# Towards System Calibration of Panoramic Laser Scanners from a Single Station

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Previous Work

#### 1.2. Aim of this Study

- The first aim is to prove that in the case of the panoramic laser scanners most of the calibration parameters can be estimated from a single scanner station, without a need for any reference information. Therefore, substantial time savings can be achieved, and all targets can be optimally oriented for a single scanner station. This eliminates the need to use measurements compromised by steep incidence angles leading to larger errors [31,32]. Moreover, it will be discussed that even all relevant calibration parameters can be estimated, if an additional reference information is introduced.
- The second aim is to describe an adaptation and implementation of the mechanically interpretable calibration parameters from [29] to the calibration algorithms described in the Section 2.3 and Section 2.5. This implementation is not straightforward and it is discussed in detail in the subsection titled Implementation of Section 2.2. The motivation for implementing these parameters is the presumption that using the mechanically explainable parameters will lead to their better stability and reusability. Namely, the stability of the parameters usually used in the user oriented self-calibration (also system calibration) approaches is still not adequately investigated, and could be questioned, as for example in [33]. This presumption was not tested within the conducted experiment, but it is the part of the ongoing investigation and will be incorporated in future publications.
- Third, the new angular parameterization is introduced for the implicit implementation of two-face measurements. This led to a development of the new calibration algorithm based on the scanning of all targets in two faces of the instrument. It can be used to estimate most of the relevant calibration parameters with a simple measurement setup and from a single scanner station. This makes it an interesting solution for a quick instrument check-up prior to the higher demanding field tasks.

## 2. Theoretical Background

#### 2.1. Instrument Geometry and Angular Parameterization

#### 2.2. Calibration Parameters

#### 2.2.1. Laser Source

_{1n}and x

_{1z}calibration parameters. Similar to the translation case, the laser source can rotate around three axes. While the rotation around the horizontal axis does not influence the measurements, the rotations around the vertical axis and collimation axis influence the measurements (Figure 3b). These misalignments are denoted as parameters x

_{5n}and x

_{5z}.

#### 2.2.2. Rotating Mirror

_{3}. The influence on the range is successfully removed in most of the laser scanners by the reference range measurement or it is absorbed in the rangefinder offset parameter, so it is not included in the functional model of the range related calibration parameters (Equation (4)). The second error is the mirror tilt (Figure 4b), denoted as x

_{6}. This error describes the case when the mirror is incorrectly placed on its mount and because of that it does not intersect the vertical axis with the inclination of 45°. The term influences only the measured horizontal angles.

#### 2.2.3. Primary Rotational or Horizontal Axis

_{2}. The second case is equal to the horizontal axis error in the total station, horizontal and vertical axes do not intersect forming an angle of 90° (Figure 5b). This case is modelled with the x

_{7}parameter. The difference between the latter error and the error represented in Figure 4b is in the different position of the horizontal axis in space, which conditions the different path of the laser beam. Hence, it has different effect on the measured angles.

#### 2.2.4. Total Station Related Parameters

_{4}), rangefinder offset (x

_{10}), and encoder related errors. A comparison between the total station model of parameters [21] and mechanical model is given in Table 1. The main focus of the research is placed on the imperfection of the instrument components assembly. Therefore, an extensive rangefinder calibration was not included. As a result, calibration parameters such as rangefinder scale and cyclic errors are omitted.

#### 2.2.5. Implementation

_{5z}) and the horizontal axis tilt (x

_{7}) are combined in one (x

_{5z−7}) due to the same functional definition. Avoiding this operation would lead to a poor condition number and to a singularity in the normal equations. We would like to specially highlight the case of the parameters x

_{1n}and x

_{2}. Although, from the theoretical stand point, they could be separately estimated without singularity in the normal equations, we strongly recommend introducing the combined parameter x

_{1n+2}. Leaving out this parameter leads to a significant bias in the final estimates of the calibration parameters. This is explained in detail in Section 4.1 through the conducted simulations.

_{5n}) affects both horizontal and vertical angles, but in practice the case is different. As it was proven in [28], this error term is partially absorbed in the exterior orientation parameters, more precisely in the rotation angle around the vertical axis. Consequently, there is no influence on the horizontal angles and this parameter can be omitted from the horizontal angles equation. Furthermore, the used TLS exploits four orthogonal reading heads for the calculation of both horizontal and vertical angles [4]. Therefore, it can be presumed that all encoder related errors (Table 1) are averaged and removed. As a result, they are omitted from the functional model.

_{…}terms are the calibration parameters describing the systematic errors and ${v}_{{r}_{j}^{i}}$, ${v}_{{\phi}_{j}^{i}},\text{}{v}_{{\theta}_{j}^{i}}$ are the adjustment residuals describing the random errors. As it can be seen from the equations, this paper uses a set of 11 calibration parameters in order to describe all relevant mechanical misalignments. It is important to note that the multiplication factor k used in the referent literature [29] was omitted herein due to the different parametrization of the polar measurements.

#### 2.2.6. Two-Face Sensitivity

_{1z}is sensitive in the horizontal angle equation, while it is not in the vertical angle equation. For the parameter x

_{1n}, the case is opposite. It can be estimated in the vertical angle equation within the parameter x

_{1n+2}and separated later (detailed explanation in Section 4.1). Therefore, we can say that the parameters x

_{1z}and x

_{1n}are partially two-face sensitive.

_{5z}is contained within the parameter x

_{5z−7}, which is a two face sensitive parameter. However, due to the same functional definition and the lack of additional information, like in the case of parameters x

**and x**

_{1n}_{2}, the value of the parameter x

_{5z}cannot be separated from the parameter x

_{5z−7}. Hence, the value of the parameter x

_{5z}cannot be estimated from two-face measurements and henceforth, we treat this parameter as the parameter insensitive to two face measurements.

_{10}) and vertical beam tilt (x

_{5z}). The parameters should be at least partially two-face sensitive in order to be estimated from a single scanner station without any reference. Otherwise, an additional reference value is mandatory for the complete TLS calibration. This will be explained on a simple example of the rangefinder offset parameter. The rangefinder offset retains the same sign in each of the two-face measurements. Therefore, it is not contained in the difference $dr$ and it cannot be estimated by observing one target from one location without a reference. There are two straightforward ways of estimating the parameter x

_{10}depicted in Figure 6 [35].

_{10}equals half of the difference between the true value and the value measured with the scanner. In the second case, we do not know the true value of the distance between targets. For this instance, an additional scan is required, with the scanner placed outside of the measured distance while still being placed in line with the targets (Figure 6b). The distance estimated from the second scan will not be influenced by the parameter x

_{10}and, therefore, it can be used as a reference distance, in the same manner as in the previous example.

_{10}. A similar logic can be applied for the parameter x

_{5z}. Therefore, the reference distance is required for its successful estimation. The optimal placement of the reference distance for estimating the parameter x

_{5z}is a part of the ongoing investigation.

#### 2.3. Functional Model of the System Calibration (Self-Calibration)

#### 2.4. Stochastic Model

#### 2.5. Two-Face Adjustment Algorithm

_{10}, the vertical beam tilt x

_{5z}and the horizontal beam offset x

_{1n}. While x

_{10}and x

_{5z}cannot be estimated using the proposed approach, x

_{1n}can be derived a posteriori. It is achieved by subtracting the value of the parameter x

_{2}estimated in Equation (17) from the value of the combined parameter x

_{1n+2}estimated in Equation (19). The detailed explanation is given in Section 4.1.

#### 2.6. Congruency Test

## 3. Experiment

#### 3.1. Empirical Experiment

#### 3.1.1. Instrument

#### 3.1.2. Calibration Field

^{2}). They were attached with adhesive tape on the predetermined locations on the building. The data acquisition was carried out within 24 h after the network assembly, guaranteeing the target stability. Additionally, 16 Leica “Tilt & Turn” targets, placed on tripods and magnetic holders, were incorporated in order to improve the network geometry at otherwise occluded parts in the back of the building.

#### 3.1.3. Obtaining Scanner Measurements

#### 3.1.4. Preprocessing Scanner Measurements (in Leica Cyclone)

#### 3.2. Simulation Experiment

_{10}was set to −2 mm, because it is expected to have the larger value. Also, the parameter x

_{7}was set to 8ʺ in order to avoid mutual elimination of the parameters x

_{7}and x

_{5z}, see Equation (5).

#### 3.3. Data Processing

## 4. Results and Discussion

#### 4.1. Simulation Results

_{5z−7}and x

_{1n+2}combine the influence of two misalignments and, therefore, their values are doubled. The first calibration attempt was conducted without introducing the additional calibration parameter x

_{1n+2}in the vertical angle equation (Equation (6)). Instead, the parameters x

_{1n}and x

_{2}are estimated separately as: $+\frac{({x}_{1n}+{x}_{2})\mathrm{cos}\left({\theta}_{j}^{i}\right)}{{r}_{j}^{i}}$. As it can be seen from Table 2, this leads to a noticeable bias in the estimate of the parameters x

_{1n}and x

_{5n}, which are highly correlated. In the second calibration attempt, the proposed parameter x

_{1n+2}is introduced, and the estimate of the mentioned parameters is evidently improved. Therefore, only the second attempt is discussed in further detail.

_{5z−7}and x

_{5n}) are determined with a lower precision of approximately 2.5ʺ, but without bias, while two parameter estimates are somewhat biased (x

_{1n}and x

_{5z}). The bias of the parameter x

_{1n}estimate can be bypassed. Namely, the parameter x

_{1n+2}contains the combined influence of the parameters x

_{1n}and x

_{2}. Additionally, the parameters x

_{1n+2}and x

_{2}are estimated with the higher accuracy and precision than x

_{1n}. Hence, the better estimate of the parameter x

_{1n}can be derived by subtracting the influence of the parameter x

_{2}from x

_{1n+2}. The corresponding precision can also be estimated using the low of error propagation [6]. The value and the precision of the parameter x

_{1n}estimated this way are: −0.25 and 0.05 mm. Therefore, the unbiased and more precise estimate is derived. For the evaluation, in one calibration attempt not provided herein, the parameter x

_{1n}was completely removed from the calibration adjustment (from Equation (5)). This led to no noticeable differences in the calibration results. Hence, we kept the parameter for sake of comparison.

_{5z}, which is almost perfectly correlated with the translation parameter in the direction of the z axis of the second scanner station. These correlations should be mitigated with a better network design, and this is the part of ongoing investigation in a further study. Finally, most of the calibration parameters sensitive to two-face measurements (Equations (17)–(19)) have low correlations with EOPs and OPs.

_{1n+2}clearly failed the statistical test. On the contrary, in the second case, the test statistic was lower than the threshold value, resulting in the test acceptance. This indicates that the network realized in the experiment is sensitive enough to estimate unbiased parameters if the proposed set of calibration parameters is used (Equations (4)–(6)). In other words, the estimated parameters do not significantly differ from their true values and the used network is proved to be valid for further analysis.

#### 4.2. Empirical Results

_{10}and x

_{5z}cannot be estimated from a single scanner station without reference information, as explained in Section 2.3. Additionally, parameter x

_{1n}needs to be estimated a posteriori as explained in the previous section (Section 4.1). We tried to directly estimate parameters x

_{10}, x

_{5z}and x

_{1n}from a single station in order to test these premises. While the inclusion of the parameters x

_{10}and x

_{1n}caused that the adjustment could not converge, the parameter x

_{5z}was estimated. However, the estimated value of −97.97° was obviously false and the parameter was perfectly correlated with one point coordinate. Removing the mentioned three parameters leads to the successful calibration.

## 5. Conclusions and Future Work

- To prove that most of the calibration parameters can be estimated from a single scanner station,
- To prove that the proposed two-face adjustment can yield similar results to more complex self-calibration based on the bundle adjustment and
- To present the adaptation of the mechanically interpretable calibration parameters to the system calibration of terrestrial laser scanners.

- Most of the parameters can be estimated from a single scanner station, without need for any reference information. More precisely nine out of 11 calibration parameters were successfully determined this way. This means that calibration time can be considerably reduced, in the present case from approximately five to two hours.
- In order to estimate all mechanical parameters from a single scanner station, reference measurements are required. Namely, this is the case for the remaining two parameters that are not sensitive to two-face measurements (x
_{10}and x_{5z}). - The proposed two-face adjustment can yield comparable results to the usual self-calibration strategies. Even though it is not rigorous, it is proven to be a fast and simple solution for the calibration from a single scanner station.
- The implementation of the new systematic error parametrization in the usual self-calibration approach requires some modifications. The most interesting one is the introduction of the calibration parameter x
_{1n+2}, which successfully eliminated the bias from some parameter estimates. - Using the same set of exterior orientation parameters for two consecutive scans from a single scanner station (two-face measurements) is a mandatory prerequisite for the scanner calibration from only one station, if the position of the scanner is not adequately controlled.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**(

**a**) Local Cartesian coordinate system of the scanner with a respect to the main instrument axes; (

**b**) Local coordinate system of the scanner transformed to the polar coordinates.

**Figure 3.**Laser source related mechanical misalignments: (

**a**) laser source offset; (

**b**) laser source tilt.

**Figure 6.**Estimating rangefinder offset parameter x

_{10}: (

**a**) with known reference; (

**b**) without known reference.

**Figure 8.**Network configuration—the scanner station locations (S1, S2, S3) with the orientation of scanner local coordinate systems and target distribution.

**Figure 9.**Histograms of the measurement residuals. The red color presents the residuals of the uncalibrated measurements, while the green color presents the residuals of the calibrated measurements (top: 3 × SS—calibration using parameters from Table 4, bottom: 1 × SS—calibration using parameters from the last two columns of Table 5).

**Figure 10.**The improvement achieved by using the different sets of the calibration parameters in: estimated precision of ranges, horizontal and vertical angles and 3D point position at 50 m.

Parameter | Description | Equivalent in the Total Station Model |
---|---|---|

x_{1n} | Horizontal beam offset | Horizontal eccentricity of collimation axis * |

x_{1z} | Vertical beam offset | None |

x_{2} | Horizontal axis offset | Laser axis vertical offset * |

x_{3} | Mirror offset | None |

x_{4} | Vertical index offset | Identical |

x_{5n} | Horizontal beam tilt | Vertical circle eccentricity error * |

x_{5z} | Vertical beam tilt | Horizontal axis & vertical circle eccentricity error |

x_{6} | Mirror tilt | Collimation axis error |

x_{7} | Horizontal axis error (tilt) | Identical |

x_{8x} | Horizontal angle encoder eccentricity | Identical |

x_{8y} | Horizontal angle encoder eccentricity | Identical |

x_{9n} | Vertical angle encoder eccentricity | Identical |

x_{9z} | Vertical angle encoder eccentricity | Identical |

x_{10} | Rangefinder offset | Identical |

x_{11a} | Second order scale error in the horizontal angle encoder | Identical |

x_{11b} | Second order scale error in the horizontal angle encoder | Identical |

x_{12a} | Second order scale error in the vertical angle encoder | Identical |

x_{12b} | Second order scale error in the vertical angle encoder | Identical |

**Table 2.**Comparison of the true and estimated values of the simulated calibration parameters. Correlations presented on the right hand side apply on the 2nd estimate, with first two columns indicating overall maximal correlations and last two columns indicating maximal correlations with respect towards exterior orientation (EOPs) and object point parameters (OPs). Notations: XYZ—N indicates correlation with one of the coordinates of the target N, while for example Tx

_{1}and Rx

_{1}indicate correlation with translation and rotation parameters with respect to x axis of the first scanner station.

Parameter | True | 1st Estimate | 2nd Estimate | Overall Corr. | w.r.t. EOPs & OPs | ||||
---|---|---|---|---|---|---|---|---|---|

x | $\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | $\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | Corr. | With | Corr. | With | |

x_{1n} [mm] | −0.20 | 0.07 | 0.02 | −0.10 | 0.06 | 0.66 | EOP-Rz_{1} | 0.66 | EOP-Rz_{1} |

x_{1z} [mm] | −0.20 | −0.18 | 0.05 | −0.18 | 0.05 | −0.98 | x_{5z−7} | −0.62 | EOP-Tz_{3} |

x_{2} [mm] | −0.20 | −0.18 | 0.02 | −0.17 | 0.02 | 0.21 | EOP-Tx_{3} | 0.21 | EOP-Tx_{3} |

x_{3} [mm] | −0.20 | −0.20 | 0.02 | −0.21 | 0.02 | 0.80 | x_{1z} | −0.49 | EOP-Tz_{3} |

x_{4} ["] | −8.00 | −8.39 | 0.38 | −7.97 | 0.30 | −0.45 | x_{5n} | −0.21 | EOP-Ry_{3} |

x_{5n} ["] | −8.00 | −15.52 | 1.54 | −6.22 | 2.50 | −0.86 | x_{1n+2} | 0.11 | XYZ-203 |

x_{5z−7} ["] | −16.00 | −17.83 | 2.22 | −17.65 | 2.22 | −0.98 | x_{1z} | 0.60 | EOP-Tz_{3} |

x_{6} ["] | −8.00 | −7.81 | 0.17 | −7.79 | 0.17 | −0.56 | x_{3} | −0.21 | EOP-Rz_{3} |

x_{5z} ["] | −8.00 | −13.09 | 1.99 | −12.55 | 1.98 | −0.95 | EOP-Tz_{2} | −0.95 | EOP-Tz_{2} |

x_{10} [mm] | −2.00 | −2.07 | 0.07 | −2.06 | 0.07 | −0.58 | EOP-Tx_{3} | −0.58 | EOP-Tx_{3} |

x_{1n+2} [mm] | −0.40 | - | - | −0.42 | 0.05 | −0.86 | x_{5n} | 0.06 | XYZ-209 |

Estimate | ${\mathit{T}}_{\mathit{c}}$ | ${\mathit{F}}_{\left(\mathit{h},\mathit{r},1-\mathit{\alpha}\right)}$ |
---|---|---|

1st estimate | 49.82 | 1.83 |

2nd estimate | 1.63 | 1.79 |

**Table 4.**Estimated calibration parameters from the field experiment. The notation 3 × SS denotes that all three scanner stations were used in the calibration process.

All Parameters | Correlations | |||||
---|---|---|---|---|---|---|

Parameter | 3 × SS | Overall Corr. | w.r.t. EOPs & OPs | |||

$\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | Corr. | With | Corr. | With | |

x_{1n} [mm] | −0.14 | 0.08 | 0.68 | EOP-Rz_{2} | 0.68 | EOP-Rz_{2} |

x_{1z} [mm] | 0.29 | 0.06 | −0.98 | x_{5z−7} | −0.64 | EOP-Tz_{3} |

x_{2} [mm] | 0.05 | 0.03 | 0.20 | EOP-Tx_{3} | 0.20 | EOP-Tx_{3} |

x_{3} [mm] | −0.07 | 0.02 | 0.81 | x_{1z} | −0.52 | EOP-Tz_{3} |

x_{4} ["] | −6.83 | 0.34 | −0.45 | x_{5n} | −0.21 | EOP-Ry_{3} |

x_{5n} ["] | −9.15 | 2.85 | −0.86 | x_{1n+2} | −0.11 | XYZ-208 |

x_{5z−7} ["] | −19.78 | 2.69 | −0.98 | x_{1z} | 0.62 | EOP-Tz_{3} |

x_{6} ["] | 3.64 | 0.21 | −0.56 | x_{3} | −0.21 | EOP-Rz_{3} |

x_{5z} ["] | −5.11 | 2.34 | −0.95 | EOP-Tz_{2} | −0.95 | EOP-Tz_{2} |

x_{10} [mm] | 0.61 | 0.09 | −0.59 | EOP-Tx_{3} | −0.59 | EOP-Tx_{3} |

x_{1n+2} [mm] | −0.17 | 0.05 | −0.86 | x_{5n} | 0.06 | XYZ-209 |

**Table 5.**Comparison of the different calibration attempts estimating only two-face sensitive calibration parameters (Equations (17)–(19)). Notation explanation: 3 × SS—calibration using measurements from all three scanner stations, 1 × SS (2 × EOP)—calibration using only data from the first scanner station and assigning different exterior orientation parameters for each scan, 1 × SS (1 × EOP)—assigning one set of exterior orientation parameters for both scans, 1 × SS (two-face adj.)—calibration using the two-face adjustment algorithm.

Two-Face Sensitive Parameters | ||||||||
---|---|---|---|---|---|---|---|---|

Parameter | 3 × SS | 1 × SS (2 × EOP) | 1 × SS (1 × EOP) | 1 × SS (Two-Face Adj.) | ||||

$\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | $\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | $\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | $\widehat{\mathit{x}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{x}}$ | |

x_{1n+2} [mm] | −0.18 | 0.05 | −0.33 | 0.08 | −0.28 | 0.08 | −0.28 | 0.10 |

x_{1z} [mm] | 0.27 | 0.06 | 0.20 | 0.09 | 0.33 | 0.09 | 0.33 | 0.11 |

x_{2} [mm] | 0.06 | 0.03 | −0.01 | 0.04 | 0.06 | 0.04 | 0.06 | 0.02 |

x_{3} [mm] | −0.08 | 0.02 | −0.15 | 0.04 | −0.13 | 0.03 | −0.13 | 0.04 |

x_{4} ["] | −6.86 | 0.35 | −5.00 | 0.75 | −6.75 | 0.51 | −6.75 | 0.61 |

x_{5n} ["] | −8.54 | 2.88 | −4.37 | 4.03 | −4.21 | 4.07 | −4.21 | 4.88 |

x_{5z−7} ["] | −18.83 | 2.71 | −16.28 | 3.92 | −20.99 | 3.71 | −20.99 | 4.59 |

x_{6} ["] | 3.65 | 0.21 | 4.51 | 0.36 | 4.50 | 0.33 | 4.50 | 0.41 |

**Table 6.**Congruency test used for evaluating if the parameter estimates from only one scanner station are significantly different from the calibration using all scanner stations (rejection signalizing the significant difference).

Calib. Attempts | ${\mathit{T}}_{\mathit{c}}$ | ${\mathit{F}}_{\left(\mathit{h},\mathit{r},1-\mathit{\alpha}\right)}$ |
---|---|---|

1 × SS (2 × EOP) | 2.26 | 1.94 |

1 × SS (1 × EOP) | 1.27 | 1.94 |

1 × SS (two-face adj.) | 1.23 | 1.94 |

$\widehat{\mathit{\sigma}}$ | $\mathit{r}$ [mm] | $\mathit{\phi}$ ["] | $\mathit{\theta}$ ["] | 3D pt.@50 m [mm] |
---|---|---|---|---|

Without calibration | 0.78 | 10.03 | 8.61 | 3.30 |

3 × SS | 0.75 | 7.98 | 7.46 | 2.75 |

1 × SS | 0.78 | 8.16 | 7.45 | 2.79 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Medić, T.; Holst, C.; Kuhlmann, H.
Towards System Calibration of Panoramic Laser Scanners from a Single Station. *Sensors* **2017**, *17*, 1145.
https://doi.org/10.3390/s17051145

**AMA Style**

Medić T, Holst C, Kuhlmann H.
Towards System Calibration of Panoramic Laser Scanners from a Single Station. *Sensors*. 2017; 17(5):1145.
https://doi.org/10.3390/s17051145

**Chicago/Turabian Style**

Medić, Tomislav, Christoph Holst, and Heiner Kuhlmann.
2017. "Towards System Calibration of Panoramic Laser Scanners from a Single Station" *Sensors* 17, no. 5: 1145.
https://doi.org/10.3390/s17051145