# A Spliced Satellite Optical Camera Geometric Calibration Method Based on Inter-Chip Geometry Constraints

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## Abstract

**:**

## 1. Introduction

## 2. Integral Imaging Model of Spliced Satellite Optical Camera

#### 2.1. Imaging Properties

#### 2.2. Geometric Error Characteristics

#### 2.3. Geometric Imaging Model

_{X}, B

_{Y}, B

_{Z}) expresses the GPS antenna eccentricity error, which defines the offset vector of the camera projection center relative to the GPS antenna phase center in the satellite body coordinate system. The multi-chip images of the spliced satellite optical camera share one set of satellite orbit and attitude measurement data in a unified camera system. The GPS position and attitude data corresponding to each line in the separate chip can be interpolated from the satellite orbit and attitude data according to the time stamp.

_{X}, B

_{Y}, B

_{Z}) can be ignored, and its influence is absorbed into the camera alignment angle $\left(\phi ,\omega ,\kappa \right)$, such that $\phi $ is the pitch angle, $\omega $ is the roll angle, and $\kappa $ is the yaw angle. The parameters $\left(\phi ,\omega ,\kappa \right)$ indicate the misalignment matrix ${\mathit{R}}_{off}$, as shown in Equation (2).

- (1)
- Calculating the chip number i. According to the column c of the image point $\left(r,c\right)$ in the original image system, calculate the point imaged on the i-th chip using Equation (4), where n is the total number of chips in different views, fix indicates the truncating operation, and N
_{s}is the length of a single chip in detector units.$$i=fix\left(\frac{c}{{N}_{s}}\right)+1\left(i=1,2,\dots ,n\right)$$ - (2)
- Converting the original image system o-rc to the single-chip system o
_{ci}-x_{ci}y_{ci}. Calculate the image point coordinates (x_{ci}, y_{ci}) in the single-chip system based on chip number, where p_{s}is the size of the detector unit in millimeters.$$\left\{\begin{array}{c}{x}_{ci}=0\\ {y}_{ci}=\left(c-\left(i-1\right)\cdot {N}_{s}\right)\cdot {p}_{s}\end{array}\right.$$

_{ci}-x

_{ci}y

_{ci}is the single-chip system in the focal plane. Each single-chip system takes the center of the left detector unit as the origin, the orbit flight direction as the x

_{ci}axis, and the scanning direction as the y

_{ci}axis. o

_{c}-x

_{c}y

_{c}z

_{c}represents the camera system, and o-xy represents the focal plane system.

- (3)
- Transforming from the single-chip system o
_{ci}-x_{ci}y_{ci}to the camera system o_{c}-x_{c}y_{c}z_{c}. Complete the conversion of the single-chip system to the focal plane system o-xy using the placement parameters of each chip in the focal plane, as shown in Equation (6).$$\left\{\begin{array}{c}x={x}_{ci}+{x}_{ci0}\\ y={y}_{ci}+{y}_{ci0}\end{array}\right.$$

_{0}, c

_{1}, c

_{2}, c

_{3}are the interior calibration parameters along x axis for each chip, and r

_{0}, r

_{1}, r

_{2}, r

_{3}are the interior calibration parameters along y axis for each chip.

## 3. Proposed Integral Geometric Calibration Method Investigation Based on Inter-Chip Geometry Constraints

_{1}(x

_{p}

_{1}, y

_{p}

_{1}) on the left chip and the corresponding image point p

_{2}(x

_{p}

_{2}, y

_{p}

_{2}) on the right chip. For image points p

_{1}and p

_{2}, the rigorous geometric imaging models are generated using Equations (9) and (10), where the multiple rotation matrices in Equation (3) can then be combined into Equation (11).

_{S}

_{1}, Y

_{S}

_{1}, Z

_{S}

_{1}) and (X

_{S}

_{2}, Y

_{S}

_{2}, Z

_{S}

_{2}) are the GPS antenna centers corresponding to the left and right chips, respectively. Combining Equations (9) and (10), the geometry constraint model between adjacent chips can be constructed, as presented in Equation (12).

**A**

_{1}and

**A**

_{2}are the corresponding coefficient matrices. Since all chips of the forward/nadir/backward view share the same set of attitude observation equipment, all chips in each view take one set of camera alignment angles to describe the external error $\left(\mathsf{\Delta}{\phi}_{1},\mathsf{\Delta}{\omega}_{1},\mathsf{\Delta}{\kappa}_{1}\right)=\left(\mathsf{\Delta}{\phi}_{2},\mathsf{\Delta}{\omega}_{2},\mathsf{\Delta}{\kappa}_{2}\right)=\left(\mathsf{\Delta}\phi ,\mathsf{\Delta}\omega ,\mathsf{\Delta}\kappa \right)$. $\left(\mathsf{\Delta}{x}_{1},\mathsf{\Delta}{y}_{1}\right)$ is obtained from the interior calibration parameters of the left chip and is the image point coordinate error for ${p}_{1}$ on the left chip. $\left(\mathsf{\Delta}{x}_{2},\mathsf{\Delta}{y}_{2}\right)$ is calculated by the interior calibration parameters of the right chip and is the image point coordinate error of ${p}_{2}$ on the right chip. The interior calibration parameters for each chip differ from each other. Equation (13) can then be simplified to the expression

- (1)
- Image matching

- (2)
- External geometric calibration process

- (3)
- Conventional internal geometric calibration processing

- (4)
- Refined internal geometric calibration processing

- (5)
- Iteration

## 4. Results

#### 4.1. Test Data

#### 4.2. Experiment Results of ZY-3 Satellite Three-Line Array Images

#### 4.2.1. Initial Positioning Accuracy Analysis of ZY-3 Satellite Three-Line Array Images

#### 4.2.2. Geometric Calibration Accuracy Analysis of ZY-3 Satellite Three-Line Array Images

#### 4.2.3. Validity Assessment of Geometric Calibration Parameters

#### 4.2.4. Mosaic Effect Verification of ZY-3 Image

#### 4.3. Experiment Results of TH-1 HR Image

#### 4.3.1. Initial Positioning Accuracy Analysis of TH-1 Satellite HR Images

#### 4.3.2. Geometric Calibration Accuracy Analysis of TH-1 HR Images

#### 4.3.3. Validity Assessment of Geometric Calibration Parameters

#### 4.3.4. Mosaic Effect Verification of TH-1 Image

## 5. Conclusions

## 6. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The ground coverage diagram of the spliced satellite optical camera at the imaging moment. (

**a**) The instantaneous imaging diagram; (

**b**) Ground coverage of each chip.

**Figure 5.**Workflow of the integral geometric calibration implementation scheme for the spliced satellite optical camera based on inter-chip geometry constraints.

**Figure 6.**ZY-3 satellite three-line array test data. (

**a**) 4 TDICCD overview images of the forward view; (

**b**) 3 TDICCD overview images of the nadir view; (

**c**) 4 TDICCD overview images of the backward view.

**Figure 7.**Initial positioning residuals distribution of ZY-3 three-line array camera. (

**a**) Positioning residuals in XY planar direction; (

**b**) Positioning residuals in the elevation direction.

**Figure 8.**The positioning residuals distribution of ZY-3 three-line array camera after geometric calibration. (

**a**) Positioning residuals in XY planar direction; (

**b**) Positioning residuals in the elevation direction.

**Figure 12.**Initial positioning residuals distribution of TH-1 HR camera. (

**a**) Positioning residuals in XY planar direction; (

**b**) Positioning residuals in the elevation direction.

**Figure 13.**Positioning residuals distribution of TH-1 HR camera after geometric calibration. (

**a**) Positioning residuals in the XY planar direction; (

**b**) Positioning residuals in the elevation direction.

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | −455.26 | 474.79 | −464.70 | −479.25 |

Y | −722.65 | 735.01 | −729.97 | −751.30 |

Z | 420.39 | 430.07 | 433.23 | 413.82 |

**Table 2.**Ground coordinates differences statistics of ZY-3 three-line array camera corresponding image points. (Unit: meter).

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | −0.227 | 0.187 | 0.285 | −0.604 |

Y | −0.064 | 0.064 | 0.420 | −0.177 |

Z | −0.144 | 0.172 | 0.152 | −0.560 |

**Table 3.**Initial positioning accuracy of the corresponding image points on the ZY-3 three-line array camera.

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X (meter) | −572.601 | 593.430 | −551.473 | −614.118 |

Y (meter) | −821.815 | 849.581 | −837.429 | −862.278 |

x (pixel) | −197.890 | 206.185 | −189.706 | −214.176 |

y (pixel) | −273.990 | 279.313 | −265.265 | −286.139 |

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | 0.003 | 0.331 | 0.642 | −0.600 |

Y | −0.008 | 0.369 | 0.563 | −0.992 |

Z | −0.026 | 0.464 | 0.571 | −0.290 |

**Table 5.**Ground coordinates differences statistics of ZY-3 three-line array camera corresponding image points after geometric calibration. (Unit: meter).

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | 0.023 | 0.064 | 0.095 | −0.089 |

Y | −0.014 | 0.032 | 0.040 | −0.078 |

Z | −0.012 | 0.052 | 0.106 | −0.096 |

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X (meter) | 0.009 | 0.219 | 0.446 | −0.444 |

Y (meter) | −0.005 | 0.240 | 0.425 | −0.661 |

x (pixel) | 0.003 | 0.077 | 0.167 | −0.159 |

y (pixel) | 0.001 | 0.089 | 0.135 | −0.297 |

**Table 7.**Direct uncontrolled positioning accuracy based on calibrated alignment angles and pointing angle files. (Unit: meter).

Data | X | Y | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | RMS | Max | Min | Mean | RMS | Max | Min | Mean | RMS | Max | Min | |

Jincheng | 0.205 | 5.327 | 8.341 | −6.749 | −0.195 | 6.824 | 9.486 | −5.904 | −0.214 | 6.936 | 6.628 | −5.420 |

Tianjin | 0.182 | 5.208 | 8.012 | −7.012 | −0.177 | 6.578 | 9.103 | −6.976 | −0.189 | 6.812 | 7.019 | −6.971 |

Shiyan | 0.198 | 5.311 | 9.154 | −7.423 | −0.186 | 6.960 | 8.999 | −7.012 | −0.197 | 7.126 | 7.238 | −6.899 |

**Table 8.**Ground coordinates differences statistics based on calibrated alignment angles and pointing angle files. (Unit: meter).

Data | X | Y | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | RMS | Max | Min | Mean | RMS | Max | Min | Mean | RMS | Max | Min | |

Jincheng | 0.023 | 0.068 | 0.112 | −0.188 | −0.019 | 0.039 | 0.103 | −0.077 | −0.021 | 0.053 | 0.106 | −0.146 |

Tianjin | 0.019 | 0.041 | 0.103 | −0.113 | −0.012 | 0.041 | 0.089 | −0.072 | −0.018 | 0.058 | 0.116 | −0.089 |

Shiyan | 0.024 | 0.063 | 0.107 | −0.109 | −0.021 | 0.033 | 0.097 | −0.080 | −0.019 | 0.059 | 0.118 | −0.112 |

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | 646.785 | 663.282 | 750.659 | 639.328 |

Y | 144.638 | 217.744 | 147.680 | −740.844 |

Z | 118.620 | 231.315 | 308.698 | −527.600 |

**Table 10.**Ground coordinates differences statistics of the corresponding image points. (Unit: meter).

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | −2.849 | 3.620 | 5.441 | −6.813 |

Y | −1.125 | 1.587 | 2.876 | −4.857 |

Z | −0.131 | 1.825 | 5.387 | −5.336 |

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | −0.000 | 2.412 | 6.139 | −6.142 |

Y | 0.004 | 1.628 | 4.313 | −4.257 |

Z | −0.000 | 1.206 | 3.069 | −3.071 |

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | −0.013 | 0.834 | 1.120 | −1.019 |

Y | 0.032 | 0.977 | 1.092 | −1.159 |

Z | 0.027 | 0.858 | 1.101 | −1.012 |

**Table 13.**Ground coordinate differences statistics of the corresponding image points after geometric calibration. (Unit: meter).

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X | −0.032 | 0.104 | 0.230 | −0.311 |

Y | −0.097 | 0.182 | 0.164 | −0.288 |

Z | 0.023 | 0.117 | 0.456 | −0.276 |

Index | Mean | RMS | Max | Min |
---|---|---|---|---|

X (meter) | 0.011 | 0.289 | 0.654 | −0.527 |

Y (meter) | 0.006 | 0.298 | 0.725 | −0.786 |

x (pixel) | 0.004 | 0.147 | 0.329 | −0.259 |

y (pixel) | 0.002 | 0.149 | 0.365 | −0.337 |

**Table 15.**Direct uncontrolled positioning accuracy based on calibrated alignment angles and pointing angle files. (Unit: meter).

Data | X | Y | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | RMS | Max | Min | Mean | RMS | Max | Min | Mean | RMS | Max | Min | |

Luoyang | 0.089 | 3.738 | 5.355 | −4.596 | 0.025 | 3.160 | 5.253 | −4.704 | 0.084 | 3.564 | 5.145 | −4.891 |

Dalian | 0.078 | 3.812 | 5.049 | −5.017 | 0.019 | 3.352 | 5.140 | −5.104 | 0.101 | 3.572 | 5.212 | −5.089 |

Kunming | 0.086 | 3.415 | 4.883 | −4.066 | 0.017 | 3.201 | 4.953 | −4.672 | 0.041 | 3.364 | 4.887 | −4.678 |

**Table 16.**Ground coordinates differences statistics based on calibrated alignment angles and pointing angle files. (Unit: meter).

Data | X | Y | Z | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | RMS | Max | Min | Mean | RMS | Max | Min | Mean | RMS | Max | Min | |

Luoyang | −0.047 | 0.137 | 0.777 | −0.387 | −0.025 | 0.280 | 0.405 | −0.139 | 0.088 | 0.168 | 0.212 | −0.418 |

Dalian | −0.051 | 0.148 | 0.890 | −0.481 | −0.031 | 0.197 | 0.414 | −0.142 | 0.079 | 0.173 | 0.232 | −0.389 |

Kunming | −0.039 | 0.152 | 0.695 | −0.492 | −0.029 | 0.128 | 0.501 | −0.152 | 0.076 | 0.182 | 0.240 | −0.392 |

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## Share and Cite

**MDPI and ACS Style**

Wang, T.; Zhang, Y.; Zhang, Y.; Zhang, Z.; Xiao, X.; Yu, Y.; Wang, L.
A Spliced Satellite Optical Camera Geometric Calibration Method Based on Inter-Chip Geometry Constraints. *Remote Sens.* **2021**, *13*, 2832.
https://doi.org/10.3390/rs13142832

**AMA Style**

Wang T, Zhang Y, Zhang Y, Zhang Z, Xiao X, Yu Y, Wang L.
A Spliced Satellite Optical Camera Geometric Calibration Method Based on Inter-Chip Geometry Constraints. *Remote Sensing*. 2021; 13(14):2832.
https://doi.org/10.3390/rs13142832

**Chicago/Turabian Style**

Wang, Tao, Yan Zhang, Yongsheng Zhang, Zhenchao Zhang, Xiongwu Xiao, Ying Yu, and Longhui Wang.
2021. "A Spliced Satellite Optical Camera Geometric Calibration Method Based on Inter-Chip Geometry Constraints" *Remote Sensing* 13, no. 14: 2832.
https://doi.org/10.3390/rs13142832