# Determination of Intensity-Based Stochastic Models for Terrestrial Laser Scanners Utilising 3D-Point Clouds

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## Abstract

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## 1. Introduction

_{e}and P

_{r}denote the power of the emitted and received signals, respectively. The scanner’s receiver aperture diameter is represented by D

_{r}while η

_{system}signifies its transmission coefficient. Atmospheric influences are denoted by η

_{atmos}whereas the range between the scanner and object point is symbolised by R. The angle of incidence is represented by α, the quotient of reflection of the object’s surface in the wavelength of the scanner is described by ρ

_{λ}. In summary, all parameters in the equation have an immediate impact onto the received signal strength P

_{r}. From another perspective it can be concluded that all relevant parameters in (1) are inherently considered in P

_{r}which makes individual consideration of the influencing factors unnecessary and hence justifies the intensity-based method suggested by Wujanz et al. [18].

- (i)
- Sensor domain: In which an optical signal with the power P
_{e}is emitted by a laser diode that is then deflected to the environment. - (ii)
- Environment: While travelling through the environment, the signal is subject to deterioration as a consequence of the range between scanner and an object’s surface as well as the atmospheric transmission coefficient η
_{atmos}. - (iii)
- Object domain: On the object’s surface the signal is additionally weakened in dependence to the local quotient of reflection ρ
_{λ}as well as the angle of incidence α. Another source which causes a loss of signal strength is provoked by the light collector of the scanner, as well as its deflection unit, that initiates the scanning process. The last two mentioned influences are summarised by η_{system}and are individual characteristics of the applied scanner that, hence, fall into to the sensor domain.

## 2. Determination of Intensity-Based Stochastic Models Based on Point Clouds

#### 2.1. Interpretation of Residuals as Rangefinder Stochastics

**v**were determined. In contrast to Wujanz et al. [18] all observations received equal weights. Figure 5 depicts the computed results that are highlighted by red circles. The standard deviation of ranges is graphed on the vertical axis while the raw intensity values can be found on the horizontal axis. The blue line represents the stochastic reference model generated by repeated range observations for a sampling rate of 508 kHz [18]. While the computed stochastic measures largely comply with the reference run, differences in the magnitude of up to 0.17 mm can be spotted for higher intensity values. The reason for this characteristic is likely to be caused by larger changes of the incidence angle for panels acquired at close range as opposed to large object distances. A solution to this problem could be to restrict the incidence angle of points on a sample to a certain range.

#### 2.2. Rangefinder Stochastics Based on Quasi-Ranges

_{p}, represented by a grey line on the right, is captured by a TLS. A representative range ρ

_{r}is tinted in cyan for which stochastic properties should be computed. The observed range ρ

_{o}is represented by the orange line. The green coloured semi-circle has a radius of ρ

_{r}. It can be seen that the discrepancy Δ between semi-circle and planar sample, highlighted by a blue line, increases towards the boundaries of the sample. Due to this effect differently sized regions have to be acquired in dependence to the object distance and the resolution/precision of the applied TLS’ rangefinder. Note that this geometrically provoked effect decreases with increasing object distance. However, if the entire sampled area would be considered, the resulting stochastic characteristics would increase as a consequence. Due to the fact that all captured ranges for a certain configuration are slightly different, the term “quasi-range” is used throughout the remainder of the paper.

_{P}denotes the extent of the sampled panel region. Discrepancies Δ larger than the range resolution of the applied laser scanner are assumed to cause erroneous results and are, hence, omitted. For the applied TLS the resolution sums up to ~0.1 mm [32]. Figure 7 illustrates the computed results where the range between scanner and sample is graphed on the horizontal axis. The vertical axis features the maximum extent of the area under consideration. Red tinted areas in the figure represent configurations where Δ exceeds the resolution of the scanner. As a consequence the corresponding range stochastics would be falsified. Green regions highlight combinations that are suitable to compute the stochastic behaviour of reflectorless rangefinders. It can be seen that the acquirable area on the panel increases with rising range.

#### 2.3.Evaluation of the Proposed Procedures

_{ρ}as fixed parameters. Thus, Equation (3) will later on form the stochastic model for the precision of ranges σ

_{ρ}. Table 1 contains a comparison among a reference model against the parameters which were derived in Section 2.1 and Section 2.2. The reference stems from Wujanz et al. [18] and was captured by repeated observation of ranges at a sampling rate of 508 kHz. Note that it is vital to introduce raw intensity values into the estimated functions that report the precision for ranges in metres.

- observations that were captured with the TLS under investigation;
- a functional model that is capable to represent the recorded data; and
- a stochastic model that describes the precision of the measurements.

_{0}, which is also referred to as theoretical reference standard deviation. The sum of weighted squared residuals Ω can also be expressed in matrix notation where

**v**denotes the residual vector and

**P**the weight matrix of the observations. The value f describes the degrees of freedom of the stated adjustment problem. The stochastic model is regarded as being appropriate if the empirical standard deviation s

_{0}falls into the interval of 0.7 < s

_{0}< 1.3 under the assumption that the standard deviation of the unit weight σ

_{0}was set to 1 as proposed, e.g., by Müller et al. [34] (p. 345).

_{0}. The horizontal axis shows intensity values on a logarithmic scale. The vertical green lines highlight the smallest and largest intensity values that were captured in the experiments for the generation of the stochastic models. The vertical axis graphs the empirical standard deviations s

_{0}that stem from the plane adjustments.

_{0}< 1.3. Hence, this outcome verifies the two suggested procedures for the generation of intensity-based stochastic models based on capturing 3D-point clouds instead of repeated range measurements. Note that the empirical standard deviations drift apart for higher intensities while a higher degree of conformity was apparent for the remaining parts.

## 3. Intensity-Based Stochastic Models for TLS with Impulse and Hybrid Rangefinders

#### 3.1. Stochastic Modelling of an Impulse Scanner

_{0}for every panel after the plane adjustment is graphed on the vertical axis while the horizontal axis features the mean signal strength of the corresponding points. Since all values s

_{0}fall into the specified boundaries 0.7 < s

_{0}< 1.3 it can be concluded that the generated intensity-based stochastic model is appropriate to describe the random characteristics of the scanner under investigation. In addition, the suggested procedure is generally capable to model rangefinders and, consequently, scanners that deploy the impulse method.

#### 3.2. A Stochastic Model for a Hybrid TLS

_{0}for every panel after the plane adjustment. Just as in Figure 11 the two green vertical lines mark the smallest and largest intensity values from the experiments that were used to generate the intensity-based stochastic model. All values fall into the specified boundaries of 0.7 < s

_{0}< 1.3 which allows drawing the conclusion that both computed stochastic models are adequate to model the stochastic characteristics of the scanner. All empirical standard deviations are smaller than 1, which means that the generated stochastic model releases values that are slightly too pessimistic by trend. Thitherto, no explanation for this characteristic was found and, hence, will be addressed in future investigations.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Characteristic ranges of intensity values versus range noise illustrated for a fictitious laser scanner.

**Figure 9.**Empirical standard deviations based on the reference stochastic model, as well as the novel approach.

**Figure 10.**Results of the experiment (blue circles) and generated intensity-based stochastic model (green line). Red circles indicate independent control measurements.

**Figure 11.**Empirical standard deviations of adjusted planes captured under different survey configurations. The two green vertical lines indicate the smallest and largest intensity values that stem from the experiments to generate the intensity-based stochastic model.

**Figure 12.**Results of the experiment (blue circles) and generated intensity-based stochastic model (green line) for speed mode (

**left**) and range mode (

**right**). Red circles highlight control measurements that were used to verify the computed stochastic model.Again, it was assumed that the intensity-based stochastic model follows the functional relationship described in Equation (3). The resulting parameters after the adjustment are gathered in Table 5. Note that the third parameter c was not of significance for the range mode.

**Figure 13.**Empirical standard deviations of adjusted planes captured under different survey configurations.

Stochastic Model from | $\hat{\mathit{a}}$ | $\hat{\mathit{b}}$ | $\hat{\mathit{c}}$ |
---|---|---|---|

Reference | 1.1742 | −0.5756 | --- |

Residuals | 4.1910 | −0.7145 | 0.0003 |

Quasi-ranges | 13.7781 | −0.8276 | 0.0005 |

Riegl VZ-400i | Leica ScanStation P40 | |
---|---|---|

Range [m] | 800 | 270 |

Angular resolution, accuracy (hz; vt) | 2.5″; 1.8″ | 8″; 8″ |

Range noise | 3 mm (1σ) at 100 m | 0.5 mm rms at 50 m |

Wavelength | Near infrared | 1550 nm |

**Table 3.**Parameters of the stochastic model for the rangefinder of a Riegl VZ-400i operated in MTA 1.

Sampling Rate (MHz) | $\hat{\mathit{a}}$ | $\hat{\mathit{b}}$ | $\hat{\mathit{c}}$ |
---|---|---|---|

1.2 | 17.5390 | −3.4689 | 0.0008 |

Range (m) | Incidence Angle (°) | Intensity (db) |
---|---|---|

20.974 | 27.8 | 24.36 |

20.977 | 34.2 | 18.74 |

20.988 | 56.4 | 17.07 |

20.989 | 57.1 | 22.26 |

42.121 | 28.8 | 14.71 |

42.123 | 30.7 | 20.47 |

42.128 | 52.9 | 19.06 |

42.140 | 53.6 | 13.23 |

62.645 | 21.8 | 12.59 |

62.651 | 45.9 | 10.95 |

62.651 | 21.2 | 16.69 |

62.663 | 46.1 | 15.41 |

101.091 | 44.8 | 13.30 |

101.091 | 24.1 | 14.02 |

101.099 | 17.7 | 12.63 |

101.114 | 44.6 | 11.35 |

**Table 5.**Stochastic model for the rangefinder of a Leica ScanStation P40 operated in speed and range mode.

Rangefinder Mode | Sampling Rate (MHz) | $\hat{\mathit{a}}$ | $\hat{\mathit{b}}$ | $\hat{\mathit{c}}$ |
---|---|---|---|

Speed | 1.0 | 1.904 × 10^{−6} | −1.1196 | 0.0001 |

Range | 0.5 | 4.072 × 10^{−6} | −1.011 | --- |

Rangefinder Mode | Range (m) | Incidence Angle (°) | Intensity |
---|---|---|---|

Speed | 12.681 | 15.1 | 0.0566 |

Speed | 23.719 | 25.0 | 0.0096 |

Speed | 61.202 | 7.3 | 0.0027 |

Speed | 76.022 | 42.5 | 0.0015 |

Speed | 91.473 | 5.5 | 0.0040 |

Speed | 105.730 | 19.9 | 0.0006 |

Speed | 116.415 | 9.8 | 0.0007 |

Range | 105.712 | 13.4 | 0.0026 |

Range | 127.089 | 17.4 | 0.0008 |

Range | 138.355 | 20.7 | 0.0009 |

Range | 149.715 | 19.8 | 0.0008 |

Range | 163.130 | 25.8 | 0.0007 |

Range | 180.638 | 36.5 | 0.0005 |

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**MDPI and ACS Style**

Wujanz, D.; Burger, M.; Tschirschwitz, F.; Nietzschmann, T.; Neitzel, F.; Kersten, T.P. Determination of Intensity-Based Stochastic Models for Terrestrial Laser Scanners Utilising 3D-Point Clouds. *Sensors* **2018**, *18*, 2187.
https://doi.org/10.3390/s18072187

**AMA Style**

Wujanz D, Burger M, Tschirschwitz F, Nietzschmann T, Neitzel F, Kersten TP. Determination of Intensity-Based Stochastic Models for Terrestrial Laser Scanners Utilising 3D-Point Clouds. *Sensors*. 2018; 18(7):2187.
https://doi.org/10.3390/s18072187

**Chicago/Turabian Style**

Wujanz, Daniel, Mathias Burger, Felix Tschirschwitz, Tassilo Nietzschmann, Frank Neitzel, and Thomas P. Kersten. 2018. "Determination of Intensity-Based Stochastic Models for Terrestrial Laser Scanners Utilising 3D-Point Clouds" *Sensors* 18, no. 7: 2187.
https://doi.org/10.3390/s18072187