1. Introduction
Airborne LiDAR is an active remote sensing method that has the benefits of 24 h operation, high penetration, precise range finding, and a quick turnaround. These characteristics have made it very popular for surveying a variety of regions, including urban areas [
1,
2], powerlines [
3,
4], forestry resources [
5,
6], glaciers and snow [
7,
8], and other areas [
9,
10,
11].
Streak tube imaging LiDAR (STIL) is a line array detection system [
12,
13,
14,
15,
16,
17]. The array operating mode increases efficiency, and the streak tube sensor provides a higher resolution. Based on these two advantages, it can work at mid-high altitudes and be applied in underwater detection [
18,
19,
20].
We have developed an STIL as an airborne laser system (ALS) to assess the local landscape and structures at a working distance of 3000 m in Zhaodong, China. According to the mapping mission requirements, the altitude accuracy should be better than 0.3 m. However, the error is more sensitive as the operating distance increases. For instance, a 0.02-degree deviation can cause a one-meter offset.
In the reconstruction process, there are three categories of error: computational, random, and systematic [
21]. During our data processing, the computational errors were consistently smaller than
m. The random error for an ALS was also less than 0.2 m within a range of 30° scan angles and at the operational height of 3000 m [
22]. Both error effects are compounded accuracy requirements. The primary element affecting the mapping mission goal is systematic error.
If installation errors were not taken into account, the elevation accuracy of the point cloud would be lower than one meter during data processing. This falls far short of the predicted outcome.
There are two essential steps to modifying the effect of installations. First, the whole point cloud generated by a single experiment is too large to be processed within a limited time, so we have to select some representative points for optimization. There are generally two options for picking points: one is from the overlapping bands [
23], and the other requires layout homonymy points on the ground [
24].
On the one hand, it is impossible to find a one-to-one correspondence on the ground from the rough point cloud that has not been checked and calibrated; on the other hand, measuring and averaging all points in an area is too time-consuming. Given these two factors, we decided to calibrate the system using data from overlapping bands.
However, the whole set of data from a flight test is too large to be processed in a given period of time. Thus, we need to match the same regions for different point clouds using target features and average them as control points to check parameters.
The second step is solving all installation errors using the points we selected. The most classic algorithm is the least square method (LSM) for this problem [
25]. Unfortunately, this method did not work for our airborne STIL. We have to use nonlinear optimization with global search capabilities.
Particle swarm optimization (PSO) is one of the best-known global search algorithms [
26]; it is widely used in many fields, such as path planning [
27] and neural network training [
28].
This algorithm, based on particle competition, is insensitive to initial values and monotonicity and thus has a strong global solution search capability. These features make it ideal for parametric calibration [
29,
30,
31,
32,
33].
Based on PSO, we finally obtained multiple sets of checking parameters, reducing the elevation error from more than 1 m to 0.3 m.
2. Background Knowledge
2.1. Streak Tube Imaging LiDAR
The STIL workflow is depicted in
Figure 1a. The laser is activated by the sequential pulse and is received by a streak tube image sensor. In this process, four subsystems generate data for reconstruction. The streak tube image detector records the time when echo photons come back, the scanning system provides the scanning angle using a grating ruler, the positioning and orientation system (POS) records the position and posture information at the same time, and the pulse sequential system triggers the three subsystems to start with a delay after the laser beam is emitted.
The STIL works as an ALS, which is operated as shown in
Figure 1b. It is located by the POS, which integrates a GPS and an inertial navigation system (INS). The POS calculates the GPS signals from satellites to its own position and obtains posture information from the INS.
In addition, the scanning mirror swings back along the perpendicular direction of flight, and the laser footprint is formed on the ground, as shown in
Figure 1c, where the green lines represent the laser streaks, and red indicates the scan trajectory. We designed different scanning speeds for forward and reverse directions and only used the forward scanning data. There are some overlapping areas between valid scans, which play an important role when checking the parameters.
The POS, delay, and angle data can be calculated for reconstruction, but the streak image should be converted to distance and field angle. The working principle of the streak tube detector is shown in
Figure 2.
Currently, the impulse delay, incident time, incidence angle, scanning angle, and POS data may all be used to reconstruct the point cloud.
However, there are inclinations among the scanner, the streak tube detector, and the POS during installation. Their installation mistakes make the point cloud reconstruction more challenging and less accurate.
For low-altitude ALS, installation errors have a low impact on the inversion accuracy, such that a rough measurement is sufficient to ensure that the effect of systematic errors is smaller than the effect of random errors. Unfortunately, as the working height increases, the installation errors are amplified during the point cloud-inversion process. The placement parameters obtained from ground measurements cannot meet the accuracy requirements of point cloud reconstruction.
Therefore, we had to check the system parameters with experimental flight data. These parameters are included in the coordinate transformation matrixes, under which the raw data are reconstructed into a point cloud. This is clearly a global nonlinear problem.
2.2. Particle Swarm Optimization
The PSO algorithm is a computational optimization technique inspired by the social behavior of birds flocking or fish schooling. It was first introduced by James Kennedy and Russell Eberhart in 1995 [
26].
Figure 3 shows an overview of how the PSO works. Firstly, it initializes a swarm of particles with random positions and velocities within the search space. Secondly, it evaluates each particle’s position and determines its fitness value based on the objective function. Then, it starts a cycle that updates the personal best and global best, as well as the velocity if the best solution is worse than the cut-off point.
4. Parameter Optimization
4.1. Point Selection
The whole point cloud is too big for an intelligence algorithm to optimize. As a result, we must choose a few checking sites for iterative calculation. First, the point cloud is reconstructed using the uncalibrated parameters, and then the composite required points are selected from it. After target point selection, the selected points need to be denoised via radius filtering. Finally, the raw data are determined one-to-one with the point cloud using subscripts and selected for self-checking.
Different areas play different roles in the self-checking process, and these points need to be selected using different methods. Target buildings have definite edges that can be boxed to an area with determinant latitudes and longitudes. Further, the flat ground’s altitude slowly changes, so we can use it to adjust the vertical accuracy.
Target buildings are real buildings in a city, which means they appear randomly in the point cloud. It is difficult to find a complete one-to-one mapping building in adjacent strips, but it is easy in two air lines. Therefore, we have three sets of control points:
: target building points between different air routes.
: flat area points among different air routes.
: flat area points between adjacent scanning bands.
4.1.1. Target Buildings
To ensure that the altitude accuracy is meaningful, it is important to have the control points aligned in the latitude and longitude directions. The role of the target buildings is to align the two sets of point clouds in the horizontal direction.
In the inspection area, the buildings suitable for a parametric check have a regular geometry and are at a certain height above the ground. A typical building suitable for checking schools is shown in
Figure 6b.
In order to choose the target building points, a rectangle should be boxed around the building. Then, the points higher than the altitude threshold in the rectangular area should be selected. It must be ensured that the rectangular selection box is flat except for the target building.
Target buildings are used to match the points from different air lines, so the strip splicing error should be eliminated through point-selection work. We have to select buildings that stand on one strip completely, which are always small, as shown in
Figure 6.
There are three sets of targets: the first contains points generated from east to west, the second from north to south, and the third from west to east. The three airlines constitute two control groups: white numbers on the first picture are compared against the second, and black numbers are subtracted by the third.
We checked the accuracy of the latitude and longitude directions using two groups, one of which is reverse-checking and the other is crosschecking. Both of them need four vertexes selected from imprecise point cloud manually.
Depending on the point of the target building, it is possible to make a judgment about the error in the horizontal direction as
where the subscript
denotes the average value of the
th set of point clouds,
correspond to the four different labeled builds in the three figures; x is the relative distance in longitude; and y is the latitude.
4.1.2. Flat Areas among Different Air Routes
There is a further characteristic of the flat mark point that distinguishes it from its building counterpoint. The flat object has no clear edge to locate its horizontal position, and we cannot even make sure that it is an object that contains the same area as another. However, this characteristic results from the slowly changing height, which means horizontal misalignment does not affect the altitude accuracy.
Thus, we selected a square area on streets, especially on crossroads, and made four points in every cloud. The square is more convenient than a target building because it is small enough to stay within the same scanning band. We can find the same area among the three clouds generated from different heading airlines, as shown in
Figure 7.
There are also three sets in
Figure 7, but we painted them in different pseudo colors to show the street contour. Every subgraph has eight marked areas, which contain approximately thirty points and are averaged to just one height. We subtracted the second and third height from the first and gained two groups of altitude difference values as follows:
where
and
are the altitude averages of the points in the
th target area in
Figure 7a–c.
4.1.3. Flat Area for Adjacent Scanning Bands
Error-checking parameters not only lead to misalignment between buildings generated from different airlines but also cause altitude differences between adjacent scanning bands. The points across two strips have a greater field angle, so their photons are incident on the sensor more marginally. Sometimes, half of them are wasted in the reconstruction process, but the continuously high trend from the middle to sides is still useful for checking the installation error.
The area is not wide enough to accommodate complete buildings on overlapped areas, so we have to select flat targets as in the second selection method.
Figure 8 shows the selection positions across scanning bands beneath the east–west airline.
We select a point that spans both strips and draw a circle horizontally with that point as the center and then average the heights of all the points within the circle. For the
th circle, we obtain an average value on each of the front and back strips, denoted as
and
; the error criterion can be expressed as
4.2. Parameter Optimization Using the PSO Algorithm
There are nine parameters that should be optimized, as listed in
Table 1, and we defined them as a checking vector
.
It is obvious that the reconstruction of STIL is a nonlinear process whose inverse problem cannot be solved using the least square method (LSM) based on the matrix form. Furthermore, it is a global optimization process, where the nonlinear LSM falls into the local optimal solution through the direction of the gradient descent.
One of the best ways to avoid local optimums is to introduce randomness, and most intelligent heuristic algorithms work with it. If we set a random step size for the nonlinear LSM, the program evolves into an intelligent algorithm.
The PSO not only uses random steps but also introduces a competition mechanism, a population-based stochastic optimization algorithm inspired by the social behavior of birds flocking or fish schooling. The PSO is commonly used to solve optimization problems where the objective is to find the optimal solution in a search space.
During the optimization of placement parameters, the checking vector
is abstracted into a particle. Since the PSO starts running, 50 checking vectors are generated randomly and comprise the particle swarm. Three types of checking points are reconstructed with every particle, and the weighted average is determined to obtain a matching error shown as
The average value is the distance between two groups of check points in point set , and is its weight.
The particle records its own minimum matching error as , and there is a best particle in the th iteration. Every particle has a in every iteration and moves along a straight line without a matching error. However, it exists in a swarm and is burdened with the task of finding an optimal solution, so it has to move to the historical optimal solution or the ethnic optimal solution .
Thus, the particle
, which is the
th particle in the swarm, in its
th iteration, would have the new speed
.
The vectors
and
are the positions of their corresponding particles;
and
are the global and local constants; and
are randoms. The pseudocode is shown in Algorithm 1.
Algorithm 1: The PSO for installation checking |
Input: 45 pairs of marking points |
Output: 8 parameters |
function pso(points, max_iter, target_precision)
Initialize particles radom(50, 8)
Initialize speed zeros(50,8) Initialize personal best particles = zeros(50, 8). zeros(50) for j = 1 to max_iter for i = 1:50 = test_one_parameter(points, ) if test_one_parameter(points, ) > end if end for if target_precision return end for return end function // calculate the error by one group of parameters function test_one_parameter(points, ) reconstruct points by the parameter return e end function
|
4.3. Optimization Result
4.3.1. Coupling of Placement Parameters
Fifteen optimizations were performed, and the nine checking parameters in each optimization are shown in
Figure 9. The maximum angle error is approximately 5.5°, which is impossible. These large angular errors imply coupling between the nine parameters. Coupling but not simple merging is the biggest difficulty in checking the error of airborne STILs.
There are three directions in the rotation operation, corresponding to the three sets of coupling relationships of the checking parameters, shown as
Their linear fitting diagram is shown as
Figure 10.
The parameter coupling phenomenon indicates a correlation before different placement angles and verifies that the particle swarm algorithm can solve the multi-parameter optimization problem.
Based on the definitions in
Table 1, the coupling parameters can be found to be related in orientation. The nine parameters come from three devices, the imaging system, the scanning system and the POS. Although the orientation of each parameter was not specified during modelling, based on the optimization results, it can be seen that
and
represent the same orientation in all three coordinate systems. The same is true for other parameters. In addition, the scanning system was modelled with redundancy, resulting in Ψ and Ω playing the same role.
The coefficients between the parameters and in these three sets of couplings converge to 1, suggesting that the two may be condensed. The linear coupling coefficients for the other two sets of relationships are not 1, indicating that these parameters cannot be optimized by fitting the relationships. However, according to the model’s definition, the parameters and are both used to describe the error of the motor rotation angle. The grating ruler has been installed co-axially to the rotating shaft so that its zero position is virtually the same physical quantity as the motor placement error in the tangential direction of the rotating shift.
When one or more of these parameters are fixed, the parameters associated with them will also converge to fixed values during the optimization process. In traditional calibration methods, it is necessary to obtain the exact placement error. The phenomenon of parameter coupling, on the other hand, makes the number of parameters that have to be obtained by measurements much lower, which is very beneficial for the traditional checking methods.
Considering that the purpose of checking is to improve the accuracy of the point cloud, the non-uniqueness of the optimum caused by parameter coupling is not a major issue. Even the exact physical meaning of the nine parameters seems less important, as long as they are substituted into the conformal equations to be able to invert a highly accurate point cloud.
4.3.2. Reconstruction Accuracy
An ALS topographic mapping task is essentially a measurement of the topographic relief of an area. In its reconstructed point clouds, latitude and longitude often act as the coordinate frame, while elevation is the measured value. Therefore, we locate the proximity points in different point clouds by giving the latitude and longitude and use the elevation difference between the two as the reconstruction error.
Though three types of points are used in global optimization, the criteria for determining the error are used for elevation.
Finally, the vertical errors are around 0.3 m, different every time when solved through the PSO, and the minimum value is as low as 0.271 m, as shown in
Figure 11. In contrast, whether assuming placement errors of 0 or optimizing using nonlinear LSM starting with random numbers, the elevation errors of the generated point cloud are all greater than 1 m.
On the one hand, the optimization results of the placement errors have huge jitter and are coupled with each other; on the other hand, the point cloud reconstruction results are relatively stable, and their reconstruction errors are controlled within a small range. In fact, although we have been referring to the parameters to be optimized as placement errors, they are clearly not true installation angles. In other words, we have achieved highly accurate point cloud reconstruction without checking parameters.
The placement error obtained by the PSO algorithm produces a significant effect on point cloud reconstruction. As far as the horizontal direction is concerned, it solves the misalignment problem between the strips and reflects the validity of the building target criterion shown in
Figure 12.
The difference before and after PSO is more intuitively shown in
Figure 13, where a clear misalignment before the nonlinear optimization is shown in
Figure 13b. The PSO has smoothed this issue out, as shown in
Figure 13c.