Centroid Extraction of Laser Spots Captured by Infrared Detectors Combining Laser Footprint Images and Detector Observation Data

Abstract

On-orbit geometric calibration of satellite-borne laser based on infrared detectors is the key tool to ensure the elevation measurement accuracy, and the accuracy of on-orbit geometric calibration is directly determined by laser spots captured by detectors. Mathematical methods, such as gray-scale barycenter, are widely applied for centroid extraction of spots captured by infrared detectors and completely depend on the energy values at points measured by detectors, which have low precision and are greatly affected by the consistency of the detectors and other factors at present. Based on the above question, considering the consistency between the real laser footprint shape and spot captured by detectors, a centroid extraction method of laser spots captured by infrared detectors combining laser footprint images and detector observation data is proposed for making up this defect to some extent. First, the self-adaptive “two-step method” is used to denoise footprint images hierarchically to obtain the real shape of footprints for constraining the spots captured by detectors, and then the centroids of spots are extracted by using the energy-weighted barycenter method based on regional blocks. In the experiment, Gaofen-7 (GF-7) satellite is taken as the research object, and the proposed method, as well as the other six methods, are used for the centroid extraction of laser spots captured by detectors, the calculation of calibration parameters based on the single-beam and dual-beam laser calibration models, the positioning of laser footprints, and cross verification. According to the results, the plane accuracy of centroid extraction using the proposed method is as follows: 0.34 grids for Beam 1 and 0.33 grids for Beam 2. In addition, on flat terrain, the elevation accuracy of Beam 1 and Beam 2 in 2021 is 5.2 cm and 5.0 cm, respectively, 0.6 cm and 4.2 cm higher than those in the most accurate one among other methods; the elevation accuracy in 2020 is 23.3 cm and 7.1 cm, respectively, 7.7 cm and 2.7 cm higher than those in the most accurate one among other methods. On slopes and gentle slopes, the method proposed is also superior to other methods. Since the change of pointing angle caused by satellite jitter, atmosphere, etc., between different years, the accuracy drops when laser footprints of 2020 are located using the parameters of 2021. In summary, under different terrains and years, the results fully demonstrate the effectiveness and accuracy of the proposed method, which has more significant advantages than other traditional methods.

1. Introduction

Geometric positioning accuracy is one of the most important indexes to measure the performance of China’s remote sensing satellites [1]. Due to its high accuracy for geometric positioning, the satellite-borne laser altimeter has been widely used in deep space exploration and earth observation [2]. The United States, for example, launched ICESat-1 (Ice, Cloud, and land Elevation Satellite-1) carrying GLAS (Geoscience Laser Altimeter System) linear laser altimeter system [3,4] and ICESat-2 (Ice, Cloud, and land Elevation Satellite-2) carrying ATLAS (Advanced Topographic Laser Altimeter System) single-photon laser altimeter system in 2003 and 2018 successively. On the other hand, with the development of China‘s laser technology, satellite-borne laser altimeters have been increasingly used in earth observation. In May 2016, Ziyuan 3-02 (ZY3-02) satellite was successfully launched, which carried an experimental laser altimeter for the first time [5]. In November 2019, the GF-7 satellite carrying China’s first civilian laser altimeter system for a formal application to earth observation was also sent into orbit successfully [6,7,8].

For the GF-7 satellite, its laser altimeter system comprises four laser devices (including two principal devices powered on by default, hereinafter referred to as “Beam 1” and “Beam 2”, and the other two are placed as auxiliaries that are not powered on by default; each device can work independently), two laser footprint cameras (LFC) and one laser optical axis surveillance camera (LOASC) [9], as shown in Figure 1, mainly used to assist the optical stereo camera in improving the positioning accuracy of uncontrolled mapping [10]. The pointing accuracy of satellite-borne laser is the key factor affecting the laser footprint positioning accuracy, and on-orbit geometric calibration of satellite-borne laser is an important means to ensure the laser pointing accuracy. During the on-orbit geometric calibration of the GF-7 satellite, the centroid of the spots captured by the infrared detectors is taken as the ground control points (GCPs) to complete the calculation of calibration parameters and the positioning of laser footprints [11,12,13]. That is, whether the centroid of the laser spot captured by the infrared detectors can be extracted with high-precision directly affects whether the calibration of the GF-7 satellite has high precision GCPs, and further affects the accuracy of the geometric calibration. Therefore, how to obtain high-precision calibration laser spot centroids becomes fundamental to improving their calibration accuracy.

Van Waerbeke et al. [14] proposed a centroid extraction algorithm based on point light sources which could obtain the spot shape with geometric parameters. However, this method presents relatively low extraction accuracy for large-spot centroids. Zhan et al. [15] put forward a spherical circle fitting algorithm based on the method of spherical fitting by mapping the peripheral points of a solar image into the object space, but its performance is poor when the noise is loud. Wang et al. [16] proposed to position centroids by using Gaussian gray diffusion model, and it shows a relatively good performance in data simulation but has low data reliability under complex circumstances. For laser spot centroid extraction of GF-7, Ren et al. [17] combined the Gaussian fitting method, Canny edge detection, and gray-scale barycenter method, but the accuracy drops when spots are largely affected by background features. Yao et al. [18] proposed the ellipse fitting centroid extraction algorithm with threshold constraints, but this method heavily depends on the experience threshold in the denoising process with relatively low accuracy in the case of low spot amplitude. Song et al. [19] put forward the centroid extraction method of least square ellipse fitting, combining the cumulative matrix and taking problems including “pseudo-spot” and “weak spot” into consideration, but it could result in large calculation errors due to asymmetrical spot intensity and blurred edges.

The above researches mainly aim to extract laser spot centroids in optical remote-sensing images. For the geometric calibration of satellite-borne laser altimeters, calibration based on infrared detectors is one of the most accurate methods [9]. However, researches on centroid extraction based on spots captured by infrared detectors is few. As the centroid of the spot captured by infrared detectors can be used as ground control to realize the geometric calibration of satellite laser, it is very important to improve the accuracy of centroid extraction. On the other hand, traditional mathematical methods, including gray-scale barycenter and Gaussian surface fitting, which are directly applied to centroid extraction of spots captured by infrared detectors, may result in the accuracy being greatly affected by factors such as detector consistency. The so-called “detector consistency” refers to the response of different detectors to the same energy density. When the quantitative results from different detectors for the same energy are identical, it means that the consistency is good, i.e., the good consistency of the detectors ensures that more accurate and real laser footprint data can be obtained. However, in reality, mass-produced detectors are subject to a certain degree of consistency error due to various factors. And the specific performance is that, theoretically, the energy in the laser footprint obeys Gaussian-like distribution. If the consistency is poor, the energy value on the side displayed by the detector will be greater than the value in the middle, leading to a decrease in the accuracy of centroid extraction. In addition, atmospheric turbulence, atmospheric refraction, and detector layout have effects on the accuracy of centroid extraction of spots captured by infrared detectors.

In order to resolve the above-said defects, a centroid extraction method of laser spots captured by infrared detectors combining laser footprint images and detector observation data is proposed. As the laser footprint image and detector observation data have the characteristics of consistency under the same platform, time, and atmospheric conditions, the proposed method can make full use of such characteristics to establish the correlation between the laser footprint and the spot, thus removing the influence of detector consistency and other problems, and improving the accuracy of spot centroid extraction. To be more specific, first, the interference of particle noise and background features on the laser footprint image is removed by hierarchical denoising with the self-adaptive “two-step method”, allowing to obtain the real geometric shape of the laser footprint. Then, the real geometrical shape is mapped to the spot captured by infrared detectors, followed by the spot being constrained. At last, the centroid of the constrained spot is extracted with the help of the energy-weighted barycenter method based on regional blocks.

2. Materials and Methods

2.1. Study Areas and Data Sources

2.1.1. Experimental Data

In this paper, GF-7 satellite-borne laser altimeter is selected as the experimental object, and the experimental data, supplemented by footprint images, are laser spots captured by infrared detectors. After comprehensively considering multiple influential factors, including terrain and weather, on the accuracy of the on-orbit geometric calibration of satellite-borne laser altimeter, the flat area near Sonid Right Banner (equivalent to a county-level administrative division), Inner Mongolia, China was selected as the experimental field. In June 2020 and September 2021, large-scale on-orbit geometric calibrations of GF-7 satellite lasers were completed using infrared detectors. GF-7 satellite-borne laser footprints were successfully captured by detectors during the process. The laser calibration and data distribution of the experimental field is shown in Figure 2, the layout of detectors is shown in Figure 2b, and the discrete values of energy captured are shown in Figure 2c. In addition, the centroids of laser footprints captured by infrared detectors in Figure 2a were extracted using seven methods, including the method proposed. The centroid extraction results of footprints 241743151.00 and 241743151.33 were used as GCPs to complete the on-orbit geometric calibration of Beam 1 by infrared detectors, and both were successfully captured in Track 10119 on 29 August 2021. Footprints 243038830.00 and 243038830.33 were successfully, captured in Track 10412 on 13 September 2021, which were used for the geometric calibration of Beam 2. Subsequently, centroid extraction for all footprints was carried out and completed based on different methods, and the results were then cross-verified with the positioning results obtained after the geometric calibration of infrared detectors.

2.1.2. Verification Data

The laser footprint positioning accuracy is an important index to evaluate the accuracy of spot centroid extraction based on infrared detectors [20]. Therefore, to verify the centroid extraction accuracy using numerous methods, including the one proposed in this paper, based on the extracted centroids of spots captured by infrared detectors, calibration parameters were calculated, and laser footprints were positioned by using single-beam and double-beam laser calibration models. The positioning accuracy of laser footprints was evaluated according to a large amount of data, indirectly verifying the accuracy of seven methods for centroid extraction, including the method proposed. First, based on the spots obtained by the on-orbit geometric calibration in the experimental field of Sonid Right Banner, as shown in Figure 2a, the results of centroid extraction and laser footprint positioning were used for cross-verification. Second, on account of all spots obtained during the periods of on-orbit geometric calibration in 2020 and 2021, respectively, GCPs were collected using real-time kinematic (RTK) technology to form 9 × 9 ground control grids around the spots, as shown in Figure 3a. Then, in terms of the ground control grids, the real surface elevation was interpolated to evaluate the elevation of the laser footprint, indirectly verifying the elevation accuracy of centroid extraction results obtained using multiple methods. At last, the calibration parameters calculated with the centroid extraction results were extrapolated to laser data in other tracks, in which the data of Track 5278 passing Kuqa, Xinjiang Uygur Autonomous Region on 14 October 2020 was selected to complete laser footprint positioning based on calibration parameters. The high-precision airborne light detection and ranging (LiDAR) improved and verified for accuracy by GCPs [21,22], as shown in Figure 3c, was utilized to evaluate the elevation positioning accuracy of laser footprints, indirectly verifying the elevation accuracy of centroid extraction results from multiple methods.

The laser data were screened within the experimental field and then obtained 23 effective laser points of Beam 1 and 24 effective laser points of Beam 2, as shown in Figure 3b.

The experimental and validation data used in this study are shown in

Table 1. Experimental and validation data used in this study.

Purpose	Data	Beam	Time	Filename	Timecode of Laser Footprint (s)
Calibration	GCPs	Beam 1	29 August 2021	SYC-GF7-20210830-010119-0000011637_1	241,743,151.00, 241,743,151.33
	GCPs	Beam 2	13 September 2021	MYC-GF7-20210914-010412-000011921_2	243,038,830.00, 243,038,830.33
Positioning verification (method one)	GCPs	Beam 1	29 August 2021	SYC-GF7-20210830-010119-0000011637_1	241,743,151.00, 241,743,151.33
	GCPs	Beam 2	13 September 2021	MYC-GF7-20210914-010412-000011921_2	243,038,830.00, 243,038,830.33
Positioning verification (method two)	CPs	Beam 1	14 June 2020 19 June 2020 24 June 2020	SYC-GF7-20200614-003402-0000003397_1 SYC-GF7-20200619-003478-0000003529_1 KSC-GF7-20200624-003555-0000003615_1	203,600,254.00, 204,032,177.00, 204,464,095.00
	CPs	Beam 2	14 June 2020 19 June 2020	SYC-GF7-20200614-003402-0000003397_1 SYC-GF7-20200619-003478-0000003529_1	203,600,253.33, 204,032,175.67
Elevation verification	RTK(CPs)	Beam 1	14 June 2020 19 June 2020 24 June 2020 29 August 2021 3 September 2021 13 September 2021	SYC-GF7-20200614-003402-0000003397_1 SYC-GF7-20200619-003478-0000003529_1 KSC-GF7-20200624-003555-0000003615_1 SYC-GF7-20210830-010119-0000011637_1 MYC-GF7-20210903-010194-000011739_1 MYC-GF7-20210914-010412-000011921_1	203,600,253.67, 203,600,254.00, 204,032,175.67, 204,032,176.00, 204,032,176.33, 204,464,095.00, 204,464,097.33, 204,464,098.00, 241,743,149.00, 241,743,149.33, 241,743,149.66, 241,743,150.00, 241,743,150.33, 241,743,150.66, 243,038,830.33, 243,038,830.66, 242,175,040.66, 242,175,041.00.
		Beam 2	14 June 2020 19 June 2020 24 June 2020 29 August 2021 13 September 2021	SYC-GF7-20200614-003402-0000003397_2 SYC-GF7-20200619-003478-0000003529_2 KSC-GF7-20200624-003555-0000003615_2 SYC-GF7-20210830-010119-0000011637_2 MYC-GF7-20210914-0104120000011921_2	203,600,253.00, 203,600,253.33, 204,032,175.33 204,032,175.67, 204,032,176.00, 204,464,097.33 204,464,099.33, 204,464,099.67, 241,743,147.33, 241,743,148.00, 241,743,148.33, 241,743,154.00, 241,743,154.33, 243,038,830.67, 243,038,831.00, 243,038,831.67, 243,038,832.00, 243,038,832.33
	Laser Footprints	Beam 1	14 October 2020	KRN-GF7-20201014-005278-0000005703_1	214,148,011.33, 214,148,011.66, 214,148,012.33, 214,148,012.66, 214,148,013.00, 214,148,013.33, 214,148,013.66, 214,148,014.00, 214,148,014.33, 214,148,014.66, 214,148,015.33, 214,148,015.66, 214,148,016.00, 214,148,016.33, 214,148,016.66, 214,148,017.00, 214,148,017.66, 214,148,018.00, 214,148,018.33, 214,148,018.66, 214,148,019.00, 214,148,019.33, 214,148,020.00.
		Beam 2	14 October 2020	KRN-GF7-20201014-005278-0000005703_2	214,148,011.00, 214,148,011.33, 214,148,011.67, 214,148,012.00, 214,148,012.33, 214,148,013.33, 214,148,013.67, 214,148,014.00, 214,148,014.33, 214,148,014.67, 214,148,015.33, 214,148,015.67, 214,148,016.00, 214,148,016.33, 214,148,016.67, 214,148,017.00, 214,148,017.33, 214,148,017.67, 214,148,018.00, 214,148,018.33, 214,148,018.67, 214,148,019.00, 214,148,019.33, 214,148,019.67
	DSM	Beam 1/ Beam 2	10 October 2020	High precision airborne LIDAR DSM	214,148,011.33, 214,148,011.66, 214,148,012.33, 214,148,012.66, 214,148,013.00, 214,148,013.33, 214,148,013.66, 214,148,014.00, 214,148,014.33, 214,148,014.66, 214,148,015.33, 214,148,015.66, 214,148,016.00, 214,148,016.33, 214,148,016.66, 214,148,017.00, 214,148,017.66, 214,148,018.00, 214,148,018.33, 214,148,018.66, 214,148,019.00, 214,148,019.33, 214,148,020.00, 214,148,011.00, 214,148,011.33, 214,148,011.67, 214,148,012.00, 214,148,012.33, 214,148,013.33, 214,148,013.67, 214,148,014.00, 214,148,014.33, 214,148,014.67, 214,148,015.33, 214,148,015.67, 214,148,016.00, 214,148,016.33, 214,148,016.67, 214,148,017.00, 214,148,017.33, 214,148,017.67, 214,148,018.00, 214,148,018.33, 214,148,018.67, 214,148,019.00, 214,148,019.33, 214,148,019.67

. Among them, the spots captured by the infrared detectors that are not used for on-orbit geometric calibration and the GCPs acquired by RTK are called control points (CPs). CPs are used as verification points to verify accuracy [23].

2.2. Centroid Extraction of Laser Spots Captured by Infrared Detectors Combining Laser Footprint Images and Detector Observation Data

At present, there are few studies on the centroid extraction methods of satellite laser calibration spots based on infrared detectors. Generally, mathematical methods such as gray-scale weighting are applied directly, which completely depend on the energy value at points in the footprints actually measured by detectors, making the accuracy affected by multiple factors such as detector consistency, detector layout spacing, and atmospheric turbulence. Therefore, as the footprint camera is able to form an image of a laser footprint in the laser altimeter system, and the fact that real laser footprints obtained based on the same platform at the same time in an ideal state are consistent with the spots captured by infrared detectors, a method named centroid extraction of laser spots captured by infrared detectors combining laser footprint images and detector observation data is proposed. The basic schematic diagram is shown in Figure 4.

As shown in Figure 4a, the original footprint images of GF-7 satellites are inevitably affected by factors such as the state of laser footprints, background features, etc., which often lead to the deviation of energy from Gaussian-like distribution and unclear boundaries [24]. And to obtain the real geometrical shape of footprints, denoising is necessary. Therefore, in the experiment, the real laser footprint shape was obtained by hierarchical denoising with a self-adaptive “two-step method”, as shown in Figure 4b. Based on the imaging characteristics of the GF-7 footprint camera, the denoised footprint image was rotated about the Z axis. Then ellipse fitting was conducted on the rotated image based on least squares, and the ellipse shown in Figure 4c was abstracted—where $a$ and $b$ are the major and minor semi-axes of the ellipse, and $\theta$ is the inclination angle of the ellipse. Based on the consistency between the real laser footprint and the spot captured by infrared detectors, the geometrical shape of the real laser footprint in Figure 4c was mapped to the spot captured by infrared detectors. In view of the close spatial relationship between points with the largest/second largest energy value in the spot and its centroid, the ellipses A and B were drawn by centering on the two points, combined with geometric parameters of the real footprint. Consequently, red, green, and yellow areas were formed by the intersection of two ellipses, as shown in Figure 4d. However, as the detectors were spaced, it could only be assumed that the centroid was positioned between the points with the largest/second largest energy value and was closer to the point with the largest value. And to further improve the centroid extraction accuracy, the relation of the distance and energy between the centroid and detectors was comprehensively considered, allowing the energy values in different areas to be given different weights to extract the spot centroid using the energy-weighted barycenter method based on regional blocks. The basic flow chart of this method is shown in Figure 5.

2.2.1. Hierarchical Denoising for Laser Footprint Images with Self-Adaptive “Two-Step Method”

Hierarchical denoising with a self-adaptive “two-step method” includes coarse denoising for footprint images based on median filtering and self-adaptive threshold and fine denoising for footprint images based on the Gaussian model.

• Coarse denoising:

There is usually a large amount of granular noise in the satellite-borne laser spot, which needs to be removed [17,25]. And commonly used methods for image denoising include Gaussian filtering, median filtering, etc., among which median filtering performs better in removing granular noise [26].

$$F’\left(x,y\right)=\mathit{median}\left[F\left(x-m,y-m\right)\right]$$

(1)

where $F’\left(x,y\right)$ is the median of pixel gray level in the neighborhood of $\left(x,y\right)$; $m$ is the length of the filtering window; $median[]$ is the median of all pixel gray levels in the neighborhood range. In view of both denoising and image detail protection, $m=3$.

After the granular noise is removed, each pixel can be treated as the overlaying effect of the spot and ground feature. The footprint image section in the place of the spot can be extracted. In the experiment, after actual calibration, the lengths of Beam 1 and Beam 2 did not exceed 25 m, the ratio of the major axis to the minor axis was less than 1.5, and the number of major axis pixels was 10–18 pixels [24]. Therefore, taking the point with the largest brightness as the center, image sections of 18 pixels × 14 pixels and 20 pixels × 16 pixels were selected, respectively. Then, the average brightness in the annular part formed by the two sections was calculated as the self-adaptive threshold for coarse denoising.

2.

Fine denoising:

The beam width of Gaussian beams usually has two definitions, i.e., the width at $\frac{1}{\mathit{exp}(2)}$ and 1/2 of the peak intensity [27]. In this paper, the beam width is defined by treating $\frac{1}{\mathit{exp}(2)}$ of the peak intensity as the threshold, and the Gaussian function is established for the footprint spot, as shown in Formula (2):

$$F\left(x,y\right)=A\mathit{exp}\left[-\frac{{\left(x-x0\right)}^2}{{\sigma }_x^2}-\frac{{\left(y-y0\right)}^2}{{\sigma }_y^2}\right]$$

(2)

where $A$ is the amplitude of the laser spot; $\left(x_0,y_0\right)$ is the corresponding centroid coordinates of the spot; ${\sigma }_x$ and ${\sigma }_y$ are standard deviations at the directions of $x$ and $y$; $F\left(x,y\right)$ is the corresponding gray value of the image point with coordinates $\left(x,y\right)$. The Gaussian peak value is obtained according to Formula (2). Fine denoising for footprint image spot is completed according to Formula (3) by combining the Gaussian peak value and $\frac{1}{\mathit{exp}(2)}$ of the peak value.

$$I\left(x,y\right)=\left\{\begin{array}{cc}\hfill 0,\hfill & F\left(x,y\right)<F\left(x_0,y_0\right)\times \frac{1}{\mathit{exp}\left(2\right)}\hfill \\ F\left(x,y\right),\hfill & F\left(x,y\right)\ge F\left(x_0,y_0\right)\times \frac{1}{\mathit{exp}\left(2\right)}\hfill \end{array}\right.$$

(3)

where $F\left(x_0,y_0\right)$ is the peak value obtained through Gaussian surface fitting for the footprint spot; $I\left(x,y\right)$ is the pixel gray value at $\left(x,y\right)$ of the footprint image after fine denoising. The result after fine denoising is shown in Figure 3b.

2.2.2. Geometric Parameters Extraction for Real Laser Footprints

Real laser footprints obtained by hierarchical denoising with the “two-step method” are combined with least squares for ellipse fitting.

$$A{X}^2+BXY+C{Y}^2+DX+EY+F=0$$

(4)

where $A,B,C,D,E and F$ are parameters of the ellipse; according to ellipse fitting based on least squares, a general equation of the ellipse is derived. Then the coordinates of the center of an ellipse are obtained as the coordinates of the spot centroid [28], and are expressed as:

$$X_0=\frac{BE-2CD}{4AC-B^2}$$

(5)

$$Y_0=\frac{BD-2AE}{4AC-B^2}$$

(6)

$$a=\sqrt{\frac{2\left(A{X}_0{}^2+C{Y}_0{}^2+B{X}_0{Y}_0-F\right)}{A+C-{({(A-C)}^2+B^2)}^{1/2}}}$$

(7)

$$b=\sqrt{\frac{2\left(A{X}_0{}^2+C{Y}_0{}^2+B{X}_0{Y}_0-F\right)}{A+C+{({(A-C)}^2+B^2)}^{1/2}}}$$

(8)

$$\theta =1/2arctan\left(B/\left(A-C\right)\right)$$

(9)

where $X_0,Y_0$ is the coordinates of the center of an ellipse obtained through a calculation based on ellipse parameters; $a$ and $b$ are the major and minor semi-axes of the ellipse.

As shown in Figure 3c, parameters including the ratio of the major axis to the minor axis of the ellipse and inclination angle obtained according to the above formula were adopted as the constraints for spots captured by infrared detectors.

2.2.3. After Constraining the Spot Captured by Detectors, the Centroid of the Spot Was Extracted by Energy Weighted Barycenter Method Based on Regional Blocks

Taking the points with the largest/second largest energy value in the spot as the centers, the ellipses were drawn by combining the geometric parameters of real laser footprints. Then the red, green, and yellow areas were formed, as shown in Figure 3d. Only detector energy values in the three areas were selected as experimental data for centroid extraction. Based on the energy and space relationship of centroid position and detector energy, different weights were given to the detector energy values in different regions, as shown in Formula (10).

$$\omega \left(i,j\right)=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill \left(i,j\right)\in \left(F_a\cap F_b\right)\hfill \\ \frac{F_{\mathit{a\_max}}}{\left(F_{\mathit{a\_max}}+F_{\mathit{b\_max}}\right),}\hfill & \left(i,j\right)\in F_a-\left(F_a\cap F_b\right)\hfill \\ \frac{F_{b\_\mathit{max}}}{\left(F_{a\_\mathit{max}}+F_{b\_\mathit{max}}\right),}\hfill & \left(i,j\right)\in F_b-\left(F_a\cap F_b\right)\hfill \end{array}\right.$$

(10)

where $\omega \left(i,j\right)$ is the corresponding weight of the detector energy value at $\left(i,j\right)$; $F_a$ is the set of coordinates contained in the ellipse with the largest energy value as the center; $F_b$ is the set of coordinates contained in the ellipse with the second largest energy value as the center; $F_{a\_\mathit{max}}$ is the largest energy value in $F_a$; $F_{b\_\mathit{max}}$ is the largest energy value in $F_b$.

Based on the above different weights, the centroid of the spot was extracted by using the energy weighted barycenter method based on regional blocks, as shown in Formula (11).

$$\begin{gathered} X_{GCP}=\frac{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^{N}i\cdot \omega \cdot I\left(i,j\right)}{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^N\omega \cdot I\left(i,j\right)} \\ {Y}_{GCP}=\frac{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^{N}j\cdot \omega \cdot I\left(i,j\right)}{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^N\omega \cdot I\left(i,j\right)} \\ {Z}_{GCP}=\frac{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^{N}h\cdot \omega \cdot I\left(i,j\right)}{{{\displaystyle \sum }}_{i=1}^M{{\displaystyle \sum }}_{j=1}^N\omega \cdot I\left(i,j\right)} \end{gathered}$$

(11)

where $\left(X_{GCP},Y_{GCP},Z_{GCP}\right)$ is the coordinates of the spot centroid; $I\left(i,j\right)$ is the energy value of the detector in $i^{th}$ row $j^{th}$ column; $h$ is the corresponding height; $\omega$ is the weight in the corresponding position; $M$ and $N$ are the largest values of the row and column number.

2.3. Verification Based on Laser Footprint Positioning

Based on the spots captured by detectors and laser data of the GF-7 satellite, the control variable method was used to locate the laser footprints under the condition that only the centroid of the spots captured by detectors was changed. The positioning accuracy of the laser data under the centroid of the spot extracted by seven methods, including the method proposed, was then compared. The seven methods are the method proposed in this paper, the Gaussian surface fitting method, the centroid method, the first-order gray-scale weighting method considering the detectors, the second-order gray-scale weighting method considering the detectors, the least squares ellipse fitting method, and the polynomial surface fitting method. The flow chart is shown in Figure 6.

2.3.1. Centroid Extraction of Spots Based on Infrared Detectors

• Centroid Extraction of Laser Spots Captured by Infrared Detectors Combining Laser Footprint Images and Detector Observation Data:

  ➣

  The GF-7 satellite laser footprint images were denoised by the “two-step method” to obtain real laser footprints. Then, the geometric parameters, such as the ratio of the major axis to the minor axis and the inclination angle, were obtained by means of the least square ellipse fitting method;

  ➣

  Based on the obtained geometric parameters, ellipses were drawn in combination with the largest value and the second largest value of the spot energy to complete the division of the spot area;

  ➣

  The centroid of the spot was extracted using the energy weighted barycenter method based on regional blocks.

• Other traditional methods for centroid extraction of spots:

  ➣

  The centroids of spots were extracted using six traditional methods, such as the Gaussian surface fitting method.

2.3.2. Calculation of Laser Pointing Angle and Ranging Parameters Based on the Single-Beam and Double-Beam Laser Calibration Models According to the Centroids Extracted

The laser pointing angle and ranging parameters were calculated by a single-beam laser rough calibration model as well as a double-beam laser fine calibration model based on centroids of spots extracted using multiple methods [9].

• Single-beam laser rough calibration:

In view of the satellite platform centroid, laser emission position, Global Positioning System (GPS) antenna, and relative position offset and rotational geometric relation of the Earth ellipsoid, the on-orbit geometric calibration model of single-beam laser altimeter was established, as shown in Formula (12):

$${\left[\begin{array}{c}{X}_{GCP1}\hfill \\ Y_{GCP1}\hfill \\ Z_{GCP1}\hfill \end{array}\right]}_{WGS84}={\left[\begin{array}{c}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right]}_{GPS}+R_{J2000}^{WGS84}{R}_{body}^{J2000}\left\{\left.\left[\begin{array}{c}d{x}_1\hfill \\ d{y}_1\hfill \\ d{z}_1\hfill \end{array}\right]+\left({\rho }_1-\Delta {\rho }_{atm}-\Delta {\rho }_{tides}-\Delta {\rho }_1\right)\left[\begin{array}{c}\mathit{cos}\left({\alpha }_1+\Delta {\alpha }_1\right)\mathit{cos}\left({\beta }_1+\Delta {\beta }_1\right)\hfill \\ \mathit{cos}\left({\beta }_1+\Delta {\beta }_1\right)\mathit{sin}\left({\alpha }_1+\Delta {\alpha }_1\right)\hfill \\ \hfill \mathit{sin}\left({\beta }_1+\Delta {\beta }_1\right)\hfill \end{array}\right]\right\}\right.$$

(12)

where ${\left[\begin{array}{c}{X}_{GCP1}\hfill \\ Y_{GCP1}\hfill \\ Z_{GCP1}\hfill \end{array}\right]}_{WGS84}$ is the ground coordinates for the spot centroid of laser Beam 1 in the coordinate system of World Geodetic System—1984 Coordinate System (WGS 84); ${\left[\begin{array}{c}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right]}_{GPS}$ is the coordinates of the satellite GPS antenna center in the coordinate system of WGS84; $\left[\begin{array}{c}d{x}_1\hfill \\ d{y}_1\hfill \\ d{z}_1\hfill \end{array}\right]$ is the offset of the laser Beam 1 relative to Global Navigation Satellite System (GNSS) antenna center; $R_{J2000}^{WGS84}$ is the rotation matrix when a J2000 coordinate system is converted into a WGS84 coordinate system; $R_{body}^{J2000}$ is the rotation matrix when a satellite-based system is converted into a J2000 coordinate system; ${\rho }_1$ is the ranging value of laser Beam 1; $\Delta {\rho }_{atm}$ and $\Delta {\rho }_{tides}$ are ranging errors triggered by atmospheric delay and tide; $\Delta {\rho }_1$ is the ranging system error of Beam 1 to be calculated; ${\alpha }_1$ and ${\beta }_1$ are the angles between reference Beam 1 and the X-axis and the Y-axis in the satellite-based system; $\Delta {\alpha }_1$ and $\Delta {\beta }_1$ are the pointing modification values of ${\alpha }_1$ and ${\beta }_1$ to be calculated. With the help of GCPs, the modification value of pointing can be iteratively calculated [9].

2.

Double-beam laser fine calibration:

Based on the single-beam laser rough calibration, the relative rotation relation of the double-beam laser was introduced to establish a double-beam joint calibration model, as shown in Formula (13) [9]:

$${\left[\begin{array}{c}{X}_{GCP2}\hfill \\ Y_{GCP2}\hfill \\ Z_{GCP2}\hfill \end{array}\right]}_{WGS84}={\left[\begin{array}{c}{X}_s\hfill \\ Y_s\hfill \\ Z_s\hfill \end{array}\right]}_{WGS84}+R_{J2000}^{WGS84}{R}_{body}^{J2000}\left\{\left.\left[\begin{array}{c}d{x}_2\hfill \\ d{y}_2\hfill \\ d{z}_2\hfill \end{array}\right]+\left({\rho }_2-\Delta {\rho }_{atm}-\Delta {\rho }_{tides}-\Delta {\rho }_2\right)\Delta R\left(r,a,b\right)\left[\begin{array}{c}\mathit{cos}\left({\alpha }_2\right)\mathit{cos}\left({\beta }_2\right)\hfill \\ \hfill \mathit{cos}\left({\beta }_2\right)\mathit{sin}\left({\alpha }_2\right)\hfill \\ \hfill \mathit{sin}\left({\beta }_2\right)\hfill \end{array}\right]\right\}\right.$$

(13)

where ${\left[\begin{array}{c}{X}_{GCP2}\hfill \\ Y_{GCP2}\hfill \\ Z_{GCP2}\hfill \end{array}\right]}_{WGS84}$ is the ground coordinates for the spot centroid of laser Beam 2 in the coordinate system of WGS84; ${\left[\begin{array}{c}{X}_s\hfill \\ Y_s\hfill \\ Z_s\hfill \end{array}\right]}_{WGS84}$ is the ground coordinates for the footprint spot centroid of laser Beam 1 in the coordinate system of WGS84; $\left[\begin{array}{c}d{x}_2\hfill \\ d{y}_2\hfill \\ d{z}_2\hfill \end{array}\right]$ is the offset of laser Beam 2 relative to GNSS antenna center; ${\rho }_2$ is the ranging value of laser Beam 2; $\Delta {\rho }_2$ is the ranging system error of Beam 2 to be calculated; $\Delta R\left(r,a,b\right)=\left[\begin{array}{c}\Delta R_1\left(r,a,b\right)\hfill \\ \Delta R_2\left(r,a,b\right)\hfill \\ \Delta R_3\left(r,a,b\right)\hfill \end{array}\right]$ is the pointing rotation matrix of Beam 2 relative to Beam 1.

By taking and following the principle that the ground distance residual from the two laser beams to the ground spot centroids $\left(X_{GCP1},Y_{GCP1},Z_{GCP1}\right)$ and $\left(X_{GCP2},Y_{GCP2},Z_{GCP2}\right)$ to be minimized, Formula (14) was written into an error Equation [9]:

$$\begin{gathered} V=AX-L= \\ \left[\begin{array}{ccccccc}\frac{\partial f_1\left(X\right)}{\partial \Delta {\alpha }_1}& \frac{\partial f_1\left(X\right)}{\partial \Delta {\beta }_1}& \frac{\partial f_1\left(X\right)}{\partial \Delta {\rho }_1}& 0& 0& 0& 0\\ \frac{\partial f_1\left(Y\right)}{\partial \Delta {\alpha }_1}& \frac{\partial f_1\left(Y\right)}{\partial \Delta {\beta }_1}& \frac{\partial f_1\left(Y\right)}{\partial \Delta {\rho }_1}& 0& 0& 0& 0\\ \frac{\partial f_1\left(Z\right)}{\partial {\Delta \alpha }_1}& \frac{\partial f_1\left(Z\right)}{\partial \Delta {\beta }_1}& \frac{\partial f_1\left(Z\right)}{\partial \Delta {\rho }_1}& 0& 0& 0& 0\\ 0& 0& 0& \frac{\partial f_2\left(X\right)}{\partial r}& \frac{\partial f_2\left(X\right)}{\partial a}& \frac{\partial f_2\left(X\right)}{\partial b}& \frac{\partial f_2\left(X\right)}{\partial \Delta {\rho }_2}\\ 0& 0& 0& \frac{\partial f_2\left(Y\right)}{\partial r}& \frac{\partial f_2\left(Y\right)}{\partial a}& \frac{\partial f_2\left(Y\right)}{\partial b}& \frac{\partial f_2\left(Y\right)}{\partial {\Delta \rho }_2}\\ 0& 0& 0& \frac{\partial f_2\left(Z\right)}{\partial r}& \frac{\partial f_2\left(Z\right)}{\partial a}& \frac{\partial f_2\left(Z\right)}{\partial b}& \frac{\partial f_2\left(Z\right)}{\partial {\Delta \rho }_2}\end{array}\right]\left[\begin{array}{c}\Delta {\alpha }_1\\ \Delta {\beta }_1\\ \Delta {\rho }_1\\ r\\ a\\ b\\ {\Delta \rho }_2\end{array}\right]-\left[\begin{array}{c}\begin{array}{c}{X}_1-X_{GCP1}\\ Y_1-Y_{GCP1}\\ Z_1-Z_{GCP1}\end{array}\\ \begin{array}{c}{X}_2-X_{GCP2}\\ Y_2-Y_{GCP2}\\ Z_2-Z_{GCP2}\end{array}\end{array}\right] \end{gathered}$$

(14)

The rough calibration value of laser pointing obtained from the single-beam laser calibration model was set as the initial value, and the laser pointing angle and ranging parameters of two laser beams were calculated based on the least squares principle by capturing the array data of two laser beams, ensuring that the calibration parameters meet the global optimum.

In practical applications, for the convenience of calculation, the pointing angle is usually projected to the three directions O-XYZ of the coordinate axis to obtain the commonly used three-axis pointing angle parameters (Angle_X, Angle_Y, Angle_Z). Figure 7 shows the relationship between laser angle direction and three-axis angle. Formula (15) is used to express the geometric relationship between the two.

$$\left.\begin{array}{c}\mathit{Angle}\_X=arccos\left[cos\left(\alpha \right)cos\left(\beta \right)\right]\\ \mathit{Angle}\_Y=arccos\left[sin\left(\alpha \right)cos\left(\beta \right)\right]\\ \mathit{Angle}\_Z=arccos\left[sin\left(\beta \right)\right]\end{array}\right\}$$

(15)

2.3.3. Accuracy Evaluation of Spot Centroid Extraction Based on Infrared Detectors

The accuracy of the spot centroid was considered the only variable affecting laser footprint positioning. The laser fine calibration results were extrapolated into laser data of other tracks, and high-precision GCPs, including RTK control points and high-precision LiDAR point clouds, were selected as verification data. According to Formula (16), the calibration parameters were combined with high-precision attitude data [28,29,30] and precise orbit data [31,32] to locate laser footprints. The centroid accuracy of captured spots extracted using different methods was evaluated indirectly through the positioning accuracy and elevation accuracy of laser footprints. Besides, the elevation measurement accuracy of the GF-7 satellite-borne laser was evaluated.

$${\left[\begin{array}{c}{X}_{Footprint}\hfill \\ Y_{Footprint}\hfill \\ Z_{Footprint}\hfill \end{array}\right]}_{WGS84}={\left[\begin{array}{c}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right]}_{GPS}+R_{J2000}^{WGS84}{R}_{body}^{J2000}\left\{\left.\left[\begin{array}{c}d{x}_1\hfill \\ d{y}_1\hfill \\ d{z}_1\hfill \end{array}\right]+\left({\rho }_1-\Delta {\rho }_{atm}-\Delta {\rho }_{tides}-\Delta {\rho }_1\right)\left[\begin{array}{c}\mathit{cos}\left(\mathit{Angle}\_X\right)\hfill \\ \mathit{cos}\left(\mathit{Angle}\_\mathrm{Y}\right)\hfill \\ \mathit{cos}\left(\mathit{Angle}\_\mathrm{Z}\right)\hfill \end{array}\right]\right\}\right.$$

(16)

where ${\left[\begin{array}{c}{X}_{Footprint}\hfill \\ Y_{Footprint}\hfill \\ Z_{Footprint}\hfill \end{array}\right]}_{WGS84}$ is the result of locating the laser footprint based on the calibration parameters.

3. Results

According to the experimental process, with the help of using seven methods, including the method proposed in this paper, the centroid extraction experiment was carried out on the laser data obtained by the on-orbit geometric calibration based on infrared detectors in 2021. The results are shown in Figure 8.

The centroid extraction results of laser footprints 241743151.00 (51.00) and 241743151.33 (51.33) in Beam 1 of Track 10119 (as shown in Figure 6a,b) were used to calculate the corresponding pointing and ranging parameters of Beam 1; the centroid extraction results of laser footprints 243038830.00 (30.00) and 243038830.33 (30.33) in Beam 2 of Track 10412 (as shown in Figure 8c,d) were used to calculate the corresponding pointing and ranging parameters of Beam 2. The distribution of centroid extraction results using different methods for getting the above-mentioned laser footprints, as well as the spatial relationship with the spots, are shown in Figure 8. According to Figure 8, it is obvious that the result (the diamond-shaped points) of the method proposed is closer to the center of the spot.

Based on the centroid coordinates obtained, the method proposed in this paper and the other six methods were used to calculate the pointing and ranging parameters of the seven laser groups by single and double-beam laser calibration models. With the help of calibration parameters, the pointing calibration of Beam 1 and Beam 2 of the GF-7 satellite laser based on infrared detectors was completed. The differences before and after parameter calibration using seven methods are shown in

Table 2. Differences before and after elevation parameter calibration with seven different methods.

Centroid Extraction Methods	Beam 1		Beam 2
	dα/(°)	dβ/(°)	dα/(°)	dβ/(°)
The proposed method	0.001464	0.002862	0.000223	0.001524
Centroid method	0.001465	0.002659	0.000206	0.001405
Gaussian surface fitting	0.001488	0.002694	0.000183	0.001312
The first-order gray-scale weighting method considering the detectors	0.001048	0.002672	0.000142	0.001264
The second-order gray-scale weighting method considering the detectors	0.001052	0.002685	0.000132	0.001273
Least Squares Fitting of Ellipses	0.00144	0.002649	0.000203	0.00131
Polynomial surface fitting	0.001484	0.002695	−0.00095	0.00104

.

4. Discussion

4.1. Verification of Relative Plane Accuracy

If the measurement method for a typical feature is selected to verify the plane accuracy of the laser footprints covering the position of the feature [33], it will be difficult to select the corresponding typical feature due to the large size and spacing of the laser footprints. Even when features are available, the cost of field surveys is enormous. Therefore, due to the limitations of the plane accuracy verification in the experiment, verification of relative plane accuracy was conducted for the calibration results only. However, the plane accuracy and elevation accuracy of GF-7 satellite laser are closely correlated [34,35]. In other words, the elevation accuracy can also reflect the plane accuracy.

The verification of relative plane accuracy of different calibration results mainly included two aspects. On the one hand, the scope of possible truth values of centroid in the spot was determined; the probability that centroids obtained using different methods were distributed in this scope was calculated; the relative centroid accuracy was verified preliminarily. On the other hand, based on the calibration parameters calculated through centroids obtained using different methods, laser data of the same track as well as different tracks in the spots, were positioned. The positioning results and centroid extraction results were compared to indirectly verify the relative plane accuracy of centroids extracted using different methods.

Based on the spatial relationship between the plane position of the centroid extracted by different methods in Figure 9 and the possible scope of the true value of the centroid, the probability that centroids obtained using different methods, which fell in this scope, was obtained, as shown in Figure 10.

As shown in Figure 10, the probability that centroids extracted using the method proposed fall in the truth value scope is 100%, while the probabilities of the other six methods range from 11.1% to 55.6%. Therefore, compared with multiple traditional methods, including the Gaussian surface fitting method, the probability of higher plane accuracy for centroid extraction is relatively greater using the method proposed.

For the second method, through the calibration parameters calculated through centroids extracted with the experimental data of 2021 based on different methods, laser data of the same track as well as different tracks in the spots captured by detectors were positioned, respectively. The positioning results were analyzed in comparison with that of centroid extraction and were then compared, indirectly verifying the relative plane accuracy of centroids extracted using different methods, as shown in Figure 11 and Figure 12.

First, in order to exclude the effect of factors such as satellite jitter, centroid extraction was completed using laser footprint 51.33 of Beam 1 and 30.00 of Beam 2 in 2021 to calculate the calibration pointing and ranging parameters. The results were then used to position laser footprints 51.00 and 30.33 of the same track. Meanwhile, cross-verification of relative plane accuracy was completed with the centroid extraction results obtained using multiple methods and laser footprint positioning results based on calibration parameters, as shown in Figure 11.

In Figure 11, the results of cross-verification obtained using different methods show a significant difference. For Beam 1, the plane accuracy of both first-order gray-scale weighting and second-order gray-scale weighting methods considering detectors is smaller than 0.2 detector grid spacing (hereinafter referred to as “grid”), showing relatively higher stability than other methods. For Beam 2, on the contrary, the plane accuracy of both exceeds 1.3 grids; and the relative plane accuracy difference of both beams under the polynomial surface fitting method exceeds 0.6 grids. Therefore, the above three methods used have relatively lower stability in the calibration plane accuracy for different beams. On the contrary, the relative plane accuracy difference of the two beams using the remaining methods does not exceed 0.3 grids. The difference between the two beams using the method proposed in this paper is only 0.01 grids; the value of Beam 1 is 0.34 grids, and Beam 2 is 0.33 grids. Hence, the method proposed in this paper has relatively higher plane accuracy and shows better stability for different beams.

The calibration parameters of 2021 were extended to the laser data of 2020, and laser footprints of different tracks in 2020 were positioned with the calibration parameters of 2021; the results are shown in Figure 12. In this figure, the plane accuracy of centroids obtained using the method proposed in this paper, the Gaussian surface fitting method, centroid method, and least square ellipse fitting method, is relatively high and stable. Meanwhile, in Figure 10, compared with other methods, the above-mentioned four methods present a relatively higher probability of obtaining centroids with higher plane accuracy. In other words, the results of the two methods to verify the relative plane accuracy are consistent. In Figure 12, however, the average value of the relative plane accuracy of centroids extracted using four better methods, including the method proposed in this paper, is 3 m–4 m, while the layout spacing of corresponding detectors is 3–8 m.

The reason why the plane error exceeds the detector layout spacing is that the laser data of 2020 was processed based on the calibration parameters in 2021. The effect of factors such as satellite jitter between different years leads to changes in pointing angles, which in turn causes differences in the positioning accuracy of laser footprints. Figure 12 is essentially a supplement to Figure 11. Figure 12, on the one hand, is to illustrate that the proposed method has better plane accuracy and stability for the centroid of laser data extraction in 2020. On the other hand, to express that even if there are objective conditions such as satellite jitter between different years, the extrapolation effect of pointing parameters detected by the proposed method is better than that of the other six methods.

4.2. Verification of Elevation Accuracy

GCPs were collected using RTK technology, forming 9 × 9 ground control grids around the spot, as shown in Figure 3a. Then, the real surface elevation was interpolated to evaluate the elevation of laser footprints.

During the period of field calibration, on the flat terrain of the non-calibration area (with a slope of less than 2° and no surface features or vegetation), GCPs of laser elevation were collected using RTK technology. Thirty-six laser footprints were selected, which were within the scope of GCPs in the data of Track 3402 on 14 June 2020, Track 3478 on 19 June, Track 3555 on 24 June, Track 10119 on 29 August 2021, Track 10194 on 3 September, and Track 10412 on 13 September, including 18 footprints for Beam 1 and Beam 2, respectively. Based on the above laser footprints, real surfaces were interpolated within the surrounding ground control grids. The elevation accuracy of laser footprint positioning results and that of corresponding results of ground control grids were compared. Then the elevation accuracy of laser footprints was evaluated, and the centroid accuracy of captured spots extracted by different methods was indirectly evaluated. The comparison results are shown in Figure 13 and Figure 14.

In Figure 13 and Figure 14, there are abnormal values in the laser footprint positioning results of Beam 1 and Beam 2 in 2021, and Beam 2 in 2020, and these are concentrated in the six traditional methods. In other words, the laser footprint positioning results for centroids extracted using the method proposed to confirm its relatively good stability in elevation. According to the width of the boxes, the fluctuation degree of laser footprint positioning results of Beam 1 in 2021 is smaller. However, if comprehensive consideration is given to the fluctuation degree and abnormal values, the method proposed in this paper is superior to the other six methods. According to Figure 14a, all the methods show relatively great fluctuation in the laser footprint positioning results of Beam 1 in 2020. However, if comprehensive consideration is given to the median and mean value, the method proposed in this paper is also superior to others. In addition, according to Figure 14b, the fluctuation degree of laser footprint positioning results obtained using different methods for Beam 2 is smaller than that for Beam 1. However, if comprehensive consideration is given to the mean and median values, the method proposed in this paper is still superior to others. According to Figure 13 and Figure 14, the method proposed in this paper has relatively better uniform variability with lower data fluctuation, which has certain advantages compared with other six traditional methods.

On these grounds, quantitative analysis was made for the elevation results of laser data processed with different methods, as shown in Figure 15 and Figure 16.

According to Figure 15 and Figure 16, either in terms of beams or years, the elevation accuracy of laser footprint positioning results using the method proposed in this paper are superior to those of the other six methods. The mean value of the elevation accuracy of positioning results obtained from Beam 1 in 2021 using the method proposed in this paper is 5.2 cm. The mean values of the elevation accuracy obtained using the Gaussian surface fitting method, centroid method, and least square ellipse fitting method are slightly higher but all less than 8.0 cm. The mean values obtained using the first-order gray-scale weighting method considering detectors, second-order gray-scale weighting method considering detectors, and polynomial surface fitting method, are not less than 10.0 cm. Therefore, the elevation accuracy of laser footprint positioning results using these methods presents a layering effect, and the method proposed in this paper has significant advantages. For Beam 2 in 2021, the mean value of the elevation accuracy of laser footprint positioning results obtained using the method proposed in this paper is 5.0 m, which is significantly better than the other six methods. According to the laser data of 2021, the processing results of this method have higher elevation accuracy and are relatively stable.

For the laser data of 2020, the mean value of the elevation accuracy of laser footprint positioning results obtained using the method proposed is 23.3 cm for Beam 1 and 7.1 cm for Beam 2, better than the other six methods but significantly lower than that of 2021. In the experiment, the pointing and ranging parameters were calculated according to the experimental data of 2021 while being verified with the laser data of 2020 and 2021, respectively. However, as objective factors such as satellite jitter led to changes in calibration parameters between different years, the elevation accuracy of laser footprint positioning results in 2020 is lower than that in 2021.

To fully verify the elevation accuracy of laser footprint positioning results using the method proposed in this paper, the data were extrapolated to laser data of other tracks. High-precision airborne LiDAR point cloud data were used for elevation accuracy verification.

In this paper, the terrain with a slope greater than 6° is defined as a slope; the terrain with a slope greater than 2° but less than 5° is defined as a gentle slope. According to the data of Track 5278 after the screening, there are 12 laser footprints of Beam 1 and 12 of Beam 2 on slopes. There are 12 laser footprints of Beam 1 and 11 of Beam 2 on gentle slopes. In the experiment, the laser footprints were positioned according to the above experimental data through the calibration parameters calculated with centroids extracted using different methods, and the elevation of the positioning results was compared with the elevation of airborne LiDAR data, as shown in Figure 17 and Figure 18.

In Figure 17 and Figure 18, for Beam 1 on slopes, the elevation accuracy of laser footprint positioning results obtained using the method proposed is 57.5 cm; except for the first-order gray-scale weighting method considering detectors (65.6 cm) and the second-order gray-scale weighting method considering detectors (65.5 cm), the elevation accuracy obtained using the method proposed in this paper is 10.0 cm higher than that of other four methods, showing significant advantages. Meanwhile, for Beam 2 on slopes, the elevation accuracy of the method proposed is 30.1 cm, which is 3.0 cm higher than that of the best of other methods. Therefore, the method proposed still has certain advantages in the elevation accuracy on slopes for Beam 2. In addition, according to the laser footprint positioning results from Beam 1 on gentle slopes obtained using different methods in Figure 18, the elevation accuracy of the method proposed is 30.5 cm, which is 10.0 cm higher than that of the other six traditional methods, showing significant advantages of this method. However, the method proposed has relatively small advantages for Beam 2, and the accuracy of elevation is similar in most of the methods; only the polynomial surface fitting method shows poor elevation accuracy.

Due to large terrain fluctuations in the area where the airborne LiDAR point cloud data are obtained by field operation, it is steeper than the area in Sonid Right Banner, where infrared detectors are deployed for on-orbit geometric calibration. Either the slopes or gentle slopes in this paper are steeper than the area in Sonid Right Banner, so the elevation accuracy of laser footprint positioning results in this area is significantly lower than that in the above-mentioned area. However, in the experiment, the increase of the gradient also enlarged the calibration difference between different methods. This highlights the advantages of this method in experimental fields of different gradients.

5. Conclusions

To solve problems including insufficient detector consistency, a small amount of detector data, and the influence of atmospheric turbulences, in the experiment, a centroid extraction method of laser spots captured by infrared detectors combining the laser footprint images and detector observation data is proposed by taking full advantage of the consistency of laser footprints and spots captured by detectors in the same platform, time and atmospheric conditions. First, preprocessing was conducted for the laser footprint images by hierarchical denoising with a self-adaptive “two-step method”, to obtain the real geometrical shape of footprints. Then the real geometric parameters were mapped to the spots captured under the same conditions to realize constraining. Finally, the distance and energy values were fully considered, and the spot captured by infrared detectors was divided into blocks after constraining. The centroid of the spot after constraining was extracted by means of the energy-weighted barycenter method based on regional blocks. After centroid extraction was completed using different methods, laser pointing and ranging parameters were calculated based on single-beam and double-beam laser calibration models to position the laser footprint data and indirectly verify the accuracy of centroids extracted using different methods.

GF-7 satellite laser was adopted as the research object, and cross-verification was conducted with laser data of different tracks, laser data in different terrains, and high-precision airborne LiDAR point cloud data. Contrastive analysis was made on the relative plane accuracy and absolute elevation accuracy of seven centroid extraction methods, including the method proposed in this paper, and the accuracy of different methods was evaluated. According to the analysis of results, the relative plane accuracy of Beam 1 obtained using the method proposed in this paper reached 0.34 grids, and that of Beam 2 reached 0.33 grids. The elevation accuracy was as follows: on flat terrain, the elevation accuracy of Beam 1 and Beam 2 in 2021 was 5.2 cm and 5.0 cm, respectively, 0.6 cm and 4.2 cm higher than those in the most accurate one among other methods; the elevation accuracy in 2020 was 23.3 cm and 7.1 cm, 7.7 cm and 2.7 cm higher than those in the most accurate one among other methods. On slopes, the elevation accuracy of two beams was 57.5 cm and 30.1 cm, respectively, 8.0 cm and 3.0 cm higher than those in the most accurate one among other methods. On gentle slopes, the elevation accuracy of the two beams was 30.5 cm and 15.4 cm, respectively, 10.4 cm and 2.8 cm higher than those in the most accurate one among other methods. Therefore, whether it is the plane or elevation accuracy, the method proposed in this paper has relatively higher accuracy and shows better stability, helping increase the satellite laser positioning accuracy. Thus, the method proposed in this paper is proven to significantly increase the geometric calibration accuracy based on infrared detectors.

In conclusion, the proposed method in this paper, by taking into account the shape of each real laser footprint and the energy information of the laser spot captured by the detectors, improves the spot centroid positioning accuracy of the satellite laser calibration and has been applied to the on-orbit geometric calibration experiments of GF-7 satellite-borne laser.