## Details

Original language | English |
---|---|

Article number | 59 |

Journal | Engineering Proceedings |

Volume | 5 |

Issue number | 1 |

Publication status | Published - 19 Jul 2021 |

## Abstract

Terrestrial laser scanners (TLS) capture a large number of 3D points rapidly, with high precision and spatial resolution. These scanners are used for applications as diverse as modeling architectural or engineering structures, but also high-resolution mapping of terrain. The noise of the observations cannot be assumed to be strictly corresponding to white noise: besides being heteroscedastic, correlations between observations are likely to appear due to the high scanning rate. Unfortunately, if the variance can sometimes be modeled based on physical or empirical considerations, the latter are more often neglected. Trustworthy knowledge is, however, mandatory to avoid the overestimation of the precision of the point cloud and, potentially, the non-detection of deformation between scans recorded at different epochs using statistical testing strategies. The TLS point clouds can be approximated with parametric surfaces, such as planes, using the Gauss–Helmert model, or the newly introduced T-splines surfaces. In both cases, the goal is to minimize the squared distance between the observations and the approximated surfaces in order to estimate parameters, such as normal vector or control points. In this contribution, we will show how the residuals of the surface approximation can be used to derive the correlation structure of the noise of the observations. We will estimate the correlation parameters using the Whittle maximum likelihood and use comparable simulations and real data to validate our methodology. Using the least-squares adjustment as a “filter of the geometry” paves the way for the determination of a correlation model for many sensors recording 3D point clouds.

## Keywords

- correlation, Hurst exponent, least-squares, surface approximation, T-splines, Whittle maximum likelihood

## ASJC Scopus subject areas

- Engineering(all)
**Mechanical Engineering**- Engineering(all)
**Electrical and Electronic Engineering**- Engineering(all)
**Industrial and Manufacturing Engineering**- Engineering(all)
**Biomedical Engineering**

## Cite this

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- Harvard
- Apa
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- BibTeX
- RIS

**Using Least-Squares Residuals to Assess the Stochasticity of Measurements: Example: Terrestrial Laser Scanner and Surface Modeling.**/ Kermarrec, Gaël; Schild, Niklas; Hartmann, Jan.

In: Engineering Proceedings, Vol. 5, No. 1, 59, 19.07.2021.

Research output: Contribution to journal › Article › Research › peer review

*Engineering Proceedings*, vol. 5, no. 1, 59. https://doi.org/10.3390/engproc2021005059

*Engineering Proceedings*,

*5*(1), Article 59. https://doi.org/10.3390/engproc2021005059

}

TY - JOUR

T1 - Using Least-Squares Residuals to Assess the Stochasticity of Measurements

T2 - Example: Terrestrial Laser Scanner and Surface Modeling

AU - Kermarrec, Gaël

AU - Schild, Niklas

AU - Hartmann, Jan

N1 - Funding Information: This study is supported by the Deutsche Forschungsgemeinschaft under the project KE2453/2-1.

PY - 2021/7/19

Y1 - 2021/7/19

N2 - Terrestrial laser scanners (TLS) capture a large number of 3D points rapidly, with high precision and spatial resolution. These scanners are used for applications as diverse as modeling architectural or engineering structures, but also high-resolution mapping of terrain. The noise of the observations cannot be assumed to be strictly corresponding to white noise: besides being heteroscedastic, correlations between observations are likely to appear due to the high scanning rate. Unfortunately, if the variance can sometimes be modeled based on physical or empirical considerations, the latter are more often neglected. Trustworthy knowledge is, however, mandatory to avoid the overestimation of the precision of the point cloud and, potentially, the non-detection of deformation between scans recorded at different epochs using statistical testing strategies. The TLS point clouds can be approximated with parametric surfaces, such as planes, using the Gauss–Helmert model, or the newly introduced T-splines surfaces. In both cases, the goal is to minimize the squared distance between the observations and the approximated surfaces in order to estimate parameters, such as normal vector or control points. In this contribution, we will show how the residuals of the surface approximation can be used to derive the correlation structure of the noise of the observations. We will estimate the correlation parameters using the Whittle maximum likelihood and use comparable simulations and real data to validate our methodology. Using the least-squares adjustment as a “filter of the geometry” paves the way for the determination of a correlation model for many sensors recording 3D point clouds.

AB - Terrestrial laser scanners (TLS) capture a large number of 3D points rapidly, with high precision and spatial resolution. These scanners are used for applications as diverse as modeling architectural or engineering structures, but also high-resolution mapping of terrain. The noise of the observations cannot be assumed to be strictly corresponding to white noise: besides being heteroscedastic, correlations between observations are likely to appear due to the high scanning rate. Unfortunately, if the variance can sometimes be modeled based on physical or empirical considerations, the latter are more often neglected. Trustworthy knowledge is, however, mandatory to avoid the overestimation of the precision of the point cloud and, potentially, the non-detection of deformation between scans recorded at different epochs using statistical testing strategies. The TLS point clouds can be approximated with parametric surfaces, such as planes, using the Gauss–Helmert model, or the newly introduced T-splines surfaces. In both cases, the goal is to minimize the squared distance between the observations and the approximated surfaces in order to estimate parameters, such as normal vector or control points. In this contribution, we will show how the residuals of the surface approximation can be used to derive the correlation structure of the noise of the observations. We will estimate the correlation parameters using the Whittle maximum likelihood and use comparable simulations and real data to validate our methodology. Using the least-squares adjustment as a “filter of the geometry” paves the way for the determination of a correlation model for many sensors recording 3D point clouds.

KW - correlation

KW - Hurst exponent

KW - least-squares

KW - surface approximation

KW - T-splines

KW - Whittle maximum likelihood

UR - http://www.scopus.com/inward/record.url?scp=85145420892&partnerID=8YFLogxK

U2 - 10.3390/engproc2021005059

DO - 10.3390/engproc2021005059

M3 - Article

VL - 5

JO - Engineering Proceedings

JF - Engineering Proceedings

IS - 1

M1 - 59

ER -