A TLS single-site cloud data density rigorous calculation model and a correction method thereof

Abstract

The application discloses a TLS single-station cloud data density rigorous calculation model and a correction method thereof. The model comprises the following steps: performing line-by-line scanning on a target by using TLS; establishing a three-dimensional space coordinate system with the center of the TLS as the origin; utilizing a vertical scanning plane where a scanning line of a target point is located, a plane formed by left and right adjacent scanning lines of the target point, and a micro plane where the target point is located to perform intersection respectively, obtaining direction vectors S2 and S3 of two intersecting straight lines, combining a scanning line direction vector S1 of the target point, obtaining angles θ1 and θ2 between the scanning line of the target point and the two intersecting straight lines respectively, solving horizontal and vertical point distances △h and △v by using a triangle sine theorem, obtaining a peripheral parallelogram area of the target point, solving a ratio of a circular neighborhood area of the target point to the peripheral parallelogram area of the target point, and obtaining a point cloud density. O P f P g P e m l P OP P m l s s ​​​​​​​​​​​​​​​​

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of point cloud density calculation, and particularly relates to a TLS single-station point cloud data density rigorous calculation model and a correction method thereof. BACKGROUND

[0002] A terrestrial laser scanning system (TLS) is an advanced active remote sensing technology that can obtain high-precision and high-density laser point cloud data of a target and its surrounding scene without contacting the target. Compared with traditional measurement technologies such as a total station, RTK, aerial photogrammetry, satellite observation, etc., the TLS can obtain millimeter-level resolution and high-precision target outer contour point cloud data, and has great advantages in measurement efficiency and work cost, and has a wide range of applications in topographic surveying, forestry investigation, agricultural investigation, disaster monitoring, etc.

[0003] The scanning data obtained by the TLS contains geometric information and physical information, wherein the geometric information refers to the three-dimensional space coordinates of the point cloud, and the physical information refers to the color information and laser intensity value of the point cloud. Most of the current research and application are based on the geometric information of the point cloud data, that is, the normal vector, curvature, rate of change of curvature, density, point-line-surface features and other geometric features of the point cloud are calculated by using the obtained three-dimensional space coordinates of the point cloud, and then the geometric features are used for target recognition, classification and extraction, etc. The point cloud density refers to the number of point clouds in a certain spatial range around a target point, and indicates the geometric features and spatial distribution of the target. As one of the most basic and important geometric features of the point cloud, the density can be used for fine classification of the point cloud and extraction of target features, and has been widely applied in researches such as separation of plant stems and leaves, extraction of building contours, etc. However, the point cloud density usually changes in different scenes, and decreases with the increase of the scanning distance. Compared with the airborne laser radar and the spaceborne laser radar, the TLS is closer to the scanning target, and the distance changes greatly, so the change is more obvious for the TLS scanning data. Therefore, the change of the point cloud density is an important factor that needs to be considered in the extraction of geometric features.

[0004] To solve the problem of varying point cloud density, many scholars have taken different strategies in their research. Chen et al set adaptive density search range according to the relationship between point cloud density and scanning distance to ensure that the search neighborhood and the point cloud density maintain similar scales, and achieve the extraction of tree trunk point cloud in forest scene. On this basis, Chen et al estimated the point spacing by scanning distance and angular resolution, and determined the density search neighborhood according to the point spacing, and used the polar coordinate grid instead of the commonly used grid to adapt to the density variation and improve the extraction accuracy of distant buildings. Che et al calculated the reference point density according to the point cloud density and scanning angle resolution, which was used for ground point filtering of TLS scanning data. Cheng et al projected the point cloud to the horizontal grid, counted the number of point cloud in each grid and took it as the projected point density (Density of Projected Points, DoPP), and then calculated the adaptive threshold of DoPP according to the distance between the farthest building and the scanner, the minimum building height and the scanning angle resolution to complete the building outline extraction. The above research can utilize the varying point cloud density to a certain extent for feature extraction, but these methods can only be used for specific features, and lack of applicability for complete scenes. At the same time, most of the algorithms need to provide more prior knowledge, and the intelligence of the algorithm needs to be improved.

[0005] Due to the influence of occlusion effect, multiple stations need to be set up to obtain complete data of the whole scene when using TLS scanning, but at the same time, it will also cause a large amount of overlap in the overall data, and the point cloud density in the overlapping area will increase significantly, which will greatly interfere with the feature extraction using point cloud density, while single station TLS scanning data will not be affected. Therefore, it is a better strategy to study the feature extraction of single station TLS point cloud density and then register multiple station data to obtain the features of the whole scene. At present, some researches have tried to establish a theoretical density calculation model of TLS single station point cloud, by eliminating the objective factors (such as distance, incident angle, etc.) in the model, to get the corrected point cloud density only related to the geometric features (such as size) of the target, and finally conduct feature classification on this basis. This method does not need complex parameter setting, and is not specific to a particular feature, so it has strong practicality. Tan et al established a theoretical model of point cloud density and distance, and then eliminated the influence of distance on point cloud density, and used the corrected point cloud density to separate forest stems and leaves. The point cloud density calculation model established in this research is relatively simple, only applicable to simple scenes (such as horizontal plane, vertical plane, etc.), and does not consider the existence of various inclinations of the target itself, so its applicability to complete and complex scenes is poor. SUMMARY

[0006] The purpose of the present application is to provide a TLS single-site cloud data density rigorous calculation model and a correction method thereof according to the deficiencies of the prior art, considering factors such as distance, horizontal and vertical angle resolution, arbitrary spatial distribution state of the target, based on the scanning principle of the scanner and the spatial distribution and internal geometric correlation between the scanning points, lines and surfaces, and then using the angle between the scanning line and the target in the horizontal and vertical directions to represent the spatial distribution state of the target, the rigorous calculation model of the single-point density in the TLS single-site cloud is derived mathematically, and the model is suitable for the theoretical density calculation of any shape and spatial distribution point in any scene obtained by any TLS. At the same time, by defining the reference distance and angle, the influence of the objective elements in the model is eliminated, based on the density rigorous calculation model, a density correction method for the TLS single-site cloud is proposed, and the corrected point cloud density is only related to the geometric size / feature of the target, which can be used for researches such as feature extraction, classification and identification.

[0007] The purpose of the present application is achieved by the following technical solutions:

[0008] A TLS single-site cloud data density rigorous calculation model, characterized in that the model comprises: performing line-by-line scanning on a target by TLS; establishing a three-dimensional coordinate system with the center O of the TLS as the origin, and using the vertical scanning plane f in which the scanning line of the target point P is located and the plane g formed by the left and right adjacent scanning lines of the target point P to intersect with the micro plane e in which the target point P is located, to obtain the direction vectors S2 and S3 of the two intersecting straight lines m and l, and combine the direction vector S1 of the scanning line OP of the target point P to obtain the angles θ1 and θ2 between the scanning line of the target point P and the two intersecting straight lines m and l, and solve the horizontal and vertical point spacings Δh and Δv by the trigonometric theorem, to obtain the area s of the target point peripheral parallelogram, and obtain the point cloud density by solving the ratio of the target point circular neighborhood area to the area s of the target point peripheral parallelogram.

[0009] The line-by-line scanning is the vertical scanning from bottom to top or from top to bottom within the vertical scanning angle range of the scanning line, and the discrete point measurement is performed with the vertical angle resolution β as the step size, and then the vertical scanning is performed again after rotating a horizontal angle resolution α, and the scanning is ended after completing the specified horizontal scanning angle or a circular scanning.

[0010] Let P={p i ,i=1,2,..,n} be a point cloud set, and the point cloud set in the spherical neighborhood range with a radius r around the target point p i is expressed as:

[0011] R(p i ,r)={p j / d(p i ,p j )≤r,p j∈P and j≠i} (1);

[0012] wherein: d(p i ,p j ) is the geometric distance from the target point p i to p j ;

[0013] The total number of points in the point cloud set R(p i ,r) is the density D(p i ,r) of the target point p i :

[0014]

[0015] The left and right adjacent point distances of the target point p i are PN and PM, respectively, and the front and back or up and down adjacent point distances are PK and PL, respectively, so that the two side lengths Δh and Δv of the target point peripheral parallelogram are:

[0016]

[0017] Let the coordinates of the target point P or p i be (x i ,y i ,z i ), and the coordinates of the TLS center O be (x0,y0,z0), so that the direction vector S1 of the straight line OP is:

[0018] S1=(x i -x0,y i -y0,z i -z0) (4);

[0019] Assume that the plane equations of the planes e, f, and g are:

[0020]

[0021] Then the straight line equation of the straight line m intersecting the plane e and the plane f is:

[0022]

[0023] The direction vector S2 of the straight line m is:

[0024]

[0025] wherein: i=(1,0,0), j=(0,1,0), and k=(0,0,1), so that S2 is:

[0026] S2=(b1c2-b2c1,-a1c2+a2c1,a1b2-a2b1) (8);

[0027] The cosine value of the angle θ1 between the scanning line OP and the straight line m can be obtained by combining the equations (4) and (8):

[0028]

[0029] Then θ1 = cos -1 (cosθ1), and ∠OPL is taken as θ1;

[0030] In ΔOPL, the following equation can be obtained by the sine theorem:

[0031]

[0032] In the equation: is the geometric distance from the target point to the center of the TLS;

[0033] In ΔOPK, the following equation can be obtained by the sine theorem:

[0034]

[0035] The Δv can be obtained by combining the equation (3), the equation (10) and the equation (11):

[0036]

[0037] The straight line equation of the straight line l intersected by the plane e and the plane g is:

[0038]

[0039] The direction vector S3 of the straight line l is obtained as:

[0040]

[0041] In the equation: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1), and S3 is obtained as:

[0042] S3 = (b1c3-b3c1, -a1c3+a3c1, a1b3-a3b1) (15);

[0043] The cosine value of the angle θ2 between the scanning line OP and the straight line l can be obtained by combining the equation (4) and the equation (15):

[0044]

[0045] Then θ2 = cos -1 (cosθ2), and ∠OPN is taken as θ2;

[0046] In ΔOPN, the following equation can be obtained by the sine theorem:

[0047]

[0048] In ΔOPM, from the sine theorem, we have:

[0049]

[0050] Combining equation (3), equation (17) and equation (18), we have Δh as:

[0051]

[0052] ∠OPL, ∠OPN and ∠LPN are considered as three vertex angles of a triangular pyramid with P as the vertex, ∠OPL and ∠OPN are θ1 and θ2 respectively, and ∠LPN is the angle between straight line m and l, which is set as θ3. According to the basic formula of tetrahedral space angle, we have:

[0053] cosθ1·cosθ2+sinθ1·sinθ2·cosγ=cosθ3 (20);

[0054] In the equation, γ is the dihedral angle between plane f and plane g; since plane f is perpendicular to plane g, γ = 90°,

[0055] Simplify equation (20) as:

[0056] cosθ1·cosθ2=cosθ3 (21);

[0057] Then we have:

[0058] Combining equation (12) and equation (19), we have the calculation formula of the area s of the target point peripheral parallelogram as:

[0059]

[0060] Substitute the area s of the target point peripheral parallelogram calculated by equation (22) into equation (2), and we get the target point p i The density calculation formula is as follows:

[0061]

[0062] Plane e is the plane where the target point p i is located. The least square method is used to fit the k nearest neighbor points around the target point, and the plane normal vector is obtained as (u i , v i , w i ). Then, combined with the fact that the plane passes through the target point p i , the plane equation coefficients of plane e are obtained as:

[0063]

[0064] Since the plane f passes through the point O(x0, y0, z0) and P(x i , y i , z i ), and is perpendicular to the XOY plane, its plane equation is converted into the straight line equation passing through (x0, y0) and (x i , y i ):

[0065]

[0066] The plane equation coefficients of the plane f are obtained from equation (25):

[0067]

[0068] When x i = 0, the plane f coincides with the YOZ plane, and at this time, its plane equation coefficients should be: a2 = 1, b2 = 0, c2 = 0, d2 = 0; since the plane g is perpendicular to the plane f and passes through the scanning line OP, the normal vector S4 = (a2, b2, c2) of the plane f and the direction vector of the scanning line OP are obtained by the outer product, and the normal vector S5 of the plane g is obtained:

[0069]

[0070] In the formula, i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1), and S5 is obtained as:

[0071] S5 = ((y i -y0)c2-(z i -z0)b2,-(x i -x0)c2+(z i -z0)a2,(x i -x0)b2-(y i -y0)a2) (28) ;

[0072] Combining equation (26) with equation (28) to simplify equation (28) to:

[0073]

[0074] Since the plane g passes through the TLS center point O, the plane equation coefficients of the plane g are obtained as:

[0075]

[0076] When x i = 0, the plane g is perpendicular to the YOZ plane, and at this time, its plane equation coefficients are: a3 = 0, b3 = z i -z0, c3 = -(y i -y0), d3 = yi z0-z i y0;

[0077] Substitute formula (24), (26) and (30) into formula (9) and (16) to obtain θ1 and θ2.

[0078] A correction method of a TLS single-site cloud data density strict calculation model, characterized in that the correction method comprises:

[0079] The density D(p i ,r) of the target point p i is corrected as follows:

[0080]

[0081] In the formula, D c (p i ,r) is the corrected density of the target point p i ,d' and θ1' and θ2' are the reference distance and the reference angles θ1 and θ2 respectively, and the space coordinates of the reference target point are set as The normal vector of the plane where the reference target point is located is d', θ1' and θ2' are calculated.

[0082] Simplify formula (23) as follows:

[0083]

[0084] The simplified density correction formula is obtained as follows:

[0085]

[0086] For the simplified density calculation model, the following three cases are divided:

[0087] 1) θ1 and θ2 are both 90°, at this time, the normal vector of the micro plane e where the target point is located coincides with the scanning line OP, and the scanning line OP is perpendicular to the plane e, and the density calculation formula at this time is obtained from formula (32) as follows:

[0088]

[0089] From formula (34), the point cloud density at this time is only inversely proportional to the square of the distance, and is corrected by the following formula:

[0090]

[0091] 2) θ1=90°, θ2∈(0, 90°], the normal vector of the micro plane e where the target point is located is on the plane g, the scanning line and the target surface always keep vertical in the vertical direction, 90°-θ2 is the laser incidence angle, the density calculation formula at this time is obtained by formula (32):

[0092]

[0093] From formula (36), the point cloud density at this time is inversely proportional to the square of the distance, and is proportional to the sine of the complementary angle of the incidence angle, and is corrected by the following formula:

[0094]

[0095] 3) θ2=90°, θ1∈(0, 90°], the normal vector of the micro plane e where the target point is located is on the plane f, the scanning line and the target surface always keep vertical in the horizontal direction, 90°-θ1 is the laser incidence angle, the density calculation formula at this time is obtained by formula (32):

[0096]

[0097] From formula (38), the point cloud density at this time is inversely proportional to the square of the distance, and is proportional to the sine of the complementary angle of the incidence angle, and is corrected by the following formula:

[0098]

[0099] The application has the advantages of good feasibility, effectiveness and universality, and has great application value in TLS point cloud data processing such as target extraction, classification and identification. BRIEF DESCRIPTION OF DRAWINGS

[0100] Figure 1 It is a point cloud density difference diagram of different geometric feature targets of the application;

[0101] Figure 2 It is a TLS scanning principle and point cloud theoretical density calculation diagram of the application;

[0102] Figure 3 It is a point cloud diagram of the experimental device of the application;

[0103] Figure 4 It is a first indoor experimental device layout diagram of the application;

[0104] Figure 5 It is a first indoor experimental scanning point cloud data diagram of the application;

[0105] Figure 6 It is a point cloud density distribution diagram of different distances of the first indoor experiment of the application;

[0106] Figure 7 Figure 1 is a diagram of the arrangement of the first indoor experiment device of the present application;

[0107] Figure 8 Figure 2 is a diagram of the arrangement of the second indoor experiment device of the present application;

[0108] Figure 9 Figure 3 is a point cloud density distribution diagram of different angles θ1 of the second indoor experiment of the present application;

[0109] Figure 10 Figure 4 is a diagram of the correlation between the theoretical density and the actual density of the second indoor experiment of the present application;

[0110] Figure 11 Figure 5 is a diagram of the arrangement of the third indoor experiment device of the present application;

[0111] Figure 12 Figure 6 is a point cloud density distribution diagram of different angles θ2 of the third indoor experiment of the present application;

[0112] Figure 13 Figure 7 is a diagram of the correlation between the theoretical density and the actual density of the third indoor experiment of the present application. DETAILED DESCRIPTION

[0113] The features and other related features of the present application are further described in detail below with reference to the accompanying drawings, which are provided to assist those skilled in the art in understanding the present application:

[0114] As shown in Figure 1, the marks 1-4 represent: laser spot 1, TLS 2, target 3, and round wood plate 4, respectively. Figures 1-13

[0115] Embodiment 1: As shown in Figure 1, the present embodiment relates to a TLS single-site point cloud data density rigorous calculation model, which includes: Figures 1-2

[0116] 1, let P = {p i ,i = 1, 2,.., n} be a point cloud set, the point cloud set within the spherical neighborhood range with a radius of r around the target point p i can be expressed as:

[0117] R(p i ,r) = {p j / d(p i ,p j ) ≤ r, p j ∈ P and j ≠ i} (1);

[0118] In the formula: d(p i ,p j ) is the geometric distance from the target point p i to p j . Then the point cloud set R(p​​i The total number of points in r) is the target point p i The density of p i , r) is denoted as D(p

[0119] In general, the point cloud density is defined as the total number of points in a spherical neighborhood with radius r, and the volume of the spherical neighborhood is However, for single-station TLS scanning, only the part of the target directly facing the TLS can be recorded, so the actual calculation neighborhood of the point cloud density can be approximated as a circle with radius r 2 For targets with different geometric characteristics, when the neighborhood radius r is defined as an appropriate size, the point cloud density will be significantly different. As shown in Figure 1 , 1 is the laser spot, and the sizes of the two objects (cylinders) in the figure are different, with radii c1 and c2, and c1 > c2. If the neighborhood radius r = r1 <= c2, that is, it is smaller than the size of both objects, the point cloud densities of the two objects are the same. However, if c2 < r = r2 <= c1, that is, the neighborhood radius is larger than the size of one object and smaller than the size of the other object, the point cloud density of the larger object is larger, and at this time the two objects can be better classified. Therefore, by defining an appropriate neighborhood radius r, the point cloud density information can represent the geometric size and spatial distribution pattern of the target, and be used for classification, identification, and other research of targets with different geometric characteristics.

[0120] 2, TLS uses a line-by-line scanning working mode. The scanning line first performs scanning from bottom to top (or from top to bottom) in the vertical direction within the vertical scanning angle range, with vertical angle resolution β as the step size for discrete point measurement, then rotates by a horizontal angle resolution α and performs vertical scanning again, until the specified horizontal scanning angle or a circular scanning is completed and the scanning ends. As shown in Figure 2As shown, 1 is the laser spot, 2 is the TLS, 3 is the target, and a three-dimensional coordinate system is established with the center O of the TLS 2 as the origin. Due to the fixed divergence angle of the laser, a laser spot 1 will be formed on the target surface, but the divergence angle is usually much smaller than a and β, so its influence can be ignored. The planes where ΔOHI, ΔOLK and ΔOQR are located are respectively three consecutive vertical scanning planes, all of which are perpendicular to the XOY plane, and the angle interval (i.e. dihedral angle or included angle) between adjacent planes is a. In a single vertical scanning plane, due to the existence of vertical angle resolution β, there is a certain spacing between adjacent laser spots. For consecutive adjacent vertical scanning planes, when the scanning angles are the same, the scanning lines in different vertical scanning planes can be considered to be located in the same plane, and the plane is perpendicular to the vertical scanning plane where the scanning line is located. For a small plane e of the target surface, points N, P, M have the same scanning angle and are located on the same straight line, and scanning lines ON, OP, OM are located in the same plane. Let the plane where ΔOLK is located be f, and the plane where ΔOMN is located be g, so the planes f and g are perpendicular to each other. Let the intersection line of plane f and plane e be m, and the intersection line of plane g and plane e be l. The scanning line OP is the intersection line of the two perpendicular planes f and g. When OP is perpendicular to plane e, OP is perpendicular to the straight lines m and l in plane e at the same time. At this time, the included angle of straight lines m and l is the dihedral angle of the two perpendicular planes f and g, so the included angle of straight lines m and l is 90°. When OP is only perpendicular to straight line m, since OP is the intersection line of the two perpendicular planes f and g, and straight line m is located on plane f, straight line m is perpendicular to plane g, so straight line m is also perpendicular to straight line l on plane g, and the included angle is 90°. Similarly, when OP is only perpendicular to straight line l, the included angle of straight lines m and l is also 90°. When OP is not perpendicular to straight lines m and l, since OP is the intersection line of the two perpendicular planes f and g, the included angle of straight lines m and l will be greater or less than the dihedral angle of planes f and g, so the included angle is obtuse or acute. Let the normal vector of the plane e where the target point P is located be (u, v, w), then the included angle between the scanning line OP and the normal vector is the incidence angle, denoted as λ. Let the included angles of the scanning line OP with straight lines m and l be θ1 and θ2 respectively. Due to the existence of various inclinations of the target surface, the normal vector of the target surface is generally not coplanar with plane f or g, that is, the incidence angle λ and θ1 or θ2 are not mutually complementary, for example, when the target is rotated around straight line m, the normal vector of the target surface will change, causing the incidence angle to change, but the scanning line OP, straight line m and plane f remain fixed, that is, the angle θ1 remains fixed. Only when the normal vector is coplanar with plane f or g, that is, one of θ1 or θ2 is 90°, the complementary angle of the other angle is the incidence angle.

[0121] For TLS, the point cloud density usually changes with the distance and other factors, which shows that the density is high near the target and low far from the target. The main idea of this model is as follows: the vertical scanning plane (plane f) of the target point's scanning line, the plane (plane g) composed of the left and right adjacent scanning lines of the target point, and the micro plane (plane e) where the target point is located are intersected, respectively, to obtain the direction vectors (S2 and S3) of the two intersecting straight lines (m and l), and then the direction vector (S1) of the target point's scanning line (straight line OP) is combined to obtain the angles (θ1 and θ2) between the target point's scanning line and the two intersecting straight lines. The horizontal and vertical point distances (Δh and Δv) can be obtained by the sine theorem of triangle, and then the area (s) of the parallelogram around the target point can be obtained. Finally, the ratio of the circular neighborhood area of the target point to the area of the parallelogram can be obtained to obtain the point cloud density:

[0122]

[0123] In the formula: s is the area of the parallelogram around the target point. This model uses a parallelogram to represent a laser spot, which can be obtained as follows: as shown in Figure 2 , first, connect the target point and its four neighboring points with straight lines, and then draw perpendicular lines through the midpoints of the four connecting lines. The area enclosed by the newly generated four straight lines is a parallelogram, which contains a laser spot. Similarly, a parallelogram can be formed around each laser spot, and each parallelogram can represent a laser point. The parallelograms are closely connected without gaps. Within a circular neighborhood with a radius of r, the areas of the parallelograms change very little and are approximately equal to the area of the parallelogram around the target point. Therefore, the point cloud density of the target point can be calculated by formula (2).

[0124] 2.1 The area of the parallelogram is closely related to the distance between the target point and the adjacent laser spot, as shown in FIG. 2. The distances between the left and right adjacent points of the target point p i are PN and PM, and the distances between the front and back (up and down) adjacent points are PK and PL. Usually, PN and PM, PK and PL are not equal, and the distance to the far end of the instrument is slightly longer. Only when the scanning line OP is perpendicular to the target surface, they are equal. From the above analysis, the two side lengths Δh and Δv of the parallelogram are:

[0125]

[0126] Let the coordinates of the target point P (or p i ) be (x i , y i , z i ), and the coordinates of the instrument (TLS) center O be (x0, y0, z0). The direction vector S1 of the straight line OP can be obtained as follows:

[0127] S1=(x i -x0,y i -y0,z i -z0) (4);

[0128] Suppose the equations of planes e, f, and g are as follows:

[0129]

[0130] Then the equation of the line m that intersects plane e and plane f is:

[0131]

[0132] Therefore, the direction vector S2 of line m can be obtained as:

[0133]

[0134] In the formula: i = (1,0,0), j = (0,1,0), k = (0,0,1), we can obtain S2 as:

[0135] S2=(b1c2-b2c1,-a1c2+a2c1,a1b2-a2b1) (8);

[0136] Therefore, combining equations (4) and (8), the cosine of the angle θ1 between the scan line OP and the straight line m is:

[0137]

[0138] Then θ1 = cos -1 (cosθ1), such as Figure 2 As shown, the calculated result of θ1 could be ∠OPL or ∠OPK. Since ∠OPL and ∠OPK are supplementary angles, choosing either one as θ1 has no effect on the calculation of the length of LK. Therefore, ∠OPL is used as θ1 for subsequent calculations. In ΔOPL, by the Law of Sines, we can obtain:

[0139]

[0140] In the formula, Let be the geometric distance from the target point to the center of the instrument. Similarly, in ΔOPK, by the law of sines, we can obtain:

[0141]

[0142] Therefore, combining equations (3), (10), and (11), we can obtain Δv as:

[0143]

[0144] The linear equation of the straight line l intersecting the plane e and the plane g is:

[0145]

[0146] Therefore, the direction vector S3 of the straight line l can be obtained as follows:

[0147]

[0148] where i=(1, 0, 0), j=(0, 1, 0), and k=(0, 0, 1), and S3 can be obtained as follows:

[0149] S3=(b1c3-b3c1,-a1c3+a3c1,a1b3-a3b1) (15);

[0150] Therefore, the cosine value of the angle θ2 between the scanning line OP and the straight line l can be obtained by combining equations (4) and (15) as follows:

[0151]

[0152] Therefore, θ2=cos -1 (cosθ2), and the calculation method of θ1 is similar, and here ∠OPN is taken as θ2 for subsequent calculation. In ΔOPN, the following can be obtained by the sine theorem:

[0153]

[0154] In ΔOPM, the following can be obtained by the sine theorem:

[0155]

[0156] Therefore, Δh can be obtained by combining equations (3), (17), and (18) as follows:

[0157]

[0158] From Figure 2 It can be seen that ∠OPL, ∠OPN, and ∠LPN can be regarded as the three vertex angles of a triangular pyramid with P as the vertex, as shown in Figure 2 ∠OPL and ∠OPN are θ1 and θ2 respectively, and ∠LPN is the angle between the straight lines m and l, which is set as θ3. According to the basic formula of tetrahedral space angle, the following relationship can be obtained:

[0159] cosθ1·cosθ2+sinθ1·sinθ2·cosγ=cosθ3 (20);

[0160] where γ is the dihedral angle of the plane f and the plane g. Since the plane f is perpendicular to the plane g, γ=90°, and equation (20) can be simplified as:

[0161] cos θ1cos θ2= cos θ3 (21)

[0162] Then Combining with formula (12) and (19), the formula of the area s of the parallelogram can be obtained as:

[0163]

[0164] 2.2 Substituting the parallelogram area s calculated by formula (22) into formula (2), the target point p i The density calculation formula is as follows:

[0165]

[0166] It can be seen from formula (23) that the point cloud density is related to the six parameters of r, α, β, d i , θ1, and θ2. The neighborhood radius r is set by human and is related to the size of the target object. The angle resolutions α and β are set according to actual work needs and are constant values in the single-station TLS scanning process. By observing formula (9), (16), and the calculation formula of the distance d i , it can be seen that the three parameters of d i , θ1, and θ2 are mainly related to the equation coefficients of the planes e, f, and g, the spatial coordinates of the target point, and the instrument center coordinates. Among them, the three-dimensional spatial coordinates of the target point and the instrument center coordinates can be directly obtained from the TLS point cloud data, and then d i can be solved.

[0167] 2.3 The plane e is the plane where the target point p i is located. Generally, the least squares method can be used to fit the plane with k neighboring points around the target point, and the plane normal vector is (u i , v i , w i ). Combined with the plane passing through the target point p i , the plane equation coefficients of the plane e can be obtained as:

[0168]

[0169] Since the plane f passes through the points O(x0, y0, z0) and P(x i , y i , z i ), and is perpendicular to the XOY plane, the plane equation of the plane f can be converted into the straight line equation passing through (x0, y0) and (x i , y i ):

[0170]

[0171] The plane equation coefficients of the plane f can be obtained from formula (25) as follows:

[0172]

[0173] In formula (26), it is worth noting that when x i = 0, the plane f coincides with the YOZ plane, and the plane equation coefficients thereof should be a2 = 1, b2 = 0, c2 = 0, and d2 = 0. Since the plane g is perpendicular to the plane f and passes through the scanning line OP, the normal vector S5 of the plane g can be obtained by calculating the outer product of the normal vector S4 of the plane f (a2, b2, c2) and the direction vector of the scanning line OP (formula (8)).

[0174]

[0175] In the formula, i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1), and S5 can be obtained as follows:

[0176] S5 = ((y i -y0)c2-(z i -z0)b2,-(x i -x0)c2+(z i -z0)a2,(x i -x0)b2-(y i -y0)a2) (28) ;

[0177] In combination with formula (26), formula (28) can be simplified as follows:

[0178]

[0179] Since the plane g passes through the instrument center point O, the plane equation coefficients of the plane g can be obtained as follows:

[0180]

[0181] In formula (30), it is worth noting that when x i = 0, the plane g is perpendicular to the YOZ plane, and the plane equation coefficients thereof should be a3 = 0, b3 = z i -z0, c3 = -(y i -y0), and d3 = y i z0-z i y0. By observing formula (24), (26), and (30), it can be seen that the plane equation coefficients of the planes e, f, and g are related to the spatial coordinates of the target point, the instrument center coordinates, and the normal vector of the plane where the target point is located. These parameters can be obtained from the TLS point cloud data, and therefore, formula (24), (26), and (30) can be substituted into formula (9) and (16) to obtain θ1 and θ2.

[0182] Embodiment 2: This embodiment relates to a correction method of a TLS single-site cloud data density rigorous calculation model, and the correction method comprises the following steps:

[0183] 1. As can be seen from formula (23), the point cloud density is related to d i , θ1 and θ2, therefore, the original density data cannot be directly used to distinguish targets with different geometric features, and the original density data can be corrected by the following formula:

[0184]

[0185] In the formula, D c (p i ,r) is the target point p i , d', θ1' and θ2' are the corrected density, reference distance, reference angle θ1 and θ2, respectively, which can be arbitrarily set. In this embodiment, the reference target point space coordinates are set as The normal vector of the plane where the reference target point is located is d', θ1' and θ2' are calculated. Formula (31) shows that D c (p i ,r) eliminates the influence of distance and angle, and is only related to the spatial geometric feature of the target and the neighborhood radius r, and can be used for classification and identification of targets with different geometric features.

[0186] The numerical size of the angle resolution α and β is closely related to the sparsity of the point cloud, and in actual work, they are usually set to a very small value (such as 0.05°) to obtain dense enough point cloud data of the target, therefore, formula (23) can be simplified as:

[0187]

[0188] Further, the simplified density correction formula is:

[0189]

[0190] 2. For the simplified density calculation model (formula (32)), when θ1 or θ2 is 90°, the following special cases can be divided:

[0191] 1) Both θ1 and θ2 are 90°. At this time, the normal vector of the micro plane e where the target point is located coincides with the scanning line OP, that is, the scanning line OP is perpendicular to the plane e, and the density calculation formula at this time can be obtained from formula (32) as:

[0192]

[0193] As can be seen from formula (34), the point cloud density at this time is only inversely proportional to the square of the distance, which is consistent with the relationship between the point cloud density and the distance in the theoretical model established by Tan et al., and can be corrected by the following formula:

[0194]

[0195] 2) θ1= 90°, θ2∈(0, 90°]. At this time, the normal vector of the micro plane e where the target point is located is on the plane g, and the scanning line and the target surface always remain perpendicular in the vertical direction. 90°- θ2 is the laser incidence angle. The density calculation formula at this time can be obtained from formula (32) as follows:

[0196]

[0197] As can be seen from formula (36), the point cloud density at this time is inversely proportional to the square of the distance and proportional to the sine of the complementary angle of the incidence angle. The correction can be made by the following formula:

[0198]

[0199] 3) θ2= 90°, θ1∈(0, 90°]. At this time, the normal vector of the micro plane e where the target point is located is on the plane f, and the scanning line and the target surface always remain perpendicular in the horizontal direction. 90°- θ1 is the laser incidence angle. The density calculation formula at this time can be obtained from formula (32) as follows:

[0200]

[0201] As can be seen from formula (38), the point cloud density at this time is inversely proportional to the square of the distance and proportional to the sine of the complementary angle of the incidence angle. The correction can be made by the following formula:

[0202]

[0203] In combination with Embodiment 1 and Embodiment 2, it is illustrated that the TLS single-station point cloud density calculation model can be applied to calculate the point cloud density of targets with various inclinations, and then correct the density, which is theoretically applicable to various types of scenes in nature (such as buildings, trees, roads, etc.). Since the model is calculated from the basic working principle of TLS, it is applicable to different types of TLS scanners. Meanwhile, when the angle resolution α and β of certain measurement data cannot be obtained, the point cloud density can also be well corrected by formula (33). After the point cloud density data is corrected by the model, it can be used for classification, identification and other related researches of targets with different geometric characteristics.

[0204] Example 3: To verify the theoretical density calculation model of the TLS single-site cloud, three indoor quantitative experiments were conducted to verify the relationship between the three variables d (distance d), θ1 (angle θ2), and θ2 in the model and the point cloud density. Specifically, a relatively smooth circular wooden board with a diameter of 20 cm and a thickness of 3 mm was connected to a rotating platform, which was then fixed on a tripod. The rotating platform allowed for precise control of the rotation angle of the circular wooden board. The circular wooden board was scanned using TLS, and the point cloud density within a 0.1 m radius neighborhood of the center point of the circular wooden board was determined by counting the number of points on the board.

[0205] The indoor experiments used a Riegl VZ-4000 scanner, a pulse-based TLS scanning system that measures distance by the time difference between the transmitted and echo signals. The Riegl VZ-4000 has a laser wavelength of 1550 nm, a laser divergence angle of 0.15 mrad, and horizontal and vertical field of view angles of 360° and 60°, respectively. It possesses excellent ranging capabilities, with a measurement range of 5–4000 m. This scanning system offers four selectable laser pulse repetition rates (PRR): 30, 50, 100, and 150 kHz; a lower PRR corresponds to better ranging performance. For the three indoor quantitative experiments, due to the small size of the circular wooden board, setting the horizontal field of view to 145°–165° and the vertical field of view to 80°–100° was sufficient for the scanning requirements. The horizontal and vertical angular resolutions were set to 0.02° and 0.03°, respectively, and the PRR was fixed at 30 kHz. After scanning, the following results were obtained: Figure 3 The point cloud shown needs to be cropped and its point cloud counted. Cloudcompare software is used here to process it, remove noise points around the wooden board, and crop it strictly according to the geometric dimensions of the wooden board.

[0206] The first indoor experiment was mainly used to verify the relationship between distance d and point cloud density. Angles θ1 and θ2 were kept constant, and the distance d was continuously varied during scanning to obtain the point cloud density corresponding to different distances, thus revealing the relationship between distance d and point cloud density. The experiment was conducted on a horizontal and flat surface, as shown in Figure 4, where 2 represents the TLS and 4 represents the circular wooden board. First, the circular wooden board was placed vertically, directly facing the TLS. Then, it was rotated upwards and to the left by a certain angle to ensure that angles θ1 and θ2 were consistent with the general case. Finally, the center of the circular wooden board was adjusted to be at the same height as the center of the TLS (the ground was scanned before the experiment to determine the height of the TLS center), ensuring that θ1 and θ2 remained constant each time the circular wooden board was moved forward. The theoretical ranging range of the Riegl VZ-4000 is 5–4000 m. Measurements at distances that are too close or too far result in significant errors. At a distance of 50 m, the distance between two points is approximately 0.03 m. In this experiment, the point cloud on a 0.2 m diameter circular wooden board was already very sparse. To ensure the accuracy of the experimental results, the distance range was set to 8–48.8 m, with a movement interval of 2.4 m. A scan was performed after each movement, resulting in 18 scans at different distances. The scanned point cloud data are shown below. Figure 5 As shown.

[0207] The angles θ1 and θ2 at the center point of the circular wooden board can be calculated according to equations (9) and (16), resulting in 18 sets of mutually approximate θ1 and θ2. Here, their average values ​​of 67.49° and 67.79° are taken as θ1 and θ2 for calculation in this experiment. The experimental results are as follows: Figure 6 and Figure 7 As shown, Figure 6 The graph shows the point cloud density distribution at different distances. The horizontal axis represents distance, the vertical axis represents point cloud density, the asterisk represents the experimentally measured point cloud density, the curve represents the point cloud density calculated by formula (32), and the circle represents the point cloud density at the corresponding experimental distance on the curve. Figure 7 A correlation diagram is shown between the theoretical point cloud density calculated by the model and the actual point cloud density measured experimentally. It can be seen that the theoretical and actual point cloud densities have good consistency, with a correlation coefficient of 0.9997, indicating a high degree of agreement. The calculated root mean square error (RMSE) is 14.37, which is also relatively small, indicating that the relationship between distance d and point cloud density is consistent with formula (32), that is, point cloud density is inversely proportional to the square of the distance. Figure 6 It can be seen that, overall, the point cloud density decreases with increasing distance, with a larger decrease in the first 20m, followed by a more gradual decrease. The above experiments demonstrate that distance has a significant impact on point cloud density. To obtain a corrected density that is only related to the target geometry, the distance effect needs to be eliminated.

[0208] The second indoor experiment primarily verified the relationship between angle θ1 and point cloud density. By keeping the distance d and angle θ2 constant, and continuously changing the angle θ1 during scanning, the point cloud density corresponding to different θ1 values ​​was obtained, thus revealing the relationship between angle θ1 and point cloud density. The experiment was conducted on a flat surface, such as... Figure 8 As shown, 2 represents the TLS (Transmission Linear Artificial Intelligence), and 4 represents the circular wooden board. First, the circular wooden board is placed 16m away from the TLS and positioned vertically facing the TLS. Then, it is rotated to the left by a certain angle to ensure that angle θ2 is within the normal range. The circular wooden board initially has a vertical position, i.e., angle θ1 is 90°. It is then rotated gradually around the axis containing line l in 5° increments, up to 10°. When θ1 is less than 10°, only a very small number of points exist on the circular wooden board, so this is not considered in this experiment. A scan is performed each time the circular wooden board is rotated. During this process, the distance d and angle θ2 are considered constant, resulting in 17 scans at different angles θ1.

[0209] The distance to the center point of the circular wooden board can be calculated from its three-dimensional spatial coordinates, and the angle θ2 can be calculated according to equation (16). A total of 17 sets of mutually approximate d and θ2 are obtained. Here, the average values ​​of 16.03m and 71.96° are taken as d and θ2 in this experiment for calculation. The experimental results are as follows: Figure 9 and Figure 10 As shown, it can be seen that the theoretical density of the point cloud calculated by changing different angles θ1 has good consistency with the actual density of the point cloud measured experimentally. The correlation coefficient between the two is 0.9990, indicating that they have a strong correlation. The calculated root mean square error (RMSE) is 11.94, which is also relatively small, indicating that the relationship between angle θ1 and point cloud density is consistent with formula (32). Figure 9 It can be seen that, overall, the point cloud density decreases as the angle θ1 decreases, with a smaller decrease between 90° and 70°, followed by a continuous decrease in a nearly linear manner. The above experiments demonstrate that the angle θ1 has a significant impact on the point cloud density. To obtain a corrected density that is only related to the target geometry, the influence of the angle θ1 needs to be eliminated.

[0210] The third indoor experiment was mainly used to verify the relationship between angle θ2 and point cloud density. With the distance d and angle θ1 fixed, and the angle θ2 continuously changed for scanning, the point cloud density corresponding to different θ2 values ​​was obtained, thus revealing the relationship between angle θ2 and point cloud density. For example... Figure 11As shown, 2 represents the TLS (Transmission of Traceability), and 4 represents the circular wooden board. The experimental procedure is similar to that for angle θ1, except that the circular wooden board is adjusted to a vertical position and then rotated upwards by a certain angle to ensure that angle θ1 is the general case. The initial angle θ2 of the circular wooden board is 90°. Then, the circular wooden board is gradually rotated around the axis containing the straight line m in 5° increments, up to 10°. A scan is performed each time the circular wooden board is rotated. During this process, the distance d and angle θ1 are considered constant, resulting in 17 scans at different angles θ2.

[0211] The distance to the center point of the circular wooden board can be calculated from its three-dimensional spatial coordinates, and the angle θ1 can be calculated according to equation (9). A total of 17 sets of mutually approximate d and θ1 are obtained. Here, their average values ​​of 16.01m and 67.32° are taken as d and θ1 in this experiment for calculation. The experimental results are as follows: Figure 12 and Figure 13 As shown, it can be seen that the actual density of the point cloud obtained by changing different θ2 measurements has good consistency with the theoretical density calculated by the model. The correlation coefficient between the two is 0.9985, which is a strong correlation. The calculated root mean square error (RMSE) is 10.33, which is also small. This shows that the relationship between angle θ2 and point cloud density is consistent with formula (32). Figure 12 It can be seen that, overall, the trend of point cloud density changing with angle θ2 is almost the same as that of angle θ1, indicating that angle θ2 also has a significant impact on point cloud density. If we want to obtain the corrected density that is only related to the target geometry, we need to eliminate the influence of angle θ2.

[0212] The three experiments above verified the relationship between the three variables d, θ1, and θ2 in the point cloud theoretical density calculation model and the point cloud density. The experimental results show that the calculated results of the point cloud theoretical density calculation model have good correlation and consistency with the actual point cloud density, with only a small error between them, indicating that the model can well represent the actual changes in point cloud density. The experiments also show that the point cloud density exhibits different trends as the three variables d, θ1, and θ2 change. Therefore, it is necessary to simultaneously eliminate the influence of these three variables to obtain a corrected point cloud density, which can then be better used in research on the identification, classification, and extraction of targets with different geometric features.

[0213] In summary, the embodiment mathematically deduces a rigorous calculation model of TLS single-site cloud theoretical density, and proposes a correction method of TLS single-site cloud density by defining reference distance and angle. By eliminating the influence of objective factors such as distance and angle, the corrected point cloud density related only to the geometric size / characteristics of the target is obtained. The relationship between the three variables of distance d, angle θ1 and θ2 in the model and the point cloud density is verified through indoor quantitative experiments, and the correlation coefficients all reach 0.99, and the root mean square error RMSE is also very small, which shows that the theoretical density model can better express the real point cloud density variation. At the same time, the model can be applied to various types of TLS scanning systems, and does not need to know the angle resolution, and has strong applicability and robustness. By setting a suitable neighborhood radius r, the original point cloud density can be corrected using the point cloud density correction model, which can be used for target classification, recognition and extraction, etc. In the future, this method can be used in the semantic segmentation research of single-site TLS scanning data, such as vegetation stem-leaf separation, urban rod-shaped object extraction, single tree segmentation, etc., which can be used to further improve the accuracy of target classification, recognition and extraction.

[0214] Although the above embodiments have been described in detail with reference to the accompanying drawings, those skilled in the art can recognize that various improvements and changes can be made to the present application without departing from the scope defined by the claims, and therefore detailed description is not repeated here.

Claims

1. A rigorous calculation model for TLS single-site cloud data density, characterized in that, The model includes: scanning the target line by line via TLS; with TLS as the center. O Establish a three-dimensional spatial coordinate system with the origin, and use the target point P The vertical scanning plane where the scan line is located f Target point P The plane formed by the left and right adjacent scan lines g Respectively with the target point P The tiny plane it occupies e Intersecting lines, find the two intersecting lines. m and l Direction vector and Combined with target point P scan lines OP Direction vector To obtain the target point P The scan lines intersect with the two aforementioned straight lines respectively. m and l The included angle and The distance between the horizontal and vertical points can be determined using the triangle sine theorem. and The area of ​​the parallelogram surrounding the target point is obtained. s Find the area of ​​the circular neighborhood of the target point and the area of ​​the parallelogram surrounding the target point. s The ratio of the two values ​​is used to obtain the point cloud density; The line-by-line scanning refers to scanning the scan lines from bottom to top or from top to bottom within the vertical scanning angle range in the vertical direction, with a vertical angular resolution. To perform discrete point measurements for the step size, rotate by a horizontal angular resolution. Then perform a vertical scan again until the specified horizontal scan angle or a circular scan is completed and the process ends.

2. The rigorous calculation model for TLS single-site cloud data density as described in claim 1, characterized in that, make For point cloud aggregation, at the target point The surrounding radius is The set of point clouds within a spherical neighborhood is represented as: ； In the formula: For target point to geometric distance; Then point cloud set The total number of points in the target point density : ； Target point The distances between the left and right adjacent points are respectively PN, PM The distances between adjacent points, either front and back or top and bottom, are respectively PK, PL The lengths of the two sides of the parallelogram surrounding the target point are obtained. and They are respectively: ； Let the target point P or The coordinates are TLS Center O The coordinates are , obtain scan lines OP The direction vector of the line for: ； Assuming a plane e , f , g The equation of the plane is: ； Then the plane e With plane f Intersecting lines m The equation of the straight line is: ； Find the straight line m Direction vector for: ； In the formula: , , ,have to for: ； Combining equations (4) and (8), the scan line can be obtained. OP With a straight line m The included angle The cosine value is: ； but ,by As ; exist In the middle, by the Law of Sines, we get: ； In the formula: The geometric distance from the target point to the center of the TLS; exist In the middle, by the Law of Sines, we get: ； Combining equations (3), (10), and (11), we get for: ； flat e With plane g Intersecting lines l The equation of the straight line is: ； Find the straight line l Direction vector for: ； In the formula: , , ,have to for: ； Combining equations (4) and (15) yields the scan line. OP With a straight line l The included angle The cosine value is: ； but ,by As ; exist In the middle, by the Law of Sines, we get: ； exist In the middle, by the Law of Sines, we get: ； Combining equations (3), (17), and (18), we get for: ； , and Considered as P The three vertices of a triangular pyramid are denoted by _____. and They are respectively and , It is a straight line m and l The included angle, let it be set as From the basic formula for the space angle of a tetrahedron, we get: ； In the formula: For plane f and plane g The dihedral angle; due to the plane f With plane g vertical, Equation (20) can be simplified to: ； but ; Combining equations (12) and (19), we obtain the area of ​​the parallelogram surrounding the target point. The calculation formula is: ； The area of ​​the parallelogram surrounding the target point calculated by equation (22) Substituting into equation (2), we obtain the target point. The density calculation formula is as follows: 。 3. The rigorous calculation model for TLS single-site cloud data density as described in claim 2, characterized in that, flat e For target point The plane in which the target point is located, utilizing the surrounding area k The least squares method is used to fit the nearest neighbor points to obtain the plane normal vector. Then, combined with the fact that the plane passes through the target point , to obtain a plane e The coefficients of the plane equation are: ； Due to the plane f Passing point and And with XOY The plane is perpendicular, so its plane equation is transformed into... and The equation of the straight line: ； From equation (25), we obtain the plane. f The coefficients of the plane equation are: ； Among them, when At that time, the plane f and YOZ If the planes coincide, then the coefficients of their plane equations should be: , , , Due to the plane g With plane f Vertical, and across the scan line OP By obtaining the plane f normal vector With scan lines OP The outer product of direction vectors yields a plane. g normal vector : ； In the formula, , , ,have to for: ； Combining equation (26) simplifies equation (28) to: ； Due to the plane g Through TLS center point O Then a plane can be obtained. g The coefficients of the plane equation are: ； Among them, when At that time, the plane g and YOZ When the planes are perpendicular, the coefficients of its plane equation are: , , , ; Substituting equations (24), (26), and (30) into equations (9) and (16) yields the following results. and .

4. The correction method for a rigorous calculation model of TLS single-site cloud data density as described in any one of claims 2-3, characterized in that, The correction method includes: For target point density Perform correction: ； In the formula: For target point Corrected density , , Reference distance and reference angle, respectively. , Set the spatial coordinates of the reference target point as The normal vector of the plane containing the reference target point is Calculations yielded , ,and ; Equation (23) can be simplified to: ； The simplified density correction formula is as follows: 。 5. The correction method for a rigorous calculation model of TLS single-site cloud data density as described in claim 4, characterized in that, The simplified density calculation model can be divided into the following three cases: 1) and Both are 90°, and at this time the target point is located on a tiny plane. e normal vector and scan line OP Overlapping, scan lines OP With plane e Vertically, the density calculation formula at this time is obtained from equation (32): ； From equation (34), we can see that the point cloud density at this point is inversely proportional to the square of the distance, which can be corrected by the following equation: ； 2) , At this time, the tiny plane where the target point is located e The normal vector in the plane g Above, the scan line remains perpendicular to the target surface in the vertical direction. Let be the laser incident angle. From equation (32), the density calculation formula at this time is: ； From equation (36), we can see that the point cloud density at this point is inversely proportional to the square of the distance and directly proportional to the sine of the complementary angle of incidence. This can be corrected using the following formula: ； 3) , At this time, the tiny plane where the target point is located e The normal vector in the plane f Above, the scan line remains perpendicular to the target surface in the horizontal direction. Let be the laser incident angle. From equation (32), the density calculation formula at this time is: ； From equation (38), we can see that the point cloud density at this point is inversely proportional to the square of the distance and directly proportional to the sine of the complementary angle of incidence. This can be corrected using the following formula: 。

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