# A Photogrammetric-Photometric Stereo Method for High-Resolution Lunar Topographic Mapping Using Yutu-2 Rover Images

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Photogrammetric-Photometric Stereo Method

#### 3.1. Modelling the Image Irradiance for Close-Range Topographic Mapping

#### 3.1.1. Coordinates in Traditional Photometric Stereo vs. Coordinates in PPS

_{P}Y

_{P}Z

_{P}, and it is well noted that the axes of X

_{P}, Y

_{P}in object space are not parallel to x, y axes of the image plane. For rover orthorectified images, the axes of X

_{P}, Y

_{P}in object space are parallel to x, y axes of the image plane, traditional photometric stereo can be used. However, new image irradiance equations should be developed for original rover images with perspective projection. In summary, the following aspects should be considered when developing a proper image irradiance equation for rover-based lunar surface mapping: (1) The pitch angle of the camera cannot be ignored, and the rotation matrix cannot be treated as identity matrix; (2) Objects are not often located very far from the camera as the camera is close to the ground; (3) The axes of X

_{P}, Y

_{P}are not parallel to x, y of the image plane. Thus the surface normal should be deduced based on the principles of photogrammetry and photometric stereo.

#### 3.1.2. Modelling of Surface Normal Based on Collinearity Equations

_{S}, Y

_{S}, Z

_{S}) represents three translation components of the exterior orientation parameters of the camera; (a

_{1}, a

_{2}, a

_{3},…, c

_{3}) stands for the elements of the rotation matrix; (x

_{0}, y

_{0}) is the principle point position, f represents the principal distance of the camera.

_{0}, y

_{0}) and f can be obtained from calibrated results of the rover camera, (a

_{1}, a

_{2}, a

_{3},…, c

_{3}) are determined using rover localization results [5]. Notice that when the rotation matrix is identity matrix, there is

#### 3.1.3. The Image Irradiance Equation

_{s}, q

_{s}) can be calculated from azimuth and solar elevation angle, so that the illumination angle and emission angle can be determined as

#### 3.2. Perspective Photometric Stereo for Close-Range Topographic Mapping

_{X}and N

_{Y}must be determined firstly. Because the reflectance represents ratio of energy reflection, its value must be greater than zero, thus the estimated value of N

_{X}, N

_{Y}are constrained so that it will not exceed the feasible range.

_{X}, N

_{Y}are essential to improve the convergence speed and determine the results. Considering that the ground photogrammetry covers small areas and there are small changes in N

_{X}, N

_{Y}for the whole images, initial values of N

_{X}, N

_{Y}are assigned as a small value. Then Levenberg–Marquardt least-squares minimization algorithm [69] is adopted to solve the unknown parameters for each pixel.

_{X}, N

_{Y}), p and q can be solved directly using Equation (12).

#### 3.3. Height Estimation from Estimated Height Gradient

^{T}and post-multiplying by U, the function becomes

## 4. Experimental Analysis

#### 4.1. Quantitative Evaluation Measures

_{calc}represents the calculated normal vector, and N

_{ref}denotes the reference normal vector, for image with M × N pixels MEANN can be expressed as

#### 4.2. Experimental Analysis for Simulated Data

#### 4.3. Experimental Analysis for Yutu-2 Data

_{1}, P

_{2}represent constant penalty coefficients for all pixels q in the neighborhood of p.

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Schematic representation of different imaging conditions (

**a**) image and object coordinates for photometric stereo under orthographic projection (PSOP), (

**b**) image and object coordinates for photometric stereo under perspective projection with identity matrix (PSPP), (

**c**) image and object coordinates for PPS.

**Figure 5.**Simulated images under different lighting conditions (

**a**) simulate image of solar azimuth angle 90°and elevation angle 55° (

**b**) simulate image of solar azimuth angle 90°and elevation angle 60° (

**c**) simulate image of solar azimuth angle 90°and elevation angle 65° (

**d**).

**Figure 6.**Height map of ground truth and results of three methods (

**a**) Height map of ground truth, (

**b**) PPS result, (

**c**) PSPP result, (

**d**) PSOP result.

**Figure 7.**ROI 1 and reconstruction results of the three methods (

**a**) Enlarged view of region 1, (

**b**) ground truth, (

**c**) PPS result, (

**d**) PSPP result, (

**e**) PSOP result.

**Figure 9.**ROI 2 and reconstruction results of the three methods (

**a**) Enlarged view of region 2, (

**b**) ground truth (

**c**) PPS result, (

**d**) PSPP result, (

**e**) PSOP result.

**Figure 11.**Navcam images of the left camera under different illumination conditions (

**a**) 94741, (

**b**) 94914, (

**c**) 95633.

**Figure 12.**Examples of shadows (

**a**) shadow inside two craters, (

**b**) shadow inside a crater, (

**c**) shadow behind a boulder.

**Figure 13.**Shadow maps (

**a**) Shadow map of image 94741, (

**b**) Shadow map of image 94914, (

**c**) Shadow map of image 95633, (

**d**) Final shadow map.

**Figure 15.**ROI 1 and reconstruction results of the three methods (

**a**) ROI 1 image, (

**b**) SGM result, (

**c**) PPS result, (

**d**) shaded SGM result, (

**e**) shaded PPS result, (

**f**) PSPP result, (

**g**) PSOP result, (

**h**) Profile of the boulder (marked by the white circle in (

**a**)) from PPS result.

**Figure 17.**ROI 2 and reconstruction results of the three methods (

**a**) ROI 2 image, (

**b**) SGM result, (

**c**) PPS result, (

**d**) shaded SGM result, (

**e**) shaded PPS result, (

**f**) PSPP result, (

**g**) PSOP result, (

**h**) Profile of the boulder (marked by the white circle in (

**a**)) from PPS result.

**Figure 19.**Orthorectified images of ROI 1 in Yutu-2 rover imagery (

**a**) ROI 1 of 94741, (

**b**) ROI 1 of 94914, (

**c**) ROI 1 of 95633.

**Figure 20.**Reconstruction DEM results of ROI1 by three methods (

**a**) SGM interpolation result, (

**b**) PPS interpolation result, (

**c**) PSOP result from orthorectified images.

**Figure 21.**Orthorectified images of ROI 2 in Yutu-2 rover imagery (

**a**) ROI 2 of 94741, (

**b**) ROI 2 of 94914, (

**c**) ROI 2 of 95633.

**Figure 22.**Reconstruction results of ROI 2 by three methods (

**a**) SGM interpolation result, (

**b**) PPS interpolation result, (

**c**) PSOP result for orthorectified images.

Image ID | Solar Azimuth Angle (°) | Solar Elevation Angle (°) | Approximated Phase Angle (°) |
---|---|---|---|

Figure 5b | 90 | 55 | 107.5 |

Figure 5c | 90 | 60 | 103.3 |

Figure 5d | 90 | 65 | 99.0 |

Methods | MEANN (°) | NFD | |
---|---|---|---|

ROI 1 | PPS | 0.494 | 0.003 |

PSPP | 59.799 | - | |

PSOP | 60.065 | - | |

ROI 2 | PPS | 0.734 | 0.092 |

PSPP | 59.998 | - | |

PSOP | 59.855 | - | |

whole image | PPS | 0.324 | 0.042 |

PSPP | 59.963 | - | |

PSOP | 59.960 | - |

Stereo baseline | 0.27 m |

Focal length | 1189 pixels |

Pixel size | 1024 × 1024 |

FOV | 46.4° × 46.4° |

Image ID | Solar Azimuth Angle (°) | Solar Elevation Angle (°) | Approximated Phase Angle (°) |
---|---|---|---|

94741 | −70.8 | 17 | 37.7 |

94914 | −71.8 | 16.2 | 38.6 |

95633 | −77.1 | 11.4 | 44.1 |

Methods | MEANN (°) | NFD | |
---|---|---|---|

ROI 1 | PPS | 8.274 | 0.336 |

PSPP | 93.792 | - | |

PSOP | 102.746 | - | |

ROI 2 | PPS | 9.459 | 0.386 |

PSPP | 86.786 | - | |

PSOP | 101.843 | - |

Methods | NIQE | BRISQUE | |
---|---|---|---|

ROI 1 | SGM | 11.15 | 45.41 |

PPS | 10.81 | 45.26 | |

PSOP for orthorectified images | 11.24 | 46.15 | |

ROI 2 | SGM | 7.819 | 47.47 |

PPS | 7.540 | 47.37 | |

PSOP for orthorectified images | 8.230 | 48.31 |

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Peng, M.; Di, K.; Wang, Y.; Wan, W.; Liu, Z.; Wang, J.; Li, L.
A Photogrammetric-Photometric Stereo Method for High-Resolution Lunar Topographic Mapping Using Yutu-2 Rover Images. *Remote Sens.* **2021**, *13*, 2975.
https://doi.org/10.3390/rs13152975

**AMA Style**

Peng M, Di K, Wang Y, Wan W, Liu Z, Wang J, Li L.
A Photogrammetric-Photometric Stereo Method for High-Resolution Lunar Topographic Mapping Using Yutu-2 Rover Images. *Remote Sensing*. 2021; 13(15):2975.
https://doi.org/10.3390/rs13152975

**Chicago/Turabian Style**

Peng, Man, Kaichang Di, Yexin Wang, Wenhui Wan, Zhaoqin Liu, Jia Wang, and Lichun Li.
2021. "A Photogrammetric-Photometric Stereo Method for High-Resolution Lunar Topographic Mapping Using Yutu-2 Rover Images" *Remote Sensing* 13, no. 15: 2975.
https://doi.org/10.3390/rs13152975