Geometric calibration of the linear-array whiskbroom optical satellites based on look-angle corrections

Linear-array whiskbroom optical satellites are equipped with scanning mirrors and compensation mirrors. They rotate and scan imaging with the satellite flight orbit as the rotation axis, and geometrically splice the scanned images along the orbit, achieving a balance between large width and high resolution of remote sensing images. The addition of scanning mirrors and compensation mirrors necessitates multiple reflections for the imaging ray to reach the focal plane. Hence, the calibration of the geometric aspects for linear-array whiskbroom optical satellites will be an intricate procedure. A geometric calibration model based on look-angle corrections is established according to the imaging characteristics of linear-array whiskbroom optical satellites. All errors are corrected using correction quantities, and cubic polynomial surfaces are used to fit correction quantities. The proposed geometric calibration method based on the look-angle corrections has a sensor positioning precision of better than 1.0 pixels for the calibrated image.


Introduction
Ground spatial resolution and imaging width are very important indicators in the design of modern optical remote sensing satellites, and linear-array push-broom optical satellites (LAPOSs) often cannot be taken into consideration.Linear-array whiskbroom optical satellites (LAWOSs) were proposed to increase the width of the image without reducing the spatial resolution.The images of LAWOSs consist of scanning mirrors, compensation mirrors, and multiple linear-array CCDs [1][2] [7].When the LAWOS is in orbit, the linear-array CCDs on the imager take the orbit as the rotation axis and rotate around the forward direction of the satellite to complete the imaging of the ground in the process of rotation [3].
Although the existence of errors has been taken into account in the design of the imager and possible errors are theoretically minimized, attitude and orbit measurements, instrument installation, and camera lens aberrations can affect the quality of the imaging.In particular, the installation of rotary scanning mirrors and compensation mirrors can also generate errors [4].These errors affect the sensor positioning accuracy [5].To improve the sensor positioning precision, the image can be geometrically calibrated.
Currently, there is a scarcity of research focused on the geometric calibration of LAWOSs.Li et al. [6] proposed a three-step estimation method using ground checkpoints to estimate the calibration parameters and used a fifth-degree polynomial to fit and correct the angular measurement error of the scanning mirror.Wang proposed a calibration method for China-YG-14 high-resolution thermal infrared images, which analyzed the errors introduced by the whiskbroom motion, and solved the model instability and low precision caused by the inconsistent pendulum sweeps and uneven angles.
Cao [7] proposed a geometric calibration method based on ELACCD for DaQi-1 and introduced a rotation matrix to describe the ELACCD-to-imager coordinate system conversion relation equivalent to the rotary scanning mirrors and compensation mirrors.
Section II describes the physical sensor model (PSM) of LAWOS, the calibration of geometric model look-angle corrections, and the calculation of parameters.Section III illustrates the analysis and validation of the feasibility and applicability of the proposed method by using five WSI image datasets.Section IV presents the findings.

PSM based on LAWOSs
To geometrically calibrate LAWOS images, the first step is to establish a suitable PSM [5].PSM precisely depicts the imaging process of LAWOS, thereby obtaining optical remote sensing satellite images with high geometric quality.Due to the complexity of the process of calculating the parameters of the scanning mirrors and the parameters of the compensation mirrors, a linear-array CCD (LACCD) rotating in the same manner as the scanning mirrors and compensation mirrors would be considered equivalent to a scanning mirror rotating [7].The expression of the established PSM is as follows: where [  ] is the coordinates of the ground points; [   ] is the coordinates of the LAWOS; λ is the scale factor. ,  ,  ,  , and  are the rotation matrices for the transformation of the subscripted coordinate system to the superscripted one, respectively. consists of the rotation angles ( ,  ,  );  consists of the mounting angles ( ,  ,  ); and  consists of the rotation angles ( ,  ,  ).tan and tan are the coordinates of the image points.
The CCD detector look angles  ,  and the LACCD rotation angles ( ,  ,  ) expressions [8][9] [10] are as follows: where  and  are the row and column coordinates of the image point respectively.

Geometric Calibration Model and Parameters Solution
In the PSM based on LAWOSs, there is a strong correlation between the three types of parameters.Moreover, the step-by-step process of estimating the parameters is tedious and the computational process of geometric calibration will be very complicated.Therefore, this study proposes not to solve these three types of parameters in steps, but to use  and  directly for correction.As shown in Figure 1, when the satellite is imaging, there is a ground point A. According to Equations ( 1) and (3), the coordinates tan and tan of the corresponding image point a can be calculated.In actual situations, there will be external orientation element errors and internal orientation element errors.The tan′ and tan′ are calculated based on Equation ( 2) to obtain the image point coordinates a'.The discrepancy between theoretical point a and actual point a' is evident in Figure 1.Therefore, this study proposes to directly use the correction quantities  and  to correct the image point coordinate errors.1), (2), and (3) as follows: where  and  are respectively correction quantities to column and row coordinates in image space. ,  ,  ,  , and  are the rotation matrices for the transformation of the superscripted coordinate system to the subscripted one, respectively.
The expressions of  and  are as follows: The parameter estimation process is as follows.1) Some uniformly distributed ground checkpoints within the image are fetched from the digital orthophoto maps (DOM) and digital elevation models (DEM).
2) The CCD detector look angle parameters ( ,  ) and ( ,  ) , the LACCD rotation angle parameters ( ,  ), and the camera mounting angles ( ,  ,  ) use the values calibrated in the laboratory as initial values The correction quantities  and  are set to 0. Based on Equations ( 1) and ( 3), tan and tan can be calculated.
3) The actual tan′ and tan′ are calculated based on Equation (2).4) Calculate the difference by  = tan − tan′ and  = tan − tan′ , and the difference is fitted by a cubic polynomial surface.The parameters( , … ,  ) and ( , … ,  ) of the expression for the corrected quantity are found.

Geometric Calibration Analysis
The experimental datasets for this study are five sets of DaQi-1 images, each comprising 260 images.Each image measures 20480×120 pixels.The calibration image was selected from the images in dataset 1 with fewer cloud and water-covered areas.The distribution of 71 ground checkpoints manually fetched from the DOMs and DEMs is shown in Figure 2. The following experiments were devised to validate the feasibility of the suggested calibration method.
1) Experiment A: No fitting of the correction quantities is performed, setting their parameters to the initial value of 0.
After each set of experimental calibrations, the position of the projection point of the ground coordinates of each ground checkpoint on the image was calculated based on the parameters of the obtained corrections, and the root-mean-square error (RMSE) of the residuals from the projection point to the image point was calculated as the sensor positioning precision.The precision of sensor positioning in the calibrated image is presented in Table 1.The fitting schematic of the correction quantities for the calibrated images is shown in Figure 3.In Experiment A, the calibrated image exhibits a sensor positioning precision of just 16.649 pixels.As shown in Figure 3(a), sensor positioning error is exhibited as curved characteristics in three dimensions.This situation indicates that the jitter of the satellite during the launching process and the installation of the camera sensors always produce errors, which lead to the reduction of the sensor positioning precision.
In Experiment B, the fitting using the primary surface resulted in a sensor positioning precision of 0.774 pixels, which is a significant improvement over the uncalibrated results.This suggests that these errors have some regularity and can be predicted by fitting a function and then correcting the error using correction quantities.
In Experiment C, the precision of sensor positioning after quadratic surface fitting is improved by 0.1 pixels.This indicates that the use of quadratic surface function fitting can fit the correction quantities more accurately.
In Experiment D, the sensor positioning precision obtained by the method of step-by-step estimation of the parameters was 0.666 pixels.
In Experiment E, the sensor positioning precision is improved by 0.026 pixels after the cubic surface fitting compared to the quadratic surface fitting.This indicates that the cubic surface fitting can improve the fitting precision to some extent.Compared with the results of Experiment D, the sensor positioning precision was improved by 0.016 pixels, which indicates that the sensor positioning precision can be improved by using the surface fitting correction amount directly.Derived from the outcomes of the aforementioned experiments, the method of calibrating images using polynomial surfaces fitted to correction quantities is feasible, and the effectiveness of the fit improves to a certain extent with the number of fitted functions.

Performance validation of calibration parameters
The parameters obtained by the geometric calibration of an image should apply to other images of the same satellite to improve the precision of the correspondence between image points and ground points.Therefore, four images I2-I5 were selected from datasets 2-5 respectively to evaluate the precision of the parameters obtained by image calibration of dataset 1. 52, 46, 53, and 36 checkpoints were fetched from the DOMs and DEMs of I2-I5, respectively, and Figure 4 shows the distribution of checkpoints in the four images.In the previous subsection, the scenarios of the five sets of experiments are noted as SA-SE, and Table 2 records the sensor positioning precision of I2-I5 under scenarios SA-SE, respectively.Figure 5 is the correction quantities fitting the surface of image I2.
In Figure 5(a), it can be seen that the errors in sensor positioning of the calibrated images exhibit curved surface features in the three-dimensional distribution, which is almost the same as the distribution in Figure 3(a).calibration parameters obtained for the images in dataset 1 have precision and validity and can improve the sensor positioning precision for images I2-I5 in datasets 2-5.
From the data in Table 2, it can be seen that there is no improvement in the sensor positioning precision of images I2-I4 compared to the quadratic surface fitting when using cubic surface fitting corrections, which suggests that the results of the calibrations using the method of surface fitting have been stabilized on the experimental images.

Conclusion
A corresponding geometric calibration model is devised to address the imaging structure and process of LAWOSs, and a viable method for geometric calibration, utilizing look-angle corrections, is put forward in this study.The parameters to be estimated are directly corrected using the correction quantities  and  , which are obtained by fitting a cubic polynomial surface based on the errors of the calibration image.
From the experimental results of five sets of WSI images of DaQi-1, the correction quantities  and  are sufficient for correcting errors in DaQi-1 positioning, the method in this study also improves the sensor positioning precision compared with the method of solving the calibration parameters step by step, thus verifying the feasibility and effectiveness of the LAWOS in-orbit geometric calibration method based on look-angle corrections.

Figure 2 .
Figure 2. Ground checkpoint distributions in Image I1.The following experiments were devised to validate the feasibility of the suggested calibration method.1)Experiment A: No fitting of the correction quantities is performed, setting their parameters to the initial value of 0.2) Experiment B: Fitting the correction quantities  and  with a primary polynomial surface, i.e., the parameters ( , … ,  ) and ( , … ,  ) are calibrated, and ( , … ,  ) and ( , … ,  ) are set to the initial value of 0.3) Experiment C: Fitting the correction quantities  and  with a quadratic polynomial surface, i.e., calibrating the parameters ( , … ,  ) and ( , … ,  ) and setting ( , … ,  ) and ( , … ,  ) to the initial value of 0.4) Experiment D: Estimate the three types of parameters separately according to the method in[7], i.e., calibrate the imager parameters ( ,  ,  ) , ( , … ,  ) , ( , … ,  ) in steps, and set the parameters ( , … ,  ), ( , . .,  ), and ( , … ,  ) to the initial value of 0.5) Experiment E: Fitting of the correction quantities  and  with cubic polynomial surfaces, the parameters( , … ,  ) and ( , … ,  ) are calibrated.After each set of experimental calibrations, the position of the projection point of the ground coordinates of each ground checkpoint on the image was calculated based on the parameters of the obtained corrections, and the root-mean-square error (RMSE) of the residuals from the projection point to the image point was calculated as the sensor positioning precision.The precision of sensor positioning in the calibrated image is presented in Table1.The fitting schematic of the correction quantities for the calibrated images is shown in Figure3.Table1.Sensor positioning accuracies of Image I1.

Figure 5 .
Figure 5. Correction quantities fitting the surface of image I2 in scenarios: (a) SA, (b) SB, (c) SC, and(d) SE.In scenario SE, the precision of the proposed method in sensor positioning in this study has been improved to a certain extent compared with that of the method in[7].This indicates that the