# On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. Functional Model

#### 2.2. Stochastic Model for TLS

#### 2.2.1. Stochastic Model for Range Measurements

^{4}[20].

^{4}to 10

^{5}and are expressed in increment (Inc). Whereas the upper panel corresponds to variations of $\beta $ for a fix $\alpha $, in the lower one $\alpha $ is varied and $\beta $ kept constant to 1.3. This value corresponds to a sampling rate of 1016 kHz in Wujanz et al. [14]. Unsurprisingly, $\beta $ acts as a rescaling parameter of the variance. When $\alpha $ is decreased however, the variations of the standard deviation are more pronounced at low intensities. ${\sigma}_{r}^{}$ approaches a constant value when $\alpha $ tends to 0.

#### 2.2.2. Note on Scaling

#### 2.2.3. Building the VCM of the TLS Measurements

#### 2.2.4. Mathematical Correlations

#### 2.3. Effect of Misspecification of the VCM

- The a posteriori variance factor, which is a key quantity in many statistical tests [2] such as the overall model test, and allows judgment of the LS solution.
- The loss of efficiency of the estimator, which is based on a ratio of mean-squared errors (MSE). It measures the performance of the LS estimator: an MSE close to 0 means that the estimator has a perfect accuracy. MSE should only be used for comparative purposes. The main advantage of the MSE formulation is its nondependency on the dataset allowing still a quantification of the mean-squared differences of the estimated parameters when VCM are changed [37].

## 3. Simulation

#### 3.1. Methodology

- The Cartesian observations $l$ are approximated with a B-splines curve by assuming that ${\widehat{Q}}_{MC}=I$. $I$ is the identity matrix. i.e., no assumption about the stochastic properties of the TLS measurements is made in this first step. The design matrix $A$ is filled using the knot vector and B-splines of order 3 [27], as seen in Equation (2).
- The computed observations are obtained from ${l}_{computed}=A\widehat{x}$, where $\widehat{x}$ are the coordinates of the estimated CP.
- The OMC observations are given by $l-{l}_{computed}$. They thus correspond to having extracted the geometry from the original observations: $l-{l}_{computed}$ is a zero-mean vector (or a straight line). The VCM of the OMC measurements corresponds exactly to ${Q}_{0,MC}$ but will be replaced by its approximation ${\widehat{Q}}_{MC}$ in the second LS adjustment, i.e., the approximation of the straight line. In this last step, the determination of the knot vector and by extension of the design matrix are straightforward and we choose an order 0 for the B-splines. $A$ is filled up with 1, and only one control point is estimated, which corresponds to the mean estimator.

#### 3.2. Intensity Vector

- Case 1: the intensity vector is the original one $In{t}_{ref}$, seen in Figure 2a. The mean intensity value is 930,000 Inc, which corresponds to a mean estimated standard deviation for the range of ${\sigma}_{r}=0.64\mathrm{mm}$.
- Case 2: the intensity is the same as case 1, rescaled by a factor 1/100. By doing so, we aim to simulate low intensity values, seen in Figure 2b. Following Figure 1, a higher variability of the range variance because of the power function is expected. A higher impact on the a posteriori variance factor or the loss of efficiency when $\alpha $ or $\beta $ are misspecified can be expected.
- Case 3: the intensity profile corresponding to case 1 is divided in the middle into two parts, seen in Figure 2c: the first part corresponds to case 1, whereas the second one to case 2. The intent here is to simulate intensity values that would vary strongly inside the same object due e.g., to different reflectivity properties.

#### 3.3. Scaling of ${\sigma}_{r}$

#### 3.4. Results of Simulations

#### 3.4.1. Varying $\alpha $

- Case 1 and 2

_{MSE}follow the standard symmetric shape of a mean squared error around 0 so that R

_{MSE}is slightly lower for $\alpha =-0.9$ than for $\alpha =0$, the difference of 0.04 (4%) being considered as non-relevant. The case $\alpha =0$ retains our attention since it corresponds to a simplification of the weighting model, which becomes independent of the intensity. The value $BIAS$ reaches in that case a negligible value of −0.046. Although it corresponds to a decrease of 9% wrt. to the 0-reference, the value in itself is not significant, i.e., an overall test [40] will not be sensitive to such a variation. Thus, a misspecification of $\alpha $ under the same metric impacts neither $BIAS$ nor R

_{MSE}significantly in the first two cases.

- Case 3

#### 3.4.2. Varying $\beta $

_{MSE}are presented in Figure 4a. As mentioned previously, $\beta $ simply act as a scaling factor, seen in Figure 1a. Because the models are scaled for the sake of comparison, its impact on the two quantities is similar for all three cases as long as $\alpha $ is kept fixed. The results are thus not presented graphically for the sake of shortness. From Equation (12), R

_{MSE}is a ratio of trace so that the scaling parameter is eliminated. Thus, $\beta $ has no impact and R

_{MSE}will stay constant when $\beta $ is varied. On the other hand, $BIAS$ increases until the zero value is reached for the reference parameters. $BIAS$ exceeds 0 for $\beta >{\beta}_{0}$. From $\beta =3$, the rate of change of $BIAS$ with $\beta $ decreases, i.e., degrading the assumed standard deviation of the range will only slightly impact the $BIAS$. A saturation to $BIAS=0.8$ occurs for $\beta >4$, which corresponds to a variation of 33% from the 0-reference value. As in 3.4.1., only values of $BIAS>-1$ are relevant, i.e., too overoptimistic values of $\beta $ should be avoided, which corresponds in our case to $\beta >1.2$. Thus, the rate of change of $BIAS$ for $\beta <{\beta}_{0}$ is much stronger than for $\beta >{\beta}_{0}$ as in the interval $\left[\begin{array}{cc}1.2,& {\beta}_{0}\end{array}\right]$, $BIAS$ varies more than 250%.

#### 3.4.3. Note on the Scaled Identity Matrix, Case 1

^{7}so that the 0-reference value for $BIAS$ (red star) could be reached inside the range of values of $\gamma $. As mentioned previously, R

_{MSE}is independent of the scaling parameter and equals 0.046 for all $\gamma $. The difference remains small (4%) wrt. the reference, but is synonymous with a slight model misspecification. As expected, it corresponds to the value found by taking $\alpha =0$, and is thus coherent with the previous results. We further note that $BIAS=0$ for ${\gamma}_{0}=3850$. This optimal value is clearly related to the mean standard deviation of the range over all intensity values, i.e., 3780 mm.

#### 3.5. Summary of the Simulations

_{MSE}would have been moved by the corresponding difference wrt. to the reference taken in this article.

## 4. Real Case Analysis

#### 4.1. Methodology

#### 4.2. Results

_{MSE}are four times smaller than in the simulations, highlighting that the loss of efficiency induced by using a scaled identity matrix ($\alpha =0$) is negligible. Varying $\beta $ in a range of values from 0.5 to 5 neither affected $BIAS$ nor R

_{MSE}significantly. Corresponding results are not presented here for the sake of shortness. As mentioned in Section 3, a good approximation of the standard deviation of the range by means of the IM remains however indispensable. As long as the intensity values are homogeneous, the factor of a scaled identity matrix can be alternatively deduced from the a posteriori variance factor.

**X**component, independently of the power factor chosen. Both the goodness of the knot vector, as well as the good approximation of ${Q}_{0}$ by means of $\widehat{Q}$ are here highlighted.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Variations of ${\sigma}_{r}$ (log plot) versus intensity with the intensity-based model (

**a**) for different the scaling factors $\beta $ and (

**b**) different power factors $\alpha $.

**Figure 2.**The three intensity vectors retained for the simulations. The intensity values in [Inc] are sorted per point, i.e., per time. 2500 points were measured. Intensity vector (

**a**) $In{t}_{ref}$ corresponds to an original vector from an arch bridge; (

**b**) $In{t}_{ref}/100$; (

**c**) a two part intensity vector built as a mix between case 1 and case 2.

**Figure 3.**Impact of variations of the power factor $\alpha $ of the intensity-based model for R

_{MSE}(

**a**) and $BIAS$ (

**b**); the three subplots correspond to the three cases of interest: normal (1), low (2) and strong varying intensities (3) respectively. Please note the different scaling for case 3 (bottom b) for the sake of readability.

**Figure 4.**(

**a**) Impact of variations of the scaling $\beta $ of the intensity-based model for the bias of the loss of efficiency R

_{MSE}(

**top**) and the a posteriori variance factor (

**bottom**). All cases of interest lead to the same values; (

**b**) $BIAS$ for $\widehat{Q}=\gamma I$ corresponding to a scaled identity matrix. Please note the log values of $\gamma $ for the x-axis. The red star corresponds to $BIAS=0$ and the intensity vector to case 1.

**Figure 5.**Simulated parametric B-splines curves. (

**a**) X component; (

**b**) Y component versus point number (i.e., time). To the geometrical observations is added a noise vector, with VCM ${Q}_{0}$. This real case study corresponds to an arch bridge with a known noise vector.

**Figure 6.**Results from a real case study with known noise vector. (

**a**) R

_{MSE}and $BIAS$ of the a posteriori variance factor. The red star corresponds to the reference parameters ${\alpha}_{0}$; (

**b**) residuals for the X (

**up**) and Y (

**bottom**) components. The Monte Carlo algorithm to compute the knot vector was used for a better accuracy of the B-splines approximation.

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**MDPI and ACS Style**

Kermarrec, G.; Alkhatib, H.; Neumann, I.
On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation. *Sensors* **2018**, *18*, 2964.
https://doi.org/10.3390/s18092964

**AMA Style**

Kermarrec G, Alkhatib H, Neumann I.
On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation. *Sensors*. 2018; 18(9):2964.
https://doi.org/10.3390/s18092964

**Chicago/Turabian Style**

Kermarrec, Gaël, Hamza Alkhatib, and Ingo Neumann.
2018. "On the Sensitivity of the Parameters of the Intensity-Based Stochastic Model for Terrestrial Laser Scanner. Case Study: B-Spline Approximation" *Sensors* 18, no. 9: 2964.
https://doi.org/10.3390/s18092964