intensity consistency of the corrected white wall at sites A and B in this paper is higher than that of his research work. The reasons are as follows: first, to eliminate the distance factor, distance-intensity data with a distance value less than 1 m are considered in this paper. Therefore, compared with Tan’s method, the method presented in this paper performs better in short-distance intensity correction. Then, a new method of spherical neighborhood search fitting plane is proposed to accurately calculate the cosine of the incident angle. In particular, the relationship between neighborhood radius and distance is discussed in the process of plane fitting. The accuracy of incident angle measurement is improved by selecting an appropriate neighborhood radius. Finally, unlike Tan’s interpolation method, this paper adopts a piecewise distance polynomial and an incident angle cosine polynomial to fit distance-intensity and incident angle-intensity data. In conclusion, compared with Tan’s research, the intensity correction model established in this paper has a higher reliability and accuracy.


## 5. Conclusion

In this paper, a new point cloud intensity correction method for 2D MLS based on theoretical derivation and empirical correction is proposed to solve the problem that intensity information cannot be directly used for target recognition. Based on the diffuse reflection Lambert body of the same target reflectance, the intensity correction model of the piecewise distance polynomial and incident angle cosine polynomial is adopted, and the model parameters are calculated by experiments. The effectiveness of the intensity correction method is verified by single-site and multisite experiments on a white wall using MLS 2D LiDAR. The experimental results show that the intensity consistency is substantially improved by 70% to 92.7% after correction within the range of the distance and incident angle [0.52 m-5.34 m, 0°-74°]. Compared with the latest research, the intensity correction model proposed in this paper has a higher fitting accuracy and can effectively eliminate the MLS intensity deviation caused by distance and incident angle.However, the intensity correction model is only suitable for the specific UTM-30LX 2D LiDAR and applicable to targets similar to the standard Lambert surface. The intensity deviation still exists after the correction, indicating that further research is needed, especially to reduce the model error. In addition, it is found in the incident angle correction experiment that the correction of the incident angle is not thorough in place with a large angle, which requires further study of a more rigorous correction model. Furthermore, considering the long running time of LiDAR, the influence of internal temperature rise on intensity can further improve the accuracy of model correction, which is also a research direction for the future.



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*Source: 3707985-2022-01-20.xml*