# Sensitivity Analysis and Minimal Measurement Geometry for the Target-Based Calibration of High-End Panoramic Terrestrial Laser Scanners

^{*}

## Abstract

**:**

## 1. Introduction

- To find the most sensitive measurement geometry for estimating each individual mechanical misalignment for the instrument under investigation and
- to prove the practicality of the conducted sensitivity analysis by deriving the minimal measurement geometry sensitive to all relevant mechanical misalignments.

## 2. Theoretical Background

#### 2.1. Instrument Geometry and Angular Parameterization

#### 2.2. Calibration Parameters

#### 2.3. Functional Model of the Target-Based Self-Calibration

#### 2.4. Stochastic Model

## 3. Sensitivity Analysis

- An absolute displacement of a single target (a spatial vector in arbitrary direction within a well-defined coordinate system);
- A length consistency of a test-length (distance change between two targets);
- A difference of two-face measurements of a single target (a vector in the direction of range, horizontal or vertical angle measurements).

- The absolute target displacement is the most commonly used information source for TLS calibration in the literature [16,52]. For tracking the spatial target displacement, it is necessary to have the reference target coordinates in a well-defined reference coordinate frame, i.e., it is necessary to define a reference datum. Defining a reference frame requires tremendous effort and can be realized in two different ways: by the bundle adjustment with a strong network of observations or by reference measurements with an instrument of higher accuracy. As we aim at cost-efficient TLS calibration, this information source will not be discussed in more detail.
- Length consistency requires observing a distance between two targets in an arbitrary coordinate system. Hence, it has the potential for a cost-efficient solution. The reference value of a length-test can be realized either with the reference instrument or with an appropriate measurement strategy. Namely, the reference value does not need to reflect the accurate length between two targets. It is sufficient that the length is observed from two TLS positions in the following way: From one position, the misalignment under investigation deforms the length as much as possible, and, from the other position, it deforms it as little as possible. From comparing these two values, it is possible to derive the calibration parameter. The problem of a length test is that a part of the misalignment’s influence can cause a translation or a rotation of the test-length instead of a length change. It is hard to avoid this phenomenon completely. As we do not rely on the known coordinates in a reference coordinate system, the rotation and translation of a length in space cannot be assessed and used for the calibration. Therefore, in many cases, we lose a part of the signal, which impacts the SNR and ultimately the sensitivity of the selected measurement geometry. To calculate the noise, we apply the variance propagation from the target centroid uncertainty to the estimated length of a test-length.
- The last information source, two-face differences, can only be used for estimating two-face sensitive calibration parameters (Figure 6). A misalignment corresponding to such a CP causes a target displacement in a measurement direction that changes its direction if the target is observed on the front side (front face) or the back side (back face) of the instrument, while the magnitude remains the same. Two-face sensitivity depends on a functional connection between the misalignment and the measurements (Equations (2)–(4)). In the case of the ranges and horizontal angles, the CP should switch the sign (due to sine or tangents functions), while in the case of the vertical angles, the sign should remain the same (due to cosines or no function). This rule stems from the way of building two-face differences, which is different for each measurement group [39].

#### 3.1. Sensitivity Analysis of Two-Face Differences

#### 3.1.1. ${x}_{2}$—Horizontal Axis Offset

_{2}is an offset parameter, the impact of the misalignment (signal) is constant with the distance and changes only with respect to the vertical angle. From Figure 7, it is apparent that the highest SNR is achieved in the horizontal plane of the instrument, where the signal is the highest ($x\xb7\mathrm{sin}{90}^{\xb0}=x$) and decreases with the sine function both in the directions of the zenith and nadir. Based on the stochastic model in Figure 3, the range measurement noise grows with the distance, except for an increased value in the proximity of the instrument (the first few meters). Additionally, it is advisable to avoid distances smaller than three meters due to unmodeled near-field systematic errors (Section 2.4). Hence, this parameter is the best estimated by the two-face measurements of a target in the instrument’s horizon at a distance of approximately three meters. With this measurement configuration, no other parameter impacts the two-face difference in the range direction, and the ${x}_{2}$ estimate is completely de-coupled from the other CPs.

#### 3.1.2. ${x}_{4}$—Vertical Index Offset

#### 3.1.3. ${x}_{5n}$—Horizontal Beam Tilt and ${x}_{1n+2}$—Horizontal Beam and Horizontal Axis Offsets

#### 3.1.4. ${x}_{6}$—Mirror Tilt and ${x}_{3}$—Mirror Offset

#### 3.1.5. ${x}_{5z-7}$—Vertical Beam Tilt and Horizontal Axis Tilt and ${x}_{1z}$—Vertical Beam Offset

#### 3.2. Sensitivity Analysis of the Length-Consistency Tests

#### 3.2.1. ${x}_{10}$—Rangefinder Offset

#### 3.2.2. ${x}_{1\mathrm{n}}$—Horizontal Beam Offset

#### 3.2.3. ${x}_{5\mathrm{z}}$—Vertical Beam Tilt

#### Length Consistency Test L1

#### Length Consistency Test L2

#### 3.3. Discussion of Sensitivity Analysis

## 4. Minimal Measurement Geometry

#### 4.1. Design of Minimal Measurement Geometry

- Two-face differences for 1 target on the instrument’s horizon at a short distance to estimate ${x}_{2}$ and ${x}_{3}$ and 1 target at a long distance to estimate ${x}_{4}$ and ${x}_{6}$;
- Two-face differences for 1 elevated target at a short distance to estimate ${x}_{1n+2}$ and ${x}_{1z}$ and 1 elevated target at a long distance to estimate ${x}_{5n}$ and ${x}_{5z+7}$;
- Length consistency tests for ${x}_{1n}$, ${x}_{5z}$, and ${x}_{10}$.

#### 4.2. Simulation Experiment

#### 4.3. Empirical Experiments

#### 4.3.1. Measurement Setup

#### 4.3.2. Experiment Results With Relation to TLS Test-Fields

#### 4.3.3. Experiment Results with Relation to Leica Check and Adjust

#### 4.4. General Considerations about Minimal Measurement Geometry

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Mukupa, W.; Roberts, G.W.; Hancock, C.M.; Al-Manasir, K. A review of the use of terrestrial laser scanning application for change detection and deformation monitoring of structures. Surv. Rev.
**2017**, 49, 99–116. [Google Scholar] [CrossRef] - Bianculli, D.; Humphries, D.; Berkeley, L. Application of terrestrial laser scanner in particle accelerator and reverse engineering solutions. In Proceedings of the 14th International Workshop Accelerator Alignment, Grenoble, France, 3–7 October 2016. [Google Scholar]
- Chan, T.O.; Lichti, D.D.; Belton, D. A rigorous cylinder-based self-calibration approach for terrestrial laser scanners. ISPRS J. Photogramm. Remote Sens.
**2015**, 99, 84–99. [Google Scholar] [CrossRef] - Bae, K.; Lichti, D.D. On-site self-calibration using planar features for terrestrial laser scanners. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci
**2007**, 36, 14–19. [Google Scholar] - Holst, C.; Medić, T.; Kuhlmann, H. Dealing with systematic laser scanner errors due to misalignment at area-based deformation analyses. J. Appl. Geod.
**2018**, 12, 169–185. [Google Scholar] [CrossRef] - Chow, J.C.K.; Teskey, W.F.; Lovse, J.W. In-situ Self-calibration of Terrestrial Laser Scanners and Deformation Analysis Using Both Signalized Targets and Intersection of Planes for Indoor Applications. In Proceedings of the 14th FIG Symposium on Deformation Measurements and Analysis, Hong Kong, China, 2–4 November 2011. [Google Scholar]
- Li, X.; Li, Y.; Xie, X.; Xu, L. Lab-built terrestrial laser scanner self-calibration using mounting angle error correction. Opt. Express
**2018**, 26, 14444. [Google Scholar] [CrossRef] [PubMed] - Neitzel, F. Investigation of Axes Errors of Terrestrial Laser Scanners. In Proceedings of the 5th International Symposium Turkish-German Joint Geodetic Days, Berlin, Germany, 29–31 March 2006. [Google Scholar]
- Medić, T.; Kuhlmann, H.; Holst, C. Automatic in-situ self-calibration of a panoramic TLS from a single station using 2D keypoints. In Proceedings of the ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Enschede, The Netherlands, 10–14 June 2019. [Google Scholar]
- Li, X.; Li, Y.; Xie, X.; Xu, L. Terrestrial laser scanner autonomous self-calibration with no prior knowledge of point-clouds. IEEE Sens. J.
**2018**, 18, 9277–9285. [Google Scholar] [CrossRef] - Zhang, Z.; Sun, L.; Zhong, R.; Chen, D.; Xu, Z.; Wang, C.; Qin, C.-Z.; Sun, H.; Li, R. 3-D deep feature construction for mobile laser scanning point cloud registration. IEEE Geosci. Remote Sens. Lett.
**2019**, PP, 1–5. [Google Scholar] [CrossRef] - Garcia-San-Miguel, D.; Lerma, J.L. Geometric calibration of a terrestrial laser scanner with local additional parameters: An automatic strategy. ISPRS J. Photogramm. Remote Sens.
**2013**, 79, 122–136. [Google Scholar] [CrossRef] - Lichti, D.D. Error modelling, calibration and analysis of an AM-CW terrestrial laser scanner system. ISPRS J. Photogramm. Remote Sens.
**2007**, 61, 307–324. [Google Scholar] [CrossRef] - Abbas, M.A.; Lichti, D.D.; Chong, A.K.; Setan, H.; Majid, Z. An on-site approach for the self-calibration of terrestrial laser scanner. Meas. J. Int. Meas. Confed.
**2014**, 52, 111–123. [Google Scholar] [CrossRef] - Reshetyuk, Y. Self-Calibration and Direct Georeferencing in Terrestrial Laser Scanning. Ph.D. Thesis, KTH Stockholm, Stockholm, Sweden, January 2009. [Google Scholar]
- Chow, J.C.K.; Lichti, D.D.; Glennie, C.; Hartzell, P. Improvements to and comparison of static terrestrial LiDAR self-calibration methods. Sensors
**2013**, 13, 7224–7249. [Google Scholar] [CrossRef] [PubMed] - Ge, X. Terrestrial Laser Scanning Technology from Calibration to Registration with Respect to Deformation Monitoring. Ph.D. Thesis, Technical University of Munich, Munich, Germany, October 2016. [Google Scholar]
- Wujanz, D.; Holst, C.; Neitzel, F.; Kuhlmann, H.; Niemeier, W.; Schwieger, V. Survey Configuration for Terrestrial Laser Scanning/Aufnahmekonfiguration für Terrestrisches Laserscanning. Allg. Vermessungs-Nachrichten
**2016**, 123, 158–169. [Google Scholar] - Soudarissanane, S.; Lindenbergh, R. Optimizing Terrestrial Laser Scanning Measurement Set-Up. ISPRS Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2012**, XXXVIII, 127–132. [Google Scholar] [CrossRef] - Jia, F.; Lichti, D. A comparison of simulated annealing, genetic algorithm and particle swarm optimization in optimal first-order design of indoor TLS networks. ISPRS Ann. Photogramm. Remote Sens. Spat. Inf. Sci.
**2017**, 4, 75–82. [Google Scholar] [CrossRef] - Medić, T.; Holst, C.; Kuhlmann, H. Improving the Results of Terrestrial Laser Scanner Calibration by an Optimized Calibration Process. Available online: https://www.researchgate.net/publication/332979231_Improving_the_results_of_terrestrial_laser_scanner_calibration_by_an_optimized_calibration_process (accessed on 27 June 2019).
- Muralikrishnan, B.; Ferrucci, M.; Sawyer, D.; Gerner, G.; Lee, V.; Blackburn, C.; Phillips, S.; Petrov, P.; Yakovlev, Y.; Astrelin, A.; et al. Volumetric performance evaluation of a laser scanner based on geometric error model. Precis. Eng.
**2015**, 40, 139–150. [Google Scholar] [CrossRef] - Medić, T.; Holst, C.; Kuhlmann, H. Towards System Calibration of Panoramic Laser Scanners from a Single Station. Sensors
**2017**, 17, 1145. [Google Scholar] [CrossRef] [PubMed] - Medić, T.; Holst, C.; Janßen, J.; Kuhlmann, H. Empirical stochastic model of detected target centroids: Influence on registration and calibration of terrestrial laser scanners. J. Appl. Geod.
**2019**. ahead of print. [Google Scholar] - Grafarend, E.W.; Sanso, F. Optimization and Design of Geodetic Networks, 1st ed.; Springer: Berlin/Heidelberg, Germany, 1985; ISBN 9783642706615. [Google Scholar]
- Schwieger, V. Sensitivity analysis as a general tool for model optimisation—Examples for trajectory estimation. J. Appl. Geod.
**2007**, 1, 27–34. [Google Scholar] [CrossRef] - Niemeier, W. Anlage von Überwachungsnetzen. In Geodätische Netze in der Landes- und Ingenieurvermessung; Pelzer, H., Ed.; Konrad Wittwer Verlag: Stuttgart, Germany, 1985. [Google Scholar]
- Tiede, C. Integration of Optimization Algorithms With Sensitivity Analysis, With Application To Volcanic Regions. Ph.D. Thesis, Technische Universität Darmstadt, Darmstadt, Germany, April 2005. [Google Scholar]
- Gordon, B. Zur Bestimmung von Messunsicherheiten Terrestrischer Laserscanner. Ph.D. Thesis, Technische Universität Darmstadt, Darmstadt, Germany, September 2008. [Google Scholar]
- Muralikrishnan, B.; Shilling, M.; Rachakonda, P.; Ren, W.; Lee, V.; Sawyer, D. Toward the development of a documentary standard for derived-point to derived-point distance performance evaluation of spherical coordinate 3D imaging systems. J. Manuf. Syst.
**2015**, 37, 550–557. [Google Scholar] [CrossRef] - Neitzel, F.; Gordon, B.; Wujanz, D. Verfahren zur Standardisierten Überprüfung von Terrestrischen Laserscannern (TLS). DVW-Merkblatt 7–2014. Available online: https://www.dvw.de/dvw-iso/17364/verfahren-zur-standardisierten-berpr-fung-terrestrischen-laserscannern-tls (accessed on 27 June 2019).
- International Organization for Standardization (ISO). Optics and Optical Instruments—Field Procedures for Testing Geodetic and Surveying Instruments—Part 9: Terrestrial Laser Scanners. 2018. Available online: https://www.iso.org/standard/68382.html (accessed on 27 June 2019).
- Kern, F. Prüfen und Kalibrieren von terrestrischen Laserscannern. In Photogrammetrie Laserscanning Optische 3DMesstechnik, Beitr¨age der Oldenburger 3D-Tage; Luhmann, T., Müller, C., Eds.; Herbert Wichmann Verlag: Heidelberg, Germany, 2008; pp. 306–316. [Google Scholar]
- Walsh, G. Leica ScanStation P-Series—Details That Matter. Details that matter. Leica ScanStation—White Paper. Available online: http://blog.hexagongeosystems.com/wp-content/uploads/2015/12/Leica_ScanStation_P-Series_details_that_matter_white_paper_en-4.pdf (accessed on 21 May 2019).
- Muralikrishnan, B.; Wang, L.; Rachakonda, P.; Sawyer, D. Terrestrial laser scanner geometric error model parameter correlations in the Two-face, Length-consistency, and Network methods of self-calibration. Precis. Eng.
**2017**, 52, 15–29. [Google Scholar] [CrossRef] - Lichti, D.D. Terrestrial laser scanner self-calibration: Correlation sources and their mitigation. ISPRS J. Photogramm. Remote Sens.
**2010**, 65, 93–102. [Google Scholar] [CrossRef] - Vosselman, G.; Maas, H.G. Airborne and Terrestrial Laser Scanning; Whittles Publishing: Scottland, UK, 2010; ISBN 9781439827987. [Google Scholar]
- Staiger, R. Terrestrial Laser Scanning Technology, Systems and Applications. In Proceedings of the 2nd FIG Regional Conference, Marrakech, Morocco, 2–5 December 2003; pp. 1–10. [Google Scholar]
- Schofield, W.; Breach, M. Engineering Surveying, 6th ed.; Elsevier: Oxford, UK, 2007; ISBN 9780750669498. [Google Scholar]
- Lichti, D.D. The impact of angle parameterisation on terrestrial laser scanner self-calibration. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci
**2009**, 38, 171–176. [Google Scholar] - Muralikrishnan, B.; Rachakonda, P.; Shilling, M.; Lee, V.; Blackburn, C.; Sawyer, D.; Cheok, G.; Cournoyer, L. Report on the May 2016 ASTM E57.02 Instrument Runoff at NIST, Part 1—Background Information and Key Findings. 2016. Available online: https://nvlpubs.nist.gov/nistpubs/ir/2016/NIST.IR.8152.pdf (accessed on 27 June 2019). [CrossRef]
- Dorninger, P.; Nothegger, C.; Pfeifer, N.; Molnár, G. On-the-job detection and correction of systematic cyclic distance measurement errors of terrestrial laser scanners. J. Appl. Geod.
**2008**, 2, 191–204. [Google Scholar] [CrossRef] - Muralikrishnan, B.; Rachakonda, P.; Lee, V.; Shilling, M.; Sawyer, D.; Cheok, G.; Cournoyer, L. Relative range error evaluation of terrestrial laser scanners using a plate, a sphere, and a novel dual-sphere-plate target. Measurement
**2017**, 111, 60–68. [Google Scholar] [CrossRef] - Wujanz, D.; Burger, M.; Mettenleiter, M.; Neitzel, F. An intensity-based stochastic model for terrestrial laser scanners. ISPRS J. Photogramm. Remote Sens.
**2017**, 125, 146–155. [Google Scholar] [CrossRef] - Schmitz, B.; Holst, C.; Medic, T.; Lichti, D.D.; Kuhlmann, H. How to Efficiently Determine the Range Precision of 3D Terrestrial Laser Scanners. Sensors
**2019**, 19, 1466. [Google Scholar] [CrossRef] [PubMed] - Janßen, J.; Medić, T.; Kuhlmann, H.; Holst, C. Decreasing the uncertainty of the target centre estimation at terrestrial laser scanning by choosing the best algorithm and by improving the target design. Remote Sens.
**2019**, 11, 845. [Google Scholar] [CrossRef] - Leica Leica ScanStation P20 Industry’s Best Performing Ultra-High Speed Scanner. Leica Scanstation P20 Datasheet 2015. Available online: https://w3.leica-geosystems.com/downloads123/hds/hds/scanstation_p20/brochures-datasheet/leica_scanstation_p20_dat_en.pdf (accessed on 26 June 2019).
- Martin, D.; Gatta, G. Calibration of total stations instruments at the ESRF. In Proceedings of the XXIII FIG Congress, Munich, Germany, 8–13 October 2006; pp. 1–14. Available online: http://fig.net/resources/proceedings/fig_proceedings/fig2006/papers/ts24/ts24_05_martin_gatta_0506.pdf (accessed on 26 June 2019).
- Giebeler, J.T. Untersuchung der Genauigkeit Terrestrischer Laserscanner im Nahbereich. Bachelor Thesis, Universität Bonn, Bonn, Germany, 2017. [Google Scholar]
- Zamecnikova, M.; Wieser, A.; Woschitz, H.; Ressl, C. Influence of surface reflectivity on reflectorless electronic distance measurement and terrestrial laser scanning. J. Appl. Geod.
**2014**, 8, 311–325. [Google Scholar] [CrossRef] - Wang, L.; Muralikrishnan, B.; Rachakonda, P.; Sawyer, D. Determining geometric error model parameters of a terrestrial laser scanner through two-face, length-consistency, and network methods. Meas. Sci. Technol.
**2017**, 28, 065016. [Google Scholar] [CrossRef] - Holst, C.; Neuner, H.; Wieser, A.; Wunderlich, T.; Kuhlmann, H. Calibration of Terrestrial Laser Scanners/Kalibrierung terrestrischer Laserscanner. Allg. Vermessungs-Nachrichten
**2016**, 123, 147–157. [Google Scholar] - Muralikrishnan, B.; Sawyer, D.; Blackburn, C.; Phillips, S.; Borchardt, B.; Estler, W.T. ASME B89.4.19 Performance Evaluation Tests and Geometric Misalignments in Laser Trackers. J. Res. Natl. Inst. Stand. Technol.
**2011**, 114, 21. [Google Scholar] [CrossRef] - Leica Leica HDS Check & Adjust—User Manual. Available online: https://kb.sccssurvey.co.uk/download/139/leica.../leica-hds-check-adjust-manual.pdf (accessed on 20 May 2019).
- Heinz, E.; Holst, C.; Kuhlmann, H. Zum Einfluss der räumlichen Auflösung und verschiedener Qualitätsstufen auf die Modellierungsgenauigkeit einer Ebene beim terrestrischen Laserscanning. Allg. Vermessungs-Nachrichten
**2019**, 126, 3–12. [Google Scholar]

**Figure 1.**(

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

**b**) perfect panoramic TLS geometry [23].

**Figure 2.**Mechanical misalignments of panoramic TLSs, (

**a**,

**b**)—laser offsets and tilts, (

**c**,

**d**)—mirror offset and tilt, (

**e**,

**f**)—horizontal axis offset and tilt [23].

**Figure 3.**Target center uncertainty for range and angular measurements, with respect to the scanning distances.

**Figure 4.**Rangefinder bias in the near-field for the Leica ScanStation P20 with respect to the measurement distance.

**Figure 6.**Calibration parameter division and overview of their two-face sensitivity based on their functional connection with the TLS measurements ($r,\phi $, $\theta $).

**Figure 7.**Signal to noise ratio (SNR) for ${x}_{2}$ on a vertical plane—full field of view (

**left**) and the close-up (

**right**). The red square—the limitations of the cross-section dimensions of the facility used in the empirical experiment.

**Figure 9.**SNR for ${x}_{5n}$ in a vertical plane—full field of view (

**left**) and the close-up (

**right**).

**Figure 10.**SNR for ${x}_{1n+2}$ in a vertical plane—full field of view (

**left**) and the close-up (

**right**).

**Figure 11.**SNR for ${x}_{3}$ and ${x}_{6}$ in a vertical plane—full field of view (

**a**,

**c**) and the close-up (

**b**,

**d**)

**Figure 12.**SNR for ${x}_{1z}$ and ${x}_{5z+7}$ in a vertical plane—full field of view (

**a**,

**c**) and the close-up (

**b**,

**d**)

**Figure 13.**The SNR ratio of the length consistency test for ${x}_{10}$ (

**a**), with the least (

**b**) and the most (

**c**) sensitive TLS positions (top view); line and arrows: blue—translated length, red—extended length.

**Figure 14.**The SNR ratio of the length consistency test for ${x}_{1\mathrm{n}}$ (

**a**), with the least (

**b**) and the most (

**c**,

**d**) sensitive TLS positions (top view); line and arrows: red—rotated length, blue—extended length, green—contracted length.

**Figure 15.**(

**a**) The SNR ratio of the length consistency tests for ${x}_{5\mathrm{z}}$ (1% of the most sensitive measurement configurations—colored lines, TLS—red rhombus, two most distinguishable length-tests—black lines L1 and L2). (

**b**,

**c**) 2 realizations of the length consistency test (both the most and the least sensitive TLS positions) for ${x}_{5\mathrm{z}}$; red lines—length extension, blue lines—length translation or rotation.

**Figure 16.**Minimal network configuration used in the simulation experiment: scanner stations (S1 and S2), targets (T1, T2, and T3).

**Figure 17.**Photo-documentation of the calibration experiment: overview image of the facility (

**left**), scanner station S1 with the target T2 (

**middle**), T1, and T3 (

**right**).

Parameter | Description |
---|---|

x_{1n} | Horizontal beam offset |

x_{1z} | Vertical beam offset |

x_{2} | Horizontal axis offset |

x_{3} | Mirror offset |

x_{4} | Vertical index offset (tilt) |

x_{5n} | Horizontal beam tilt |

x_{5z} | Vertical beam tilt |

x_{6} | Mirror tilt |

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

x_{10} | Rangefinder offset |

**Table 2.**The results of the simulation experiment: CP values (simulated and estimated) and standard deviation.

Par. | ${\mathit{x}}_{\mathit{t}\mathit{r}\mathit{u}\mathit{e}}$ | $\widehat{\mathit{x}}$ | ${\mathit{\sigma}}_{\mathit{x}}$ |
---|---|---|---|

${\mathit{x}}_{\mathbf{10}}$ [mm] | −2.00 | −1.98 | 0.04 |

${\mathit{x}}_{\mathbf{2}}$ [mm] | −0.20 | −0.23 | 0.02 |

${\mathit{x}}_{\mathbf{1}\mathbf{z}}$ [mm] | −0.20 | −0.24 | 0.03 |

${\mathit{x}}_{\mathbf{3}}$ [mm] | −0.20 | −0.20 | 0.01 |

${\mathit{x}}_{\mathbf{5}\mathbf{z}+\mathbf{7}}\left[\text{}\u2033\text{}\right]$ | −16.00 | −14.74 | 1.04 |

${\mathit{x}}_{\mathbf{6}}$$\left[\text{}\u2033\text{}\right]$ | −8.00 | −7.99 | 0.09 |

${\mathit{x}}_{\mathbf{1}n}$ [mm] | −0.20 | −0.21 | 0.01 |

${\mathit{x}}_{\mathbf{4}}$$\left[\text{}\u2033\text{}\right]$ | −8.00 | −8.06 | 0.13 |

${\mathit{x}}_{\mathbf{5}\mathbf{n}}\text{}\left[\text{}\u2033\text{}\right]$ | −8.00 | −8.48 | 1.05 |

${\mathit{x}}_{\mathbf{5}z}$$\left[\text{}\u2033\text{}\right]$ | −8.00 | −7.89 | 0.62 |

${\mathit{x}}_{\mathbf{1}n+\mathbf{2}}$ [mm] | −0.40 | −0.36 | 0.03 |

**Table 3.**Attempt of estimating all parameters: CP values (repetitions 1–4, mean), standard deviation out of 4 repetitions, and correlations with all unknowns and only between the CPs.

Par. | ${\widehat{\mathit{x}}}_{1}$ | ${\widehat{\mathit{x}}}_{2}$ | ${\widehat{\mathit{x}}}_{3}$ | ${\widehat{\mathit{x}}}_{4}$ | ${\widehat{\mathit{x}}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}}$ | ${\mathit{\sigma}}_{\mathit{x}}$ | ${\mathit{\rho}}_{\mathit{x}}$ | With | ${\mathit{\rho}}_{\mathit{x}}$ | With |
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{x}}_{\mathbf{10}}$ [mm] | 4.20 | 5.06 | 4.82 | 4.61 | 4.67 | 0.36 | 0.71 | T3 | 0.24 | ${x}_{5\mathrm{z}}$ |

${\mathit{x}}_{\mathbf{2}}$ [mm] | −0.07 | −0.03 | −0.07 | −0.07 | −0.06 | 0.02 | 0.00 | ${x}_{3}$ | 0.00 | ${x}_{3}$ |

${\mathit{x}}_{\mathbf{1}z}$ [mm] | −0.15 | −0.34 | 0.18 | −0.09 | −0.10 | 0.21 | −0.97 | ${x}_{5\mathrm{z}-7}$ | −0.97 | ${x}_{5\mathrm{z}-7}$ |

${\mathit{x}}_{\mathbf{3}}$ [mm] | −0.04 | 0.01 | −0.03 | −0.05 | −0.03 | 0.03 | −0.69 | ${x}_{6}$ | −0.69 | ${x}_{6}$ |

${\mathit{x}}_{\mathbf{5}z+\mathbf{7}}\left[\text{}\u2033\text{}\right]$ | −1.55 | 5.77 | −17.89 | −9.38 | −5.76 | 10.18 | −0.97 | ${x}_{1\mathrm{z}}$ | −0.97 | ${x}_{1\mathrm{z}}$ |

${\mathit{x}}_{\mathbf{6}}$$\left[\text{}\u2033\text{}\right]$ | 2.32 | 0.71 | 2.40 | 2.71 | 2.04 | 0.90 | 0.70 | ${x}_{1\mathrm{z}}$ | 0.70 | ${x}_{1\mathrm{z}}$ |

${\mathit{x}}_{\mathbf{1}n}$ [mm] | 6.91 | −8.12 | 8.80 | 9.39 | 4.25 | 8.31 | 0.99 | T1 | −0.09 | x_{10} |

${\mathit{x}}_{\mathbf{4}}$$\left[\text{}\u2033\text{}\right]$ | −7.70 | −6.54 | −7.41 | −6.98 | −7.16 | 0.51 | −0.65 | ${x}_{5\mathrm{n}}$ | −0.65 | x_{5n} |

${\mathit{x}}_{\mathbf{5}n}\text{}\left[\text{}\u2033\text{}\right]$ | −21.11 | −29.87 | −11.98 | −14.93 | −19.47 | 7.91 | −0.96 | ${x}_{1\mathrm{n}+2}$ | −0.96 | ${x}_{1\mathrm{n}+2}$ |

${\mathit{x}}_{\mathbf{5}z}$$\left[\text{}\u2033\text{}\right]$ | 4.72 | 8.45 | 3.69 | 2.22 | 4.77 | 2.66 | 0.76 | T2 | −0.25 | ${x}_{1\mathrm{z}}$ |

${\mathit{x}}_{\mathbf{1}n+\mathbf{2}}$ [mm] | 0.15 | 0.36 | −0.19 | −0.07 | 0.06 | 0.24 | −0.96 | ${x}_{5\mathrm{n}}$ | −0.96 | ${x}_{5\mathrm{n}}$ |

**Table 4.**Estimating a subset of parameters: CP values (repetitions 1–4, mean), standard deviation out of 4 repetitions, correlations with all unknowns and only between the CPs.

Par. | ${\mathit{x}}_{\mathit{t}\mathit{r}\mathit{u}\mathit{e}}$ | ${\widehat{\mathit{x}}}_{1}$ | ${\widehat{\mathit{x}}}_{2}$ | ${\widehat{\mathit{x}}}_{3}$ | ${\widehat{\mathit{x}}}_{4}$ | ${\widehat{\mathit{x}}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}}$ | ${\mathit{\sigma}}_{\mathit{x}}$ | ${\mathit{\rho}}_{\mathit{x}}$ | With | ${\mathit{\rho}}_{\mathit{x}}$ | With |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{x}}_{\mathbf{5}z-\mathbf{7}}\left[\text{}\u2033\text{}\right]$ | −8.40 | −7.51 | −6.34 | −12.36 | −13.59 | −9.95 | 3.56 | −0.98 | ${x}_{6}$ | −0.98 | ${x}_{6}$ |

${\mathit{x}}_{\mathbf{6}}$$\left[\text{}\u2033\text{}\right]$ | 1.56 | 2.38 | 1.33 | 1.95 | 2.63 | 2.07 | 0.57 | −0.98 | ${x}_{5\mathrm{z}-7}$ | −0.98 | ${x}_{5\mathrm{z}-7}$ |

${\mathit{x}}_{\mathbf{4}}$$\left[\text{}\u2033\text{}\right]$ | −6.21 | −8.11 | −7.53 | −6.90 | −6.79 | −7.33 | 0.61 | −0.67 | ${x}_{5\mathrm{n}}$ | −0.67 | ${x}_{5\mathrm{n}}$ |

${\mathit{x}}_{\mathbf{5}n}\text{}\left[\text{}\u2033\text{}\right]$ | −15.21 | −16.25 | −17.80 | −18.08 | −17.21 | −17.34 | 0.81 | −0.67 | ${x}_{4}$ | −0.67 | ${x}_{4}$ |

${\mathit{x}}_{\mathbf{5}z}$$\left[\text{}\u2033\text{}\right]$ | - | −12.25 | −11.53 | −14.01 | −16.01 | −13.45 | 2.00 | 0.93 | T2 | 0.00 | ${x}_{5\mathrm{n}}$ |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Medić, T.; Kuhlmann, H.; Holst, C.
Sensitivity Analysis and Minimal Measurement Geometry for the Target-Based Calibration of High-End Panoramic Terrestrial Laser Scanners. *Remote Sens.* **2019**, *11*, 1519.
https://doi.org/10.3390/rs11131519

**AMA Style**

Medić T, Kuhlmann H, Holst C.
Sensitivity Analysis and Minimal Measurement Geometry for the Target-Based Calibration of High-End Panoramic Terrestrial Laser Scanners. *Remote Sensing*. 2019; 11(13):1519.
https://doi.org/10.3390/rs11131519

**Chicago/Turabian Style**

Medić, Tomislav, Heiner Kuhlmann, and Christoph Holst.
2019. "Sensitivity Analysis and Minimal Measurement Geometry for the Target-Based Calibration of High-End Panoramic Terrestrial Laser Scanners" *Remote Sensing* 11, no. 13: 1519.
https://doi.org/10.3390/rs11131519