## 4 Results
### Characterization  
To characterize the model-based algorithm in more detail, Monte Carlo simulations with thermograms of the size (\(140\times 140\)) pixels\({}^{2}\) are utilized, with which the dependency of the model-based algorithm on the CNR value as well as on the size of the turbulence wedges is examined. The deviations of the detected number \(N\) of turbulence wedges, the measured position \(x_{i}\) and the size parameters \((h_{i},w_{i})\) as well as the area \(A_{i}\) are examined. First, the dependency of the measurement results on the CNR value is investigated. The CNR value is varied in a range of \(2\leq\text{CNR}\leq 20\) by changing the pixel intensity of the \(\bar{N}=1\) turbulence wedge present in the simulated thermogram. For each CNR value, \(n=100\) thermograms are investigated. Note that the other parameters used in the simulation, such as the wedge size and the intensity gradient, are stated in Section 3.2. The number \(N\) of turbulence wedges that are detected in a thermogram is assessed and compared to the true value \(\bar{N}\). Here, the relative error  
\[F_{N}=\frac{N_{i}-\bar{N}_{i}}{\bar{N}_{i}} \tag{15}\]  
is investigated, where \(N_{i}\) is the number of detected wedges in the \(i\)th thermogram and \(\bar{N}_{i}=1\) is the true number of wedges. In Figure 9, the systematic error \(\langle F_{N}\rangle\) is shown as a function of the CNR. Only for the lowest CNR value (\(\text{CNR}=2\)), the number of detected wedges is not equal to the true number of wedges, where 7% of turbulence wedges are missed. In all other cases, \(\langle F_{N}\rangle=0\), which means that the number of found wedges is equivalent to the true number of wedges in the thermogram, i.e., the detection error rate is zero. Furthermore, the position error \(\Delta x\) normalized by the average true width \(\langle\bar{w}\rangle\) of the wedges, is investigated:  
\[\frac{\Delta x}{\langle\bar{w}\rangle}=\frac{1}{\langle\bar{w}\rangle}(x_{i} -\bar{x}_{i}). \tag{16}\]  
Here, \(x_{i}\) is the \(x\)-position of the found wedge in the \(i\)th thermogram, \(\bar{x}_{i}\) is the true \(x\)-position and \(\langle\bar{w}\rangle=15.01\) pixels is the average true width of all wedges used. Only the \begin{table}
\begin{tabular}{c c c c} \hline  & **True Value** & **GBA** & **MBA** \\ \hline Wedge I & 389.5 & 512.0 & 370.5 \\ Wedge II & 187.5 & - & 180.0 \\ Wedge III & 256.5 & - & 270.0 \\ Total & 833.5 & 512.0 & 846.5 \\ Measurement error normalized by the true value in \% & 0 & \(-\)38.6 & 1.5 \\ \hline \end{tabular}
\end{table} 
Table 2: Areas determined by the model-based algorithm (MBA) as well as the state-of-the-art, gradient-based algorithm (GBA) for the wedges in Figure 8. The area values are given in unit pixel\({}^{2}\). Figure 9: Mean relative error \(\langle F_{N}\rangle\) for different CNR values. For each CNR value, \(n=100\) thermograms were investigated. The gray region marks CNR values found in real thermograms, \(\text{CNR}=10\pm 1\,\sigma\), with \(\sigma=6\). The error bar indicates the standard deviation of the mean. \(x\)-position needs to be considered as the algorithm detects the wedges along the natural transition line, i.e., the \(y\)-position follows from the \(x\)-position and the determined natural transition line. Figure 10 shows \(\frac{\Delta x}{\left\langle\overline{w}\right\rangle}\) as a function of the CNR. For the lowest CNR value of \(\text{CNR}=2\), seven thermograms were excluded from the analysis as no turbulence wedge was found and, therefore, \(x_{i}\) is unknown. The mean relative error \(\left\langle\frac{\Delta x}{\overline{w}}\right\rangle\) stays between 2% and 4% for all CNR. The standard deviation of \(\frac{\Delta x}{\overline{w}}\) in Figure 10 (right), which represents the random error, is larger than the systematic error for most CNR values by a factor of 1.5. However, for \(\text{CNR}<3\), the systematic error is larger than the random error. Therefore, the systematic error plays a significant role at low \(\text{CNR}<3\), where it should be corrected.