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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

On-orbit geometric calibration is a key technology to guarantee the geometric quality of high-resolution optical satellite imagery. In this paper, we present an approach for the on-orbit geometric calibration of high-resolution optical satellite imagery, focusing on two core problems: constructing an on-orbit geometric calibration model and proposing a robust calculation method. First, a rigorous geometric imaging model is constructed based on the analysis of the major error sources. Second, we construct an on-orbit geometric calibration model through performing reasonable optimizing and parameter selection of the rigorous geometric imaging model. On this basis, the calibration parameters are partially calculated with a stepwise iterative method by dividing them into two groups: external and internal calibration parameters. Furthermore, to verify the effectiveness of the proposed calibration model and methodology, on-orbit geometric calibration experiments for ZY1-02C panchromatic camera and ZY-3 three-line array camera are conducted using the reference data of the Songshan calibration test site located in the Henan Province, China. The experimental results demonstrate a certain deviation of the on-orbit calibration result from the initial design values of the calibration parameters. Therefore, on-orbit geometric calibration is necessary for optical satellite imagery. On the other hand, by choosing multiple images, which cover different areas and are acquired at different points in time to verify their geometric accuracy before and after calibration, we find that after on-orbit geometric calibration, the geometric accuracy of these images without ground control points is significantly improved. Additionally, due to the effective elimination of the internal distortion of the camera, greater geometric accuracy was achieved with less ground control points than before calibration.

On-orbit geometric calibration is critical to realizing full performance of high-resolution optical remote sensing satellite geometry in subsequent processes and applications [

Due to the rapid growth in the number of satellites in China, the improvement of geometric resolution, the radiometric quality, and increasingly wide range of applications, users demand a better geometric quality of the image product. In particular, China’s first three-line array stereo-mapping satellite—ZY-3 satellite—was successfully launched, which has filled the gap in China’s civilian mapping satellites and serves as a milestone in enhancing China’s independent access to high-resolution geospatial information [

Although previous studies have made progress in optical satellite on-orbit geometric calibration to a certain extent, some key scientific problems remain unsolved. In this paper, we discuss in detail the related theory and method of optical satellite on-orbit geometric calibration. First, the sources of major error are sought and a rigorous geometric imaging model is built for optical satellite images. Secondly, through performing a reasonable optimizing and parameter selection of the rigorous geometric imaging model, we build an on-orbit geometric calibration model and study the corresponding calculation method of the calibration parameters. Finally, an on-orbit geometric calibration experiment is performed and verified for ZY1-02C panchromatic camera and ZY-3 three-line array camera with the reference data offered by the Songshan calibration test site. Our proposed calibration model and calculation method has been proven to be stable and effective and could significantly improve the geometric accuracy of optical satellite imagery.

Establishment of a rigorous geometric imaging model is the first step of on-orbit geometric calibration for optical satellite imagery [

The attitude data observed by attitude determination devices, such as star tracker and gyro, is the rotation angle between the satellite’s body-fixed coordinate system and the inertial spatial coordinate system. Normally an angle bias is designed between the satellite’s body-fixed coordinate system and the camera coordinate system, which is called “Camera installation angle”. In order to determine the attitude of the camera in the inertial spatial coordinate system, an accurate camera installation angle in the satellite’s body-fixed coordinate system is necessary. As shown in _{C}_{C}_{C}_{B}_{B}_{B}_{C}_{C}_{C}_{B}_{B}_{B}

The orbit data observed by GPS receivers is the coordinate of its antenna phase center in WGS84 geocentric euclidean coordinate system. Confined by the size of equipment in the satellite design, there is a position bias between the GPS antenna phase center and the camera's project center, which is called “GPS eccentric vector”, as shown in _{X}_{Y}_{Z}_{body} is demanded.

Due to the limitation in measurement accuracy, a measurement error often occurs in the attitude data observed by star sensors and gyros, which is called “attitude measurement error”. (Strictly speaking, in addition to the attitude measurement error, an orbit measurement error would appear due to the limitation in measurement accuracy of GPS receiver. However, when considering the optical satellite image, the orbit measurement error has a minor influence on its geometric accuracy compared to that of the attitude measurement error. As the external angle element and line element are strongly correlated, we only take the attitude measurement error into account and treat the orbit measurement error as a part of the attitude measurement error.) In a single scene of image the attitude measurement error can be considered as a constant, whereas there is a degree of random deviation between different scenes of images acquired at different times [

There are two groups of distortion errors in an optical camera. One is from the optical distortion of the camera lens, and the other from CCD translation, scale, rotation, and change of the principle distance [_{o}_{o}_{y}_{1} the radial distortion coefficient of the camera lens; _{1} and _{2} the tangential distortion coefficients of the camera lens; and (

Therefore, a rigorous geometric imaging model can be constructed (as described in _{g}, _{g}, _{g}) is the coordinate of the object point of corresponding image point in the WGS84 coordinate system. (_{gps}, _{gps}, _{gps}) represents the coordinates of the phase center of GPS antenna in the WGS84 coordinate system, which is obtained by satellite-based GPS.
_{X}_{Y}_{Z}_{body} represents the coordinate of the eccentric vector from the sensor’s projection center to the GPS antenna phase center in the satellite’s body-fixed coordinate system;
_{X}_{Y}_{Z}_{body} are the external system error parameters of the camera, and (Δ

Although the rigorous geometric imaging model takes the major errors into consideration in theory, it cannot be used in practice as an on-orbit geometric calibration model for optical satellite imagery due to over-parameterization. Because of a special imaging condition (

Obviously, the accuracy of laboratory calibration of the GPS eccentric vector as a distance can easily reach centimeter level or higher. Also, during orbiting it changes little, meaning that a geometric positioning error can be neglected, and, therefore, we can use the laboratory calibration result as true value without performing an on-orbit geometric calibration. In contrast, there is always a considerable error in the camera installation angle due to the limitation of the laboratory calibration technology and the change during launching and orbiting, which has a relatively greater influence on the geometric positioning accuracy because of the optical satellite’s high orbit. Related research has proven that the camera installation angle error is the main factor that affects geometric positioning accuracy of the optical satellite imagery and that it must be calibrated periodically. The above analysis of the external errors of an optical satellite camera indicates that we only need to perform on-orbit calibration for the camera installation angle using multiple scenes of image simultaneously and apply the laboratory calibration result directly as true value of the GPS eccentric vector.

At present, a strict physical model similar to _{x}_{y}

ZY-3 three-line array camera has a focal plane design with four CCDs in Forward/Backward and three CCDs in Nadir camera. Strictly speaking, an individual internal calibration model should be built for each CCD. However, by adopting an optical mosaic method with a half transparent and half reflecting mirror, the multiple CCDs can be directly mosaiced into a line with an accuracy of up to 0.3 pixels in Forward, Backward and Nadir cameras, according to the related technical data provided by the satellite designer. After satellite launching, two experiments are conducted for the Forward, Backward, and Nadir camera, respectively. Taking the Nadir camera as an example, in the first experiment each of the three CCDs has its own individual internal calibration model, and in the other experiment, a single internal calibration model is shared by all three CCDs as a whole CCD, and the two experiments are performed using the same ground control points and image. Calculating the differences of the directional angles of all the detectors of the three CCDs determined by two internal calibration results, we found that the differences of all detectors were within 0.2 pixels, which proves that the two experimental results of the three CCDs were very close. Therefore, it is practical to conduct an on-orbit calibration with the three CCDs as a whole CCD, as it is the same for all Forward/Backward and Nadir cameras.

The characteristics of the model are as follows: (1) Using the directional angle of every linear array CCD detector in the camera’s coordinate system as the calibration parameter can skillfully solve the problem that the physical distortion parameters of the camera could not be calibrated precisely because the principle optical axis cannot be accurately determined due to close coupling between the external and internal system errors. Although the internal system errors are partially included in the external calibration results, resulting in inaccurate measurements in the direction of the three axes of the camera’s coordinate system, this does not hamper the calculation of the internal calibration parameters. Via coupling with the internal and external calibration parameters, the systematic geometric error can still be precisely eliminated to restore the precise direction of every CCD detector in space (somewhat of a combination and redistribution of the systematic errors); (2) A cubic polynomial is adopted to describe the directional angle of every CCD detector, the internal calibration parameters are coefficients of a cubic polynomial, and have high orthogonality and low correlation.

According to the analysis above, an on-orbit geometric calibration model is constructed for the optical satellite image, as expressed in _{E} =
_{I} = (_{0}, _{1}, _{2}, _{3}, _{0}, _{1}, _{2}, _{3}) is used to determine the directional angles of all CCD detectors in the camera coordinate system.

In literature [

Auto measurement of control points: By matching the satellite image with the orthophoto and the corresponding DEM of high accuracy provided by the ground calibration test site with the image correlation technique, we can automatically measure K evenly distributed and highly accurate ground control points. The coordinate of every control point in the WGS84 geocentric euclidean coordinate system is (_{i}_{i}_{i}_{i}_{i}

The following procedure must be followed:

Initialize the internal and external calibration parameters _{E} and _{I} with the pre launch calibration

Determining the external calibration parameters: Consider the current internal calibration parameter as the “true value” and the external calibration parameter as the unknown parameter. Insert (_{E}, _{I}) into

_{i}_{i}_{i}

Calculate

Next, update the current value of the external calibration parameter _{E}

Determining the internal calibration parameters: Consider the current value of the external calibration parameter as the “true value” and the internal calibration parameter as an unknown parameter. The process is similar to that of the external calibration parameter in Step 3 and so the specific equation and process are omitted here;

Set _{E}_{E} and _{I}_{I}

Update the camera’s parameter file according to the calibration result.

Strictly speaking, to average down the attitude measurement error, we should calculate the camera installation angle using multiple scenes of images. In this case, the external calibration parameter for each scene of image is simply calculated individually, and then all external calibration results are averaged to the final combined external calibration result. Moreover, if the internal calibration has been performed earlier, there is less internal geometric distortion of all these images in the camera, and thus for each image only five ground control points, evenly distributed in its four corners and the center, are needed to calculate the external calibration parameters.

To verify the correctness and feasibility of the proposed calibration model and calculation method, an on-orbit geometric calibration experiment for ZY1-02C panchromatic camera and ZY-3 three-line array camera is conducted. As the available reference data is limited, in this paper we perform an on-orbit calibration experiment for ZY1-02C panchromatic camera only by using a scene of image covered by the reference data provided by the Henan Songshan geometric calibration test site, as well as a ZY-3 three-line array camera. Detailed information about the satellite image data and reference data are listed in

By matching the ZY1-02C’s panchromatic image with the reference data provided by the Henan Songshan geometric calibration test site, a number of corresponding points are automatically acquired that can be used as control points to determine the internal calibration parameters. The corresponding object coordinates can be directly obtained from the reference DOM and DEM. According to the methodology, we should choose control points distributed in a region that is as short as possible along the orbit orientation and uniformly on the whole CCD along the direction vertical to the orbit. The red triangulations are composed of 48,833 pervasive corresponding points that were obtained by auto-matching (

The calibration parameters for the ZY1-02C panchromatic camera were calculated using the control points shown in

To analyze the characteristics of the camera’s internal distortion objectively and quantitatively, one out of every 50 detectors in the camera’s CCD was sampled, from which differences in directional angles of those samples pre- and post-calibration were calculated and then plotted in a distortion curve (

To evaluate the effectiveness of the calibration results, we choose multiple ZY1-02C panchromatic images in the regions of Zhengzhou in Henan, Huairou in Beijing, and Aramon in France. To verify the geometric accuracy of these images before and after calibration, two sets of RPC are generated respectively based on the design value and calibrated value of the calibration parameters. An translation model and an affine transform model on the image space of the RPC is used respectively for correction with one GCPs and not less than three GCPs. The experimental results are listed in

Using the same method as above, on-orbit geometric calibration experiments are also performed for ZY-3 three-line array camera with the reference data of the Songshan calibration test site. The following provides a brief description of the image matching result, the calibration result and the geometric accuracy before and after the calibration under different control conditions in the experiments.

By matching the Nadir, Forward and Backward imagery of ZY-3 three-line array camera with the reference data, 76,893, 61,270, and 63,012 corresponding image points are acquired, respectively. Using these corresponding image points as control points, the calibration parameters of ZY-3 three-line array camera are calculated.

The following conclusions can be drawn from the results of the experiment:

The camera installation angle changed to some extent during satellite on-orbit operation (

From the distortion curve shown in

In

In this paper, we proposed an on-orbit geometric calibration model and corresponding calculation method for optical satellite imagery. Experiments were performed for ZY1-02C panchromatic camera and ZY-3 three-line array camera using the proposed model and method with high-precision reference data supported by the Songshan calibration test site that was built by Wuhan University. The experimental results show that although the ZY1-02C panchromatic camera and the ZY-3 three-line array camera were calibrated in the laboratory before launch, the laboratory calibration parameters would experience a certain amount of deviation during satellite on-orbit operation. This is likely caused by many factors, including installation, launching, and orbiting. Therefore, on-orbit geometric calibration is necessary. Otherwise, the geometric quality of images will not be able to meet requirements of subsequent processes and applications. Meanwhile, we chose multiple images that covered different areas and were obtained at various times to verify their geometric accuracy before and after calibration. The result indicates that with the proposed on-orbit calibration model and calculation method, the geometric accuracy of the imagery is significantly improved and stabilized with or without ground control points. Additionally, as the internal distortion error of the images is eliminated effectively, requirements for the number of the ground control points are greatly reduced while the geometric accuracy with ground control is higher after calibration.

Noticing that as the available reference data is limited, in this paper we perform a calibration experiment by only using a scene of image. Therefore analysis on the variation with time in the external calibration parameters during on-orbit operation remains insufficient. More in-depth studies and comprehensive experiments using different terrain and globally distributed reference areas will be carried out in the future.

This work was supported in part by the National Basic Research Program of China 973 Program under grant 2012CB719902, 2014CB744201; the Program for New Century Excellent Talents in University; the National High Technology Research and Development Program of China (No. 2011AA120203); the Fundamental Research Funds for the Central Universities (No. 2012619020205); the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (FANEDD, No. 201249); the National Natural Science Foundation of China (No. 41371430); and the Program for Changjiang Scholars and Innovative Research Team in University under grant IRT1278.

The first author conceived the study and designed the experiments; The second author developed the algorithm and wrote the program; The third author performed the experiments; The fourth author helped perform the analysis with constructive discussions and contributed to manuscript preparation.

The authors declare no conflict of interest.

Camera’s coordinate system and satellite’s body-fixed coordinate system.

Relationship between GPS antenna center and projection center of camera.

Directional angle of charge coupled device (CCD) detector.

Matching result for ZY1-02C’s panchromatic image and the reference orthophoto: (

CCD distortion curve of the ZY1-02C panchromatic camera.

CCD distortion curve of ZY-3 three-line array Camera: (

Specific information about the satellite imagery data.

GSD (m) | 5 | Nadir: 2.1 Forward/Backward: 3.5 |

Image Size (pixels) | 12,000 × 12,000 | Nadir: 24,530 × 24,575 |

Focal Length (mm) | 1010 | 1700 |

Acquisition time | 23 January 2012 | 3 February 2012 |

Area Covered | Range: 60 km × 60 km |

Specific information about the reference data.

GSD (m) | 0.2 | 1 |

Geometric Precision (RMSE/m) | Planimetric accuracy: 1 | Height accuracy: 2 |

Area Covered | Range: 100 km × 100 km |

External calibration parameter values before and after the calibration of the ZY1-02C panchromatic camera.

pitch | 0.0 | 0.097078 | 0.097078 |

roll | 0.0 | −0.046805 | −0.046805 |

yaw | 0.0 | −0.090407 | −0.090407 |

Geometric positioning accuracy before and after the calibration of ZY1-02C panchromatic camera.

| ||||||
---|---|---|---|---|---|---|

ZhenZhou, HeNan (2012.4.18) | 0 | 22 | 282.2 | 103.0 | 18.6 | 7.7 |

1 | 21 | 6.6 | 8.1 | 1.6 | 0.96 | |

3 | 19 | 6.3 | 7.7 | 1.2 | 0.85 | |

4 | 18 | 5.8 | 7.4 | 1.1 | 0.7 | |

8 | 14 | 5.6 | 6.8 | 0.95 | 0.7 | |

HuaiRou, Beijing (2012.2.1) | 0 | 21 | 267.2 | 95.8 | 6.4 | 11.7 |

1 | 20 | 6.4 | 6.9 | 2.0 | 1.3 | |

3 | 18 | 6.1 | 6.6 | 1.4 | 1.1 | |

5 | 16 | 6.0 | 6.5 | 1.3 | 1.0 | |

7 | 14 | 5.9 | 6.5 | 1.3 | 0.9 | |

Aramon, France (2012.6.14) | 0 | 24 | 268.7 | 92.3 | 5.0 | 14.2 |

1 | 23 | 7.8 | 7.4 | 2.0 | 1.4 | |

3 | 21 | 7.4 | 7.1 | 1.8 | 1.2 | |

5 | 19 | 6.9 | 6.9 | 1.7 | 1.1 | |

7 | 17 | 6.1 | 6.5 | 1.3 | 1.1 |

External calibration parameter values before and after calibration of ZY-3 three-line array camera.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

pitch | 0.0 | 0.076662 | 0.076662 | 22.0 | 21.879654 | −0.120346 | −22.0 | −22.097722 | −0.097722 |

roll | 0.0 | −0.098950 | −0.098950 | 0.0 | −0.109721 | −0.109721 | 0.0 | −0.111211 | −0.111211 |

yaw | 0.0 | −0.184034 | −0.184034 | 0.0 | −0.198358 | −0.198358 | 0.0 | −0.194175 | −0.194175 |

Stereo geometric positioning accuracy before and after calibration of ZY-3 three-line array camera.

NanYang, Henan (3 July 2012) | 0 | 19 | 1495.5 | 272.9 | 3.3 | 10.2 |

1 | 18 | 61.8 | 33.2 | 3.2 | 2.9 | |

3 | 16 | 65.5 | 33.7 | 2.9 | 2.8 | |

5 | 14 | 7.2 | 4.2 | 2.6 | 2.6 | |

8 | 11 | 3.0 | 2.9 | 2.3 | 2.6 | |

LuoYang, Henan (24 January 2012) | 0 | 24 | 1492.1 | 265.8 | 13.9 | 6.1 |

1 | 23 | 38.8 | 19.6 | 4.2 | 3.6 | |

3 | 21 | 40.9 | 20.1 | 3.3 | 3.1 | |

5 | 19 | 5.8 | 3.2 | 3.2 | 2.8 | |

8 | 16 | 3.1 | 2.7 | 2.6 | 2.5 | |

TaiYuan, Shanxi (13 May 2012) | 0 | 21 | 1486.4 | 305.9 | 12.6 | 7.2 |

1 | 20 | 38.2 | 20.5 | 4.4 | 4.1 | |

3 | 18 | 49.7 | 26.6 | 3.1 | 2.9 | |

5 | 16 | 6.4 | 5.4 | 2.9 | 2.9 | |

8 | 13 | 3.2 | 3.1 | 2.7 | 2.6 |