# Calibration of a Rotating or Revolving Platform with a LiDAR Sensor
## 4 Evaluation
### Quality Measures  
The lack of knowledge about a ground truth real-world pose of the LiDAR makes it necessary to evaluate the quality of the optimised extrinsic parameters in an indirect way. In our experiments, this  
Figure 5: Two possible target arrangements. The left depicts an optimal placement with high target distances \(L_{02}\) and \(L_{13}\) as well as an angle between them of \(\omega=90^{\circ}\). For the arrangement on the right, however, the targets are close to the LiDAR and an there is an angle \(\omega^{\prime}\
eq 90^{\circ}\) between \(L^{\prime}{}_{02}\) and \(L^{\prime}{}_{13}\). The latter is expected to lead to suboptimal calibration results.  
is done by rating the quality of recorded point clouds utilising the optimisation results from several calibration runs. Therefore, we introduce a quality measure to rate the recorded point clouds in an offline phase.  
There would be an obvious quality measure of the resulting point cloud by comparing the points of the redundant measurements, e.g., by averaging the distance to their nearest redundant neighbour. However, since there is no correspondence between the single points of the different clouds defined, it is not possible to measure their distance or something like that. As the experiments are done in an artificial environment, there are many planar surfaces in the scene. How crisp or sharp a point cloud is can therefore be measured in a more semantic way. The effect on planar surfaces, like walls, were already shown in Figure 1 and thus the point cloud quality corresponds with how noisily those planar regions have been sampled.  
To get this information, the largest planes are subsequently detected by fitting planes to the point cloud via a Random Sample Consensus (RANSAC) [18] method followed by a region growing taking only the Euclidean point distance into account. This separates planar unconnected patches lying in the same plane, such as several tables of the same height. This results in a set of smaller point clouds \(P_{k}=\{\vec{p}_{0},\vec{p}_{1},...\vec{p}_{N_{k}-1}\}\), \(k=0,1,2,\ldots,M-1\), each containing the samples of the respective planar surface patch. Afterwards, the centroid \(\vec{\mu}_{k}\) of each patch is calculated as  
\[\vec{\mu}_{k}=\frac{1}{N_{k}}\sum_{i=0}^{N_{k}-1}\vec{p}_{i},\ \ \ \vec{p}_{i}\in\mathbb{R}^{3\times 1},i=0,1,2,\ldots,N_{k}-1 \tag{14}\]  
and used to compute the covariance matrix \(C_{k}\) of this point set \(P_{k}\) as  
\[C_{k}=\frac{1}{N_{k}}\sum_{i=0}^{N_{k}-1}(\vec{\mu}_{k}-\vec{p}_{i})^{T}*(\vec {\mu}_{k}-\vec{p}_{i}),\ \ \ \vec{\mu}_{k},\vec{p}_{i}\in\mathbb{R}^{3\times 1},i=0,1,2, \ldots,N_{k}-1. \tag{15}\]  
The covariance matrix \(C_{k}\) encodes the squared deviations in the dataset with respect to \(\vec{\mu}_{k}\). Since, the data points have three dimensions, \(C_{k}\) also has three eigenvectors \(\vec{v}_{k0},\vec{v}_{k1},\vec{v}_{k2}\in\mathbb{C}^{3\times 1}\), with  
\[C_{k}*\vec{v_{ki}}=\lambda_{ki}*\vec{v_{ki}},\ \ \ i=0,1,2, \tag{16}\]  
where the eigenvalues \(\lambda_{k0},\lambda_{k1},\lambda_{k2}\in\mathbb{C}\) with \(\lambda_{k0}\geq\lambda_{k1}\geq\lambda_{k2}\) are associated with the eigenvectors \(\vec{v}_{k0},\vec{v}_{k1},\vec{v}_{k2}\), and represent how strong the deviation in the respective direction is. Because a plane only expands in two dimensions, the eigenvector \(\vec{v}_{k2}\), associated with the smallest eigenvalue \(\lambda_{k2}\), is perpendicular to the plane, if the sampled data contains no additional systematic noise in other directions. In fact, the magnitude of change in this direction has to be the sensor noise or systematic deviations like they appear when points are erroneously transformed from sensor- to common frame (Figure 1). Summing up this error for every plane \(P_{k}\) found in the scene and normalising it with the number of planes performs as a measure \(E(P)\) for the cloud sharpness:  
\[E(P)=\frac{1}{M}\sum_{k=0}^{M-1}\lambda_{k2}. \tag{17}\]