# Estimating Control Points for B-Spline Surfaces Using Fully Populated Synthetic Variance–Covariance Matrices for TLS Point Clouds

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## Abstract

**:**

## 1. Introduction

## 2. Functional and Stochastic Model

#### 2.1. Point Cloud Approximation with B-Spline Surfaces

**U**and

**V**.

**ε**(u,v). Here, the point-related $\mathit{S}\left(u,v\right)$ stands for the coordinate triple $x\left(u,v\right),y\left(u,v\right)$ and $z\left(u,v\right)$ and, thus, Equation (1) is set-up point-wise for all points of the approximated point cloud. Only the control points’ positions $\widehat{\mathit{P}}$ are usually estimated in a linear Gauß–Markov model, where the coordinate-wise defined design matrix

#### 2.2. Synthetic Variance–Covariance Matrix as TLS Stochastic Model

- 1.
- The original TLS point clouds were transformed from Cartesian to polar coordinates;
- 2.
- The SVCM in observation space ${\mathit{\Sigma}}_{ll}^{\left(\lambda \theta R\right)}$ was computed as stated in Equation (4);
- 3.
- Finally, ${\mathit{\Sigma}}_{ll}^{\left(\lambda \theta R\right)}$ was transformed in Cartesian coordinates by multiplication with the Jacobian matrix $\mathit{J}$ (not given here) that contains partial derivatives of the Cartesian coordinates with respect to the polar coordinates.

#### 2.3. Example for Simulated Data

## 3. Measurements and Evaluation Concept

#### 3.1. Experiment Set-Up

#### 3.2. Evaluation

- (a)
- ${\mathit{\Sigma}}_{ll}$ is the identity matrix
**I**; - (b)
- ${\mathit{\Sigma}}_{ll}$ is based on the main diagonal of the SVCM;
- (c)
- ${\mathit{\Sigma}}_{ll}$ is the fully populated SVCM (see Section 2.2).

- A posteriori variance factor.

- the functional model was incomplete;
- the stochastic model was chosen inappropriately;
- the observations contained gross errors.

- 2.
- A posteriori standard deviations of the estimated control points.

- 3.
- Differences of nominal and estimated control points

## 4. Results and Analysis

#### 4.1. Comparison of Different Stochastic Models

`.`

#### 4.2. Investigation of Individual Errors and Their Impact

#### 4.2.1. Zero Point Error ${x}_{10}$

#### 4.2.2. Horizontal Beam Offset ${x}_{1n}$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Vertical Index Error ${\mathit{x}}_{\mathbf{4}}$

SVCM_11 | SVCM_12 | SVCM_13 | SVCM_14 | |
---|---|---|---|---|

${\sigma}_{x4}$ (mgon) | 0.45 | 0.90 | 1.79 | 3.58 |

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**Figure 1.**Differences of the positions of the estimated control points with respect to the nominal coordinates ${\mathit{P}}_{N}$ using different stochastic models (top: identity matrix; bottom: fully populated VCM).

**Figure 3.**(

**a**) Nominal point cloud stored in CAD coordinate system (${z}_{CAD}$ axis is orthogonal to the ${x}_{CAD}-{y}_{CAD}$ plane); (

**b**) test specimen of known B-spline form with reference points.

**Figure 6.**Overview of the resulting ${\widehat{\sigma}}_{0}^{2}$ for SVCM_1-6 and the four TLS station points.

**Figure 7.**(

**a**) A posteriori standard deviations of estimated control points (S3 and SVCM_3); (

**b**) a posteriori standard deviations of estimated control points (S3 and SVCM_6).

**Figure 8.**Difference between nominal and estimated control points for stochastic models (I, D, SVCM) at S1 and SVCM_1.

CP | Tilts/Angular Errors | CP | Offsets/Metric Errors |
---|---|---|---|

${x}_{4}$ | Vertical index error | ${x}_{1n}$ | Horizontal beam offset |

${x}_{5n}$ | Horizontal beam tilt | ${x}_{1z}$ | Vertical beam offset |

${x}_{5z}$ | Vertical beam tilt | ${x}_{2}$ | Horizontal axis offset |

${x}_{6}$ | Collimation axis error | ${x}_{3}$ | Mirror offset |

${x}_{7}$ | Horizontal axis error (tilt) | ${x}_{10}$ | Zero point error |

CP | Tilts/Angular Errors (mgon) | CP | Offsets/Metric Errors (mm) |
---|---|---|---|

${\sigma}_{x4}$ | 0.45 | ${\sigma}_{x1n}$ | 0.14 |

${\sigma}_{x5n}$ | 1.79 | ${\sigma}_{x1z}$ | 0.22 |

${\sigma}_{x5z}$ | 1.60 | ${\sigma}_{x2}$ | 0.02 |

${\sigma}_{x6}$ | 0.27 | ${\sigma}_{x3}$ | 0.13 |

${\sigma}_{x7}$ | 1.93 | ${\sigma}_{x10}$ | 0.06 |

CP | SVCM_1 | SVCM_2 | SVCM_3 | SVCM_4 | SVCM_5 | SVCM_6 |
---|---|---|---|---|---|---|

${\sigma}_{\lambda}\left(\mathrm{mgon}\right)$ | 4.2 | 0.7 | 1.4 | 0.8 | 1.0 | 3.1 |

${\sigma}_{\theta}\left(\mathrm{mgon}\right)$ | 4.2 | 0.7 | 1.4 | 0.8 | 1.0 | 3.1 |

${\sigma}_{R}\left(\mathrm{mm}\right)$ | 0.5 | 0.6 | 0.5 | 0.5 | 0.5 | 0.5 |

**Table 4.**Comparison of${\widehat{\sigma}}_{0}^{2}$ of different stochastic models (a–c) for the four TLS station points.

${\widehat{\mathit{\sigma}}}_{0}^{2}$ | SVCM_1 | SVCM_2 | SVCM_3 | SVCM_4 | SVCM_5 | SVCM_6 | |
---|---|---|---|---|---|---|---|

S1 | ${\Sigma}_{ll}=I$ | 0.21 | |||||

${\Sigma}_{ll}=D$ | 0.88 | 1.94 | 1.75 | 2.05 | 1.92 | 1.10 | |

${\Sigma}_{ll}=SVCM$ | 1.00 | 18.92 | 5.27 | 15.12 | 10.39 | 1.46 | |

S2 | ${\Sigma}_{ll}=I$ | 0.09 | |||||

${\Sigma}_{ll}=D$ | 0.26 | 0.41 | 0.36 | 0.46 | 0.40 | 0.27 | |

${\Sigma}_{ll}=SVCM$ | 0.27 | 1.00 | 0.46 | 0.89 | 0.66 | 0.38 | |

S3 | ${\Sigma}_{ll}=I$ | 0.30 | |||||

${\Sigma}_{ll}=D$ | 0.59 | 0.96 | 0.82 | 1.07 | 0.91 | 0.63 | |

${\Sigma}_{ll}=SVCM$ | 0.61 | 2.08 | 1.00 | 1.89 | 1.40 | 0.65 | |

S4 | ${\Sigma}_{ll}=I$ | 0.27 | |||||

${\Sigma}_{ll}=D$ | 0.68 | 0.64 | 0.74 | 0.82 | 0.72 | 0.69 | |

${\Sigma}_{ll}=SVCM$ | 0.70 | 0.87 | 0.79 | 1.00 | 0.82 | 0.71 |

SVCM_7 | SVCM_8 | SVCM_9 | SVCM_10 | |
---|---|---|---|---|

${\sigma}_{x10}$ (mm) | 0.06 | 0.24 | 0.48 | 0.72 |

SVCM_15 | SVCM_16 | SVCM_17 | SVCM_18 | |
---|---|---|---|---|

${\sigma}_{x1n}$ (mm) | 0.14 | 0.28 | 0.56 | 1.12 |

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**MDPI and ACS Style**

Raschhofer, J.; Kerekes, G.; Harmening, C.; Neuner, H.; Schwieger, V. Estimating Control Points for B-Spline Surfaces Using Fully Populated Synthetic Variance–Covariance Matrices for TLS Point Clouds. *Remote Sens.* **2021**, *13*, 3124.
https://doi.org/10.3390/rs13163124

**AMA Style**

Raschhofer J, Kerekes G, Harmening C, Neuner H, Schwieger V. Estimating Control Points for B-Spline Surfaces Using Fully Populated Synthetic Variance–Covariance Matrices for TLS Point Clouds. *Remote Sensing*. 2021; 13(16):3124.
https://doi.org/10.3390/rs13163124

**Chicago/Turabian Style**

Raschhofer, Jakob, Gabriel Kerekes, Corinna Harmening, Hans Neuner, and Volker Schwieger. 2021. "Estimating Control Points for B-Spline Surfaces Using Fully Populated Synthetic Variance–Covariance Matrices for TLS Point Clouds" *Remote Sensing* 13, no. 16: 3124.
https://doi.org/10.3390/rs13163124