CN113625300A - Crown total leaf area measurement method based on multi-echo LiDAR - Google Patents

Crown total leaf area measurement method based on multi-echo LiDAR Download PDF

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CN113625300A
CN113625300A CN202110930216.8A CN202110930216A CN113625300A CN 113625300 A CN113625300 A CN 113625300A CN 202110930216 A CN202110930216 A CN 202110930216A CN 113625300 A CN113625300 A CN 113625300A
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CN113625300B (en
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薛玉玺
李秋洁
徐俞阳
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Nanjing Forestry University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S17/00—Systems using the reflection or reradiation of electromagnetic waves other than radio waves, e.g. lidar systems
    • G01S17/88—Lidar systems specially adapted for specific applications
    • G01S17/89—Lidar systems specially adapted for specific applications for mapping or imaging
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/48—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S17/00
    • G01S7/4802—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S17/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section

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Abstract

A method for measuring total leaf area of a crown based on multi-echo LiDAR comprises the following steps: loading a laser radar on a moving vehicle to obtain multi-echo point cloud data of trees; establishing a three-dimensional coordinate system for the scanned multi-echo point cloud data based on the initial position of the laser radar, and screening crown point cloud data of the tree to be detected; dividing the crown point cloud data into three types, calculating the grid area corresponding to each type of crown point cloud data, and accumulating to obtain the total grid area; and substituting the total grid area of the tree to be detected into a linear regression model to obtain the total leaf area. The method utilizes the LiDAR multi-echo principle to process and classify the crown point cloud data acquired by the mobile laser radar scanning system, respectively calculates the grid area of the classified point cloud data by a grid area method, and then obtains the total leaf area by bringing in a trained regression model.

Description

Crown total leaf area measurement method based on multi-echo LiDAR
Technical Field
The invention relates to a leaf area measuring method, in particular to a crown total leaf area measuring method based on multi-echo LiDAR.
Background
The method is characterized in that a laser radar (LiDAR) active remote sensing technology is used for rapidly acquiring high-resolution and high-precision three-dimensional point cloud data of target plants, and then calculating the leaf area of the target plants according to the point cloud data is an important step for researching plant productivity and ecological system material circulation. However, due to the complexity of the plant's internal structure, accurate measurement of leaf area is very challenging. Therefore, the accurate measurement method research of the leaf area of the vegetation has very important value.
Some existing methods estimate the leaf area based on the number of canopy point clouds acquired by the LiDAR. The leaf area and canopy point cloud number are in linear relation, and the point cloud number and the leaf number are subjected to fitting verification through a Gaussian function, a polynomial function and an exponential function. However, because the point cloud number is greatly influenced by the moving speed and the measuring distance of the LiDAR, a grid area method based on LiDAR single echo is used for measuring the leaf area to solve the problem. However, the method for calculating the leaf area by only using the single echo point cloud ignores the point cloud of the secondary echo. A plurality of leaf parts are overlapped in the tree crown, when laser points are arranged on the overlapped edges, two times of echoes occur, the inaccurate calculation of the leaf area can be caused by only calculating the grid area of a single echo, and therefore the grid area needs to be classified and calculated to obtain the total leaf area.
Disclosure of Invention
The invention aims to provide a multi-echo LiDAR-based method for measuring total leaf area of a crown, aiming at the problem that the result is inaccurate due to the fact that the existing leaf area measuring method only utilizes LiDAR single echoes.
The technical scheme of the invention is as follows:
the invention provides a method for measuring total leaf area of a crown based on multi-echo LiDAR, which comprises the following steps:
s1, loading the laser radar on the moving vehicle, enabling the moving vehicle to pass through the tree to be detected, and acquiring multi-echo-point cloud data of the tree by using the laser scanning sensor, wherein the multi-echo-point cloud data comprises the distance r between two echoes1、r2Intensity of echo I1、I2And an angle θ;
s2, establishing a three-dimensional coordinate system for the scanned multi-echo point cloud data based on the initial position of the laser radar, judging whether the three-dimensional coordinates of the multi-echo point cloud data belong to an area of interest, and screening out crown point cloud data of the tree to be detected;
s3, dividing the crown point cloud data into three types according to the screened crown point cloud data, calculating the grid area corresponding to each type of crown point cloud data, and accumulating to obtain the total grid area;
and S4, substituting the total grid area of the tree to be detected into a linear regression model to obtain the total leaf area.
Further, in step S1, the laser radar scans an angle ranging from-135 ° to 135 °, and a two-dimensional cross section, i.e., a scanning line, is obtained by scanning, the scanning plane is perpendicular to the linear movement track of the laser radar, and the scanning resolution is Δ θ.
Further, in step S1, the moving speed of the moving vehicle is smaller than the maximum moving speed vmax(ii) a The measuring distance between the moving vehicle and the tree to be measured is less than the maximum value d of the scanning distancemaxMaximum value v of said moving speedmaxAnd maximum value of scanning distance dmaxRespectively adopting the following formulas to calculate:
Figure BDA0003211052490000021
Figure BDA0003211052490000022
wherein, Δ leaf represents the smaller value of the length and the width of the medium-sized blade on the tree to be measured, Δ α represents the radian corresponding to the scanning resolution of the laser radar Δ θ, and Δ t represents the scanning period of the laser radar.
Further, step S2 specifically includes the following steps:
s2-1, establishing a rectangular coordinate system O-xyz by taking the initial position of the laser radar as a coordinate origin O, wherein the x-axis direction is the moving direction of the laser radar along with the vehicle, the y-axis direction is the scanning depth direction of the laser radar, and the z-axis direction is the height direction of the scanned target perpendicular to the ground; converting the multi-echo point cloud data into a three-dimensional coordinate system according to the following formula:
Figure BDA0003211052490000031
wherein j represents the frame number of the point cloud, i represents the frame number, v represents the moving speed of the laser radar, i.e. the moving vehicle, delta t represents the scanning period of the laser radar, rn(i, j) represents the distance of the nth echo at the ith position in the jth frame, n is 1 and 2, theta (i) represents the radian corresponding to the point cloud at the ith position in each frame, and xn(i, j) represents the x-axis coordinate of the nth echo at the ith position in the jth frame, yn(i, j) represents the y-axis coordinate of the nth echo at the ith position in the jth frame, zn(i, j) represents the z-axis coordinate of the nth echo at the ith position in the jth frame;
s2-2, for the established three-dimensional coordinate system, according to the region of interest ROI of the x axis, the y axis and the z axis, judging whether the ith position in the jth frame is the crown point cloud position by adopting the following formula: if yes, recording as crown point cloud data;
Figure BDA0003211052490000032
wherein x isminAnd xmaxIn the range of the x-axis, yminAnd ymaxIn the range of the y-axis, zminAnd zmaxIs the z-axis range.
Further, xminIs 0, xmaxIs the transverse width of the crown, yminIs the minimum depth distance, y, from the outer edge of the crown to the vehicle-mounted two-dimensional laser scanning sensormaxIs the maximum depth distance of the crown to the sensor, zminIs the distance difference between the bottom of the crown and the sensor in the vertical direction, and has a negative value and a range of zmaxIs the distance difference between the top of the crown and the sensor in the vertical direction.
Further, step S3 is specifically:
s3-1, dividing crown point cloud data into three types according to the following formula:
Figure BDA0003211052490000041
wherein C (I, j) represents the total point cloud of two echoes of the ith position in the jth frame, I1(I, j) represents the intensity of the first echo at the ith position in the jth frame, I2(i, j) represents the intensity of the second echo at the ith position in the jth frame, p2(i, j) represents the point cloud of the second echo at the ith position in the jth frame, and the type I represents that the laser point completely hits on the same blade and only has one echo; type II shows that laser points are partially shot on the leaves, partially penetrate through the tree crown and have two echoes; type iii indicates that the laser spot hits the edge of the overlapping blade. S3-2, calculating the area of each grid in the crown point cloud data by adopting a grid area method:
Figure BDA0003211052490000042
wherein A isSGThe area of a single grid is represented, and the Delta alpha represents the radian corresponding to the scanning resolution of the laser radar in Delta theta;
s3-2, accumulating the areas of all grids of the three types to obtain the total grid area ATG：
ATG＝∑AsG。
Further, in step S4, the linear regression model is obtained by:
s4-1, selecting a plurality of trees as training samples in the area to be tested, and acquiring the total grid area of each training sample according to the steps S1-S3;
s4-2, respectively obtaining a true value of a leaf area for each training sample;
s4-3, substituting the real values of the total grid area and the leaf area of the training sample into a first-order linear regression model to obtain k and b through fitting;
ATL＝kATG+b
wherein A isTLRepresents the true value of the leaf area, ATGRepresenting the total grid area, and k and b represent the coefficients of the first order linear model employed.
The invention has the beneficial effects that:
the method for measuring the total leaf area of the crown based on the multi-echo LiDAR utilizes the LiDAR multi-echo principle to process and classify crown point cloud data acquired by a mobile laser radar scanning system, respectively calculates the grid area of the classified point cloud data by a grid area method, and then obtains the total leaf area by introducing a trained regression model
Additional features and advantages of the invention will be set forth in the detailed description which follows.
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The above and other objects, features and advantages of the present invention will become more apparent by describing in more detail exemplary embodiments thereof with reference to the attached drawings, in which like reference numerals generally represent like parts throughout.
Fig. 1 shows one of the scanning modes of the mobile lidar in accordance with the present invention.
Fig. 2 shows a second schematic diagram of the scanning mode of the mobile lidar in the present invention.
FIG. 3 shows a schematic diagram of a target tree crown point cloud in a three-dimensional coordinate system of a mobile laser radar according to the present invention.
FIG. 4 shows a crown point cloud echo situation classification diagram in the invention.
Fig. 5 shows a schematic diagram of the grid area determination in the present invention.
FIG. 6 shows a general tree crown point cloud grid.
FIG. 7 shows a plot of the grid area and leaf area truth fits of the present invention.
Detailed Description
Preferred embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. While the preferred embodiments of the present invention are shown in the drawings, it should be understood that the present invention may be embodied in various forms and should not be limited to the embodiments set forth herein.
The invention provides a method for measuring total leaf area of a crown based on multi-echo LiDAR, which comprises the following steps:
s1, loading the laser radar on the moving vehicle, enabling the moving vehicle to pass through the tree to be detected, and acquiring multi-echo-point cloud data of the tree by using the laser scanning sensor, wherein the multi-echo-point cloud data comprises the distance r between two echoes1、r2Intensity of echo I1、I2And an angle θ; the scanning angle range of the laser radar is-135 degrees, a two-dimensional section, namely a scanning line, is obtained through scanning, the scanning plane is perpendicular to the linear moving track of the laser radar, and the scanning resolution is delta theta.
As shown in fig. 1 and 2, in step S1, the moving speed of the moving vehicle is smaller than the maximum moving speed vmax(ii) a The measuring distance between the moving vehicle and the tree to be measured is less than the maximum value d of the scanning distancemaxMaximum value v of said moving speedmaxAnd maximum value of scanning distance dmaxRespectively adopting the following formulas to calculate:
Figure BDA0003211052490000061
Figure BDA0003211052490000062
wherein, Δ leaf represents the smaller value of the length and the width of the medium-sized blade on the tree to be measured, Δ α represents the radian corresponding to the scanning resolution of the laser radar Δ θ, and Δ t represents the scanning period of the laser radar.
S2, for the scanned multi-echo point cloud data, establishing a three-dimensional coordinate system based on the starting position of the laser radar, as shown in fig. 3, determining whether the three-dimensional coordinate of the multi-echo point cloud data belongs to an area of interest, and screening out crown point cloud data of the tree to be detected, which specifically includes the following steps:
s2-1, establishing a rectangular coordinate system O-xyz by taking the initial position of the laser radar as a coordinate origin O, wherein the x-axis direction is the moving direction of the laser radar along with the vehicle, the y-axis direction is the scanning depth direction of the laser radar, and the z-axis direction is the height direction of the scanned target perpendicular to the ground; converting the multi-echo point cloud data into a three-dimensional coordinate system according to the following formula:
Figure BDA0003211052490000063
wherein j represents the frame number of the measurement point of the point cloud, i represents the frame number, taking the speed of the moving vehicle as 0.104m/s as an example, when the moving vehicle scans the distance of 1.5m at the speed, the laser radar acquires 560 frames of scanning data of a single sample tree, and 1081 pieces of point cloud data of the sequence number per frame; v represents the speed of movement of the laser radar, i.e. the moving vehicle, Δ t represents the scanning period of the laser radar, rn(i, j) represents the distance of the nth echo at the ith position in the jth frame, n is 1 and 2, theta (i) represents the radian corresponding to the point cloud at the ith position in each frame, and xn(i, j) represents the x-axis coordinate of the nth echo at the ith position in the jth frame, yn(i, j) represents the y-axis coordinate of the nth echo at the ith position in the jth frame, zn(i, j) represents the z-axis coordinate of the nth echo at the ith position in the jth frame;
s2-2, for the established three-dimensional coordinate system, according to the region of interest ROI of the x axis, the y axis and the z axis, judging whether the ith position in the jth frame is the crown point cloud position by adopting the following formula: if yes, recording as crown point cloud data;
Figure BDA0003211052490000071
wherein x isminAnd xmaxIn the range of the x-axis, yminAnd ymaxIn the range of the y-axis, zminAnd zmaxIs the z-axis range; x is the number ofminIs 0, xmaxIs the transverse width of the crown, yminIs the minimum depth distance, y, from the outer edge of the crown to the vehicle-mounted two-dimensional laser scanning sensormaxIs the maximum depth distance of the crown to the sensor, zminIs the distance difference between the bottom of the crown and the sensor in the vertical direction, and has a negative value and a range of zmaxThe top of the crown is vertical to the sensorDifference in upward distance.
S3, dividing the crown point cloud data into three types according to the screened crown point cloud data, calculating the grid area corresponding to each type of crown point cloud data, and accumulating to obtain the total grid area, wherein the method specifically comprises the following steps:
s3-1, as shown in FIG. 4, when the two-dimensional laser radar scans the crown, the crown point cloud data is divided into three types according to the following formula:
Figure BDA0003211052490000072
wherein C (I, j) represents the total point cloud of two echoes of the ith position in the jth frame, I1(I, j) represents the intensity of the first echo at the ith position in the jth frame, I2(i, j) represents the intensity of the second echo at the ith position in the jth frame, p2(i, j) represents the point cloud of the second echo at the ith position in the jth frame, and the type I represents that the laser point completely hits on the same blade and only has one echo; type II shows that laser points are partially shot on the leaves, partially penetrate through the tree crown and have two echoes; type iii indicates that the laser spot hits the edge of the overlapping blade.
S3-2, as shown in fig. 5 and 5, approximating the measurement area of each point cloud to a rectangle, where a rectangle represents a grid, the area of the grid is the sum of the areas of the rectangles, and the length and width of the rectangle are;
Figure BDA0003211052490000081
where Δ i represents the width of the rectangle and Δ j represents the length of the rectangle;
calculating the area of each grid in the crown point cloud data by adopting a grid area method;
Figure BDA0003211052490000082
wherein A isSGRepresenting the area of the individual grids, Delta alpha representing the laserThe radian corresponding to the scanning resolution delta theta is achieved;
s3-2, accumulating the areas of all grids of the three types to obtain the total grid area ATG：
ATG＝∑ASG。
S4, substituting the total grid area of the tree to be detected into a linear regression model to obtain the total leaf area, wherein the linear regression model is obtained by the following steps:
s4-1, selecting a plurality of trees as training samples in the area to be tested, and acquiring the total grid area of each training sample according to the steps S1-S3;
s4-2, respectively obtaining a true value of a leaf area for each training sample;
s4-3, substituting the real values of the total grid area and the leaf area of the training sample into a first-order linear regression model to obtain k and b through fitting;
ATL＝kATG+b
wherein A isTLRepresents the true value of the leaf area, ATGRepresenting the total grid area, and k and b represent the coefficients of the first order linear model employed.
In the specific implementation:
the test was carried out using a two-dimensional laser radar of UTM-30LX, manufactured by Hokuyo corporation of Japan. The two-dimensional laser radar adopts 905nm infrared rays, obtains measured values at different angles through motor swinging, and has the measuring distance of 0.1-30 m, the measuring precision of +/-30 mm, the scanning range of 270 degrees, the angle resolution of 0.25 degrees and the scanning period of 25 ms. The UTM-30LX acquires 1 frame of data per scan, including 1081 different angular distances and laser reflection intensities, represented by 4 bytes and 2 bytes, respectively.
In the mobile laser radar measuring system adopted in the test, the two-dimensional laser radar UTM-30LX is arranged on a conveyor belt of a 2.5-meter linear guide rail which is 0.55 meter away from the ground. The conveyor belt was driven by a stepper motor (57HBP112AL4) with a constant speed controlled by a PLC (CM 36L-30). The scanning mode is shown in fig. 1 and 2.
Fig. 3 is an example of visualizing the region of interest of the collected point cloud in the established three-dimensional coordinate system by using MATLAB, and effectively removing the point cloud which is not the crown by setting the coordinate range. Wherein x ranges from 0m to 1.35m, y ranges from 0.1m to 3m, and z ranges from-0.05 m to 1.2 m.
Dividing the screened crown point clouds into three categories as shown in fig. 4: the first is that the laser completely hits on one blade and only has one echo; the second condition is that the laser partially hits the blade, partially penetrates the tree crown and has two echoes; the third situation is that the laser partially hits on the first blade and partially hits on the second blade which is blocked, and two echoes exist.
The grid area method is used to approximate the measurement area of each point cloud to a rectangle, as shown in fig. 5 and 6, the tree-shaped total grid composed of rectangles well fits the outline of the tree to be measured, and the total grid area can be obtained by summing the areas of the rectangles.
And fitting the measured total grid area of the training tree with the corresponding true value of the total leaf area, obtaining polynomial coefficients k of 37684.22 and b of-4371.48 of a linear regression model according to a fitting curve shown in fig. 7, and obtaining the total leaf area of the test tree through the regression model.
Having described embodiments of the present invention, the foregoing description is intended to be exemplary, not exhaustive, and not limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments.

Claims (7)

1. A method for measuring total leaf area of a crown based on multi-echo LiDAR is characterized by comprising the following steps:
s1, loading the laser radar on the moving vehicle, enabling the moving vehicle to pass through the tree to be detected, and acquiring multi-echo-point cloud data of the tree by using the laser scanning sensor, wherein the multi-echo-point cloud data comprises the distance r between two echoes1、r2Intensity of echo I1、I2And an angle θ;
s2, establishing a three-dimensional coordinate system for the scanned multi-echo point cloud data based on the initial position of the laser radar, judging whether the three-dimensional coordinates of the multi-echo point cloud data belong to an area of interest, and screening out crown point cloud data of the tree to be detected;
s3, dividing the crown point cloud data into three types according to the screened crown point cloud data, calculating the grid area corresponding to each type of crown point cloud data, and accumulating to obtain the total grid area;
and S4, substituting the total grid area of the tree to be detected into a linear regression model to obtain the total leaf area.
2. The method according to claim 1, wherein in step S1, the laser radar scans an angle ranging from-135 ° to 135 °, and the scanning obtains a two-dimensional section, i.e. a scanning line, the scanning plane is perpendicular to the linear movement track of the laser radar, and the scanning resolution is Δ θ.
3. The method for multi-echo LiDAR-based crown total leaf area measurement according to claim 1, wherein in step S1, the moving speed of the moving vehicle is less than the maximum moving speed vmax(ii) a The measuring distance between the moving vehicle and the tree to be measured is less than the maximum value d of the scanning distancemax。vmaxIs 3.0m/s, dmaxIs 5 m.
4. The method for measuring total leaf area of a crown based on multi-echo LiDAR according to claim 1, wherein step S2 specifically comprises the following steps:
s2-1, establishing a rectangular coordinate system O-xyz by taking the initial position of the laser radar as a coordinate origin O, wherein the x-axis direction is the moving direction of the laser radar along with the vehicle, the y-axis direction is the scanning depth direction of the laser radar, and the z-axis direction is the height direction of the scanned target perpendicular to the ground; converting the multi-echo point cloud data into a three-dimensional coordinate system according to the following formula:
Figure FDA0003211052480000021
wherein j represents the frame number of the point cloud, i represents the frame number, v represents the moving speed of the laser radar, i.e. the moving vehicle, delta t represents the scanning period of the laser radar, rn(i, j) represents the distance of the nth echo at the ith position in the jth frame, n is 1 and 2, theta (i) represents the radian corresponding to the point cloud at the ith position in each frame, and xn(i, j) represents the x-axis coordinate of the nth echo at the ith position in the jth frame, yn(i, j) represents the y-axis coordinate of the nth echo at the ith position in the jth frame, zn(i, j) represents the z-axis coordinate of the nth echo at the ith position in the jth frame;
s2-2, for the established three-dimensional coordinate system, according to the region of interest ROI of the x axis, the y axis and the z axis, judging whether the ith position in the jth frame is the crown point cloud position by adopting the following formula: if yes, recording as crown point cloud data;
Figure FDA0003211052480000022
wherein x isminAnd xmaxIn the range of the x-axis, yminAnd ymaxIn the range of the y-axis, zminAnd zmaxIs the z-axis range.
5. The method of claim 4, wherein x is the total leaf area of the crown based on multiple-echo LiDARminIs 0, xmaxIs 1 to 1.1 times of the transverse width of the crown, yminIs 1 to 1.1 times of the minimum depth distance from the outer edge of the crown to the vehicle-mounted two-dimensional laser scanning sensor, and y ismaxIs 1 to 1.1 times of the maximum depth distance from the crown to the sensor, zminThe absolute value is 1-1.1 times of the distance between the bottom of the crown and the sensor in the vertical direction, and z isminIs a negative value, zmaxIs 1-1.1 times of the distance between the top of the crown and the sensor in the vertical direction.
6. The method for measuring total leaf area of a crown based on multi-echo LiDAR according to claim 4, wherein step S3 specifically comprises:
s3-1, dividing crown point cloud data into three types according to the following formula:
Figure FDA0003211052480000031
wherein C (I, j) represents the total point cloud of two echoes of the ith position in the jth frame, I1(I, j) represents the intensity of the first echo at the ith position in the jth frame, I2(i, j) represents the intensity of the second echo at the ith position in the jth frame, p2(I, j) represents the point cloud of the second echo at the ith position in the jth frame, and the type I represents that the laser point is completely hit on the same blade and only has one echo; the type II shows that the laser points are partially shot on the leaves, partially penetrate through the tree crown and have two echoes; type III indicates that the laser spot hits the edge of overlapping leaves.
S3-2, calculating the area of each grid in the crown point cloud data by adopting a grid area method:
Figure FDA0003211052480000032
wherein A isSGThe area of a single grid is represented, and the Delta alpha represents the radian corresponding to the scanning resolution of the laser radar in Delta theta;
s3-2, accumulating the areas of all grids of the three types to obtain the total grid area ATG：
ATG＝∑ASG。
7. The method for measuring total leaf area of a crown based on multiple-echo LiDAR according to claim 1, wherein in step S4, the linear regression model is obtained by:
s4-1, selecting a plurality of trees as training samples in the area to be tested, and acquiring the total grid area of each training sample according to the steps S1-S3;
s4-2, respectively obtaining a true value of a leaf area for each training sample;
s4-3, substituting the real values of the total grid area and the leaf area of the training sample into a first-order linear regression model to obtain k and b through fitting;
ATL＝kATG+b
wherein A isTLRepresents the true value of the leaf area, ATGRepresenting the total grid area, and k and b represent the coefficients of the first order linear model employed.
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