# Automatic Extrinsic Self-Calibration of Mobile Mapping Systems Based on Geometric 3D Features

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## Abstract

**:**

## 1. Introduction

#### 1.1. Geometric Features

#### 1.2. Contributions and Structure

## 2. Related Work

#### 2.1. Extrinsic Calibration of Mobile Mapping Systems

#### 2.2. Geometric Features

- A linear (1D) structure is given for ${\lambda}_{1,i}\gg {\lambda}_{2,i},{\lambda}_{3,i}$, since the respective points in the local neighborhood are mainly spread along one principal axis.
- A planar (2D) structure is given for ${\lambda}_{1,i},{\lambda}_{2,i}\gg {\lambda}_{3,i}$, since the considered points spread within a plane spanned by two principal axes.
- A volumetric (3D) structure is given for ${\lambda}_{1,i}\approx {\lambda}_{2,i}\approx {\lambda}_{3,i}$, since the considered points are similarly spread in all directions.

## 3. Methodology

#### 3.1. Input and Output Parameters

- The first category consists of a single parameter that represents the initial size a of a voxel grid filter. The size is defined by the edge length of each voxel. Section 3.2.2 gives more details on the downsampling.
- The second category contains the measurements of the mapping sensor. For each timestep t the mapping sensor measures a point or a point set ${}^{m}\mathbf{X}_{t}$ in the Cartesian mapping frame m. The set of point sets for each utilized time step is the input to the self-calibration.
- The third category contains the measurements of the pose estimation sensor, analogous to the second category. For each timestep t the pose estimation sensor provides a 6-DOF pose ${}_{n}{}^{w}\mathbf{M}_{t}$ that is associated to a single point set. The 6-DOF pose ${}_{n}{}^{w}\mathbf{M}_{t}$ represents the rigid transformation from the navigation frame n to the world frame w. For this simplified formal description, synchronized sensors and the interpolation of movements during the scanning are assumed.
- The fourth category consists of the six initial calibration parameters. The initial calibration matrix ${}_{m}{}^{n}\mathbf{C}$ is a reparametrization of these parameters. In Section 4.1.2 we investigate how accurate the initial calibration parameters must be for the self-calibration to be applicable.

#### 3.2. Optimization

#### 3.2.1. Computation of the Point Cloud $\mathcal{P}$

#### 3.2.2. Downsampling

#### 3.2.3. Determination of the Geometric Feature

#### 3.2.4. Computation of the Cost Function and Parameter Estimation

#### 3.3. Multi-Scale Approach

## 4. Experiments

#### 4.1. Synthetic Data

#### 4.1.1. Suitability of the Cost Functions

#### 4.1.2. Radius of Convergence

#### 4.1.3. Single-Scale vs. Multi-Scale

#### 4.2. Real Data

#### 4.2.1. Small-Scale Indoor Dataset

#### 4.2.2. Large-Scale Outdoor Dataset

## 5. Discussion

#### 5.1. Suitability of the Cost Function

#### 5.2. Radius of Convergence

#### 5.3. Single-Scale vs. Multi-Scale

#### 5.4. Real data

#### 5.4.1. Small-Scale Indoor Dataset

#### 5.4.2. Large-Scale Outdoor Dataset

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Point cloud as a result of a mobile mapping in an indoor environment. Due to the use of a line scanning device and non-uniform movements of the mobile mapping platform, the points are inhomogeneously sampled. The color of each point illustrates its height. Top: Inaccurate guess of the extrinsic calibration parameters. Bottom: Calibrated using the Robust Automatic Detection in Laser Of Calibration Chessboards (RADLOCC) [1,2,3], which is a target calibration approach for cameras and laser scanners.

**Figure 2.**Workflow of the self-calibration. Besides the scan points and associated poses, initial calibration parameters and an initial voxel grid size are the inputs to a nonlinear least squares optimization. This optimization is performed recursively with multiple scales to be robust as well as accurate. It estimates the calibration parameters by minimizing an cost function which is based on a selected geometric feature.

**Figure 3.**Calibration results depending on the ${f}_{RQE}$ and the features of planarity (${g}_{P}$), sphericity (${g}_{S}$), omnivariance (${g}_{O}$), eigenentropy (${g}_{E}$) and change of curvature (${g}_{C}$). (

**a**) shows the translation error, (

**b**) the rotation error. The feature of linearity has turned out to be unsuitable for the calibration approach and is not shown for display reasons. The features of omnivariance and eigenentropy clearly outperform ${f}_{RQE}$ and all other features.

**Figure 4.**Required accuracy of the initial calibration parameters ((

**a**): Initial translation, (

**b**): Initial rotation). Up to an initial translation error of $2.2m$ or an initial rotation error of ${31}^{\circ}$ the results are in the range of sub-millimeters and sub-degrees. More inaccurate values for the initial calibration parameters mean that the optimization no longer converges to the global minimum and the errors clearly grow.

**Figure 5.**Single-scale vs. multi-scale calibration. (

**a**) shows the translation error after performing the calibration depending on the initial translation error. (

**b**) shows the rotation error after performing the calibration depending on the initial rotation error. The coarse-scale approach is more robust to large initial calibration errors than the fine-scale approach. However, the fine-scale approach is more accurate for small initial calibration errors. The multi-scale approach combines the advantages of both single-scale approaches.

**Figure 6.**Real data results on the small-scale indoor dataset. The self-calibration based on the RQE achieves comparable results to the TC. The self-calibrations based on the features of linearity ${g}_{L}$ and planarity ${g}_{P}$ failed. The self-calibration based on the feature of sphericity ${g}_{S}$, however, results in a sharper point cloud than the target calibration.

**Figure 7.**Real data results on the small-scale indoor dataset. The self-calibrations based on the features of omnivariance ${g}_{O}$, eigenentropy ${g}_{E}$ and change of curvature ${g}_{C}$ lead to sharper point clouds compared to the target calibration.

**Figure 8.**Calibration results of the different approaches for the small-scale indoor dataset. (

**a**–

**c**) show the translation parameters, (

**d**–

**f**) show the rotation parameters. For the TC, only a single set of parameters is available without information about the precision. The medians of multiple calibrations differ significantly from the results of the target-calibration. The self-calibration based on omnivariance ${g}_{O}$ estimates all parameters with higher precision compared to the calibration based on $RQE$.

**Figure 9.**Results on the publicly available KITTI dataset [44]. The color of each point depicts its height. Both calibrations improved the initial calibration parameters such that objects and basic structures are identifiable in the point clouds. Moreover, the RC and the calibration based on the feature of omnivariance lead to very similar point clouds. The calibration based on the ${f}_{RQE}$, however, results in a more noisy point cloud, which can be seen especially at the vertical walls and at artifacts next to the two cars on the right.

**Figure 10.**Calibration results of the reference calibration (RC), the self-calibration based on the $RQE$ and our self-calibration based on the the feature of omnivariance for the large-scale outdoor dataset. (

**a**–

**c**) show the translation parameters, (

**d**–

**f**) show the rotation parameters. The estimation of the vertical component Y of the translation fails for the self-calibration based on $RQE$. The estimation of the calibration parameters is more precise for the self-calibration based on the omnivariance ${g}_{O}$, however, $\Phi $ and $\Psi $ is estimated with a larger deviation from the reference calibration.

**Table 1.**Comparison of the median of the RQE (med $\left(RQE\right)$) and the redefined features (med $\left({g}_{L}\right)$, …, med $\left({g}_{C}\right)$) for different calibration approaches. TC is the target calibration based on RADLOCC. $RQE$ is the self-calibration based on the RQE. ${g}_{L}$, …, ${g}_{C}$ are the self-calibrations based on the corresponding feature. Lower feature values are better. The lowest feature value in each column is marked in bold. X: Calibration failed. The calibration based on the RQE leads to slightly higher values than the TC. The self-calibrations based on ${g}_{S}$, ${g}_{O}$, ${g}_{E}$ and ${g}_{C}$ outperform the TC and the calibration based on $RQE$. Omnivariance ${g}_{O}$ achieves the best results.

Approach | med $\left({\mathit{f}}_{\mathit{RQE}}\right)$ | med $\left({\mathit{g}}_{\mathit{L}}\right)$ | med $\left({\mathit{g}}_{\mathit{P}}\right)$ | med $\left({\mathit{g}}_{\mathit{S}}\right)$ | med $\left({\mathit{g}}_{\mathit{O}}\right)$ | med $\left({\mathit{g}}_{\mathit{E}}\right)$ | med $\left({\mathit{g}}_{\mathit{C}}\right)$ |
---|---|---|---|---|---|---|---|

TC | $0.161$ | $0.842$ | $0.389$ | $0.215$ | $0.275$ | $0.734$ | $0.106$ |

${f}_{RQE}$ | $0.205$ | $0.845$ | $0.450$ | $0.268$ | $0.288$ | $0.787$ | $0.129$ |

${g}_{L}$ | X | X | X | X | X | X | X |

${g}_{P}$ | X | X | X | X | X | X | X |

${g}_{S}$ | $0.092$ | $\mathbf{0}.\mathbf{826}$ | $0.281$ | $0.087$ | $0.217$ | $0.546$ | $0.047$ |

${g}_{O}$ | $\mathbf{0}.\mathbf{082}$ | $0.830$ | $\mathbf{0}.\mathbf{264}$ | $\mathbf{0}.\mathbf{072}$ | $\mathbf{0}.\mathbf{205}$ | $\mathbf{0}.\mathbf{514}$ | $\mathbf{0}.\mathbf{039}$ |

${g}_{E}$ | $0.088$ | $0.828$ | $0.271$ | $0.079$ | $0.211$ | $0.529$ | $0.042$ |

${g}_{C}$ | $0.095$ | $\mathbf{0}.\mathbf{826}$ | $0.283$ | $0.090$ | $0.220$ | $0.552$ | $0.048$ |

**Table 2.**Mean point-to-plane distances for different parts of the point cloud (in mm). X: Calibration failed. The smallest point-to-plane distance in each column is marked in bold. Again, all self-calibrations except for ${g}_{L}$ and ${g}_{P}$ achieve better results than the TC and the calibration based on $RQE$. For five of the six planes, the omnivariance ${g}_{O}$ achieves the lowest mean distances.

Approach | Plane 1 | Plane 2 | Plane 3 | Plane 4 | Plane 5 | Plane 6 |
---|---|---|---|---|---|---|

TC | $14.1$ | $21.7$ | $13.6$ | $15.1$ | $14.5$ | $22.8$ |

${f}_{RQE}$ | $14.3$ | $33.0$ | $17.0$ | $25.0$ | $28.4$ | $26.3$ |

${g}_{L}$ | X | X | X | X | X | X |

${g}_{P}$ | X | X | X | X | X | X |

${g}_{S}$ | $7.1$ | $11.4$ | $9.5$ | $7.5$ | $\mathbf{9}.\mathbf{9}$ | $6.7$ |

${g}_{O}$ | $\mathbf{6}.\mathbf{3}$ | $\mathbf{8}.\mathbf{3}$ | $\mathbf{7}.\mathbf{7}$ | $\mathbf{7}.\mathbf{4}$ | $10.7$ | $\mathbf{6}.\mathbf{6}$ |

${g}_{E}$ | $10.2$ | $\mathbf{8}.\mathbf{3}$ | $8.1$ | $9.3$ | $10.1$ | $8.0$ |

${g}_{C}$ | $8.6$ | $11.2$ | $7.8$ | $7.9$ | $11.6$ | $9.7$ |

**Table 3.**Mean point-to-plane distances for different parts of the point cloud of the large-scale outdoor dataset (in cm). The smallest point-to-plane distance in each column is marked in bold. For three out of six planes the RC achieves the best results. For the other three planes the self-calibration based on the feature of omnivariance achieves the best results.

Approach | Plane 1 | Plane 2 | Plane 3 | Plane 4 | Plane 5 | Plane 6 |
---|---|---|---|---|---|---|

$RC$ | $5.4$ | $\mathbf{3}.\mathbf{0}$ | $5.4$ | $\mathbf{8}.\mathbf{3}$ | $\mathbf{8}.\mathbf{2}$ | $16.4$ |

${f}_{RQE}$ | $11.4$ | $10.3$ | $7.9$ | $8.6$ | $10.0$ | $23.1$ |

${g}_{O}$ | $\mathbf{5}.\mathbf{0}$ | $4.5$ | $\mathbf{4}.\mathbf{1}$ | $8.7$ | $9.4$ | $\mathbf{13}.\mathbf{3}$ |

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**MDPI and ACS Style**

Hillemann, M.; Weinmann, M.; Mueller, M.S.; Jutzi, B.
Automatic Extrinsic Self-Calibration of Mobile Mapping Systems Based on Geometric 3D Features. *Remote Sens.* **2019**, *11*, 1955.
https://doi.org/10.3390/rs11161955

**AMA Style**

Hillemann M, Weinmann M, Mueller MS, Jutzi B.
Automatic Extrinsic Self-Calibration of Mobile Mapping Systems Based on Geometric 3D Features. *Remote Sensing*. 2019; 11(16):1955.
https://doi.org/10.3390/rs11161955

**Chicago/Turabian Style**

Hillemann, Markus, Martin Weinmann, Markus S. Mueller, and Boris Jutzi.
2019. "Automatic Extrinsic Self-Calibration of Mobile Mapping Systems Based on Geometric 3D Features" *Remote Sensing* 11, no. 16: 1955.
https://doi.org/10.3390/rs11161955