## 2 Materials and Methods
### Evaluation of the Images  
To distinguish complete cuts from incomplete cuts during the cutting process, spectral and geometric information from the melt pool in the kerf was evaluated for the image analysis, as shown in Figure 2. Each recorded image _I(u,v,c)_ was a two-dimensional function with the column in the x-direction and rows in the y-direction for each red, green, and blue color channel. However, since the information from the red color channel appears similar to the information from the green color channel, only the green and blue parts were used for further calculations. As common for digital images, the origin is located at the top-left. Consequently, the x-axis is oriented from left to right, whereas the y-axis is oriented from top to bottom. The information about the individual red, green, and blue color channels was correspondingly provided with a data depth of eight bits, i.e., each color channel was provided with 256 gradients. To reduce the amount of data, only the spatial region of interest was used, and the area outside of the nozzle's field of view was cut out. In the first step of the algorithm, the image was binarized _bw(u,v,c)_ separately for each channel with the threshold \(t_{B}\) = 50. For a lower threshold, a larger area of the melt pool was detected up to the edges of the kerf or the edge of the nozzle's field of view. At high thresholds, smaller or no amount of melt was detected in the kerf, especially for the blue. This leads to almost identical melt pool sizes for different cutting experiments. It is also worth noting that the threshold value might need to be adjusted depending on the previous image processing steps, process parameters, and camera settings. \[bw(u,v,c)=\left\{ \begin{array}{ll}1,&if\ I\ (u,v,c)\geq t_{B}\\ 0,&otherwise\end{array} \right. \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Sheet Thickness** & **Laser Power** & **Laser Power** & **Feed Rate** & **Feed Rate** \\  & **(mm)** & **(kW)** & **Steps (kW)** & **(mm\(\cdot\)s\({}^{-1}\))** & **Steps (mm\(\cdot\)s\({}^{-1}\))** \\ \hline Stainless steel & 1.5 & & & 60–220 & 20 \\ \hline Zinc coated steel & 1.0 & & & 150–300 & 25 \\  & 3.0 & 2.0–4.0 & 0.5 & 40–100 & 10 \\ \hline  & 1.0 & & & 180–300 & 20 \\ Aluminum & 3.0 & & & 40–160 & 20 \\  & 5.0 & & & 30–90 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variation of the cut parameters. The binarized image was then used to calculate the features, including the area, ferret diameter, and position of the melt pool. To calculate the area of the melt \(A_{melt}\), all white pixels of the binarized images were summed. \[A(c)_{melt}=\sum_{i=1}^{u}\sum_{j=1}^{v}bw(i,j,c) \tag{2}\]  
The ferret diameter, in general, describes the distance between two parallel tangents of an object. In our image analysis, it describes the minimum and maximum distance of the melt pool area. In parallel, the edges of the melt pool were extracted by a canny algorithm in order to subsequently calculate the geometry of an ellipse in the edge of the melt in the direction of the feed rate. For edge detection, the binarized image _bw(u,v,c)_ was initially filtered with a gaussian filter for noise reduction with the filter function \(H(i,j)\) and a standard deviation \(\sigma\) of 0.5. \[bw^{\prime}(u,v,c)\leftarrow\sum_{i=-1}^{i=1}\sum_{j=-1}^{j=1}bw(u+i,v+j,c) \cdot H(i,j) \tag{3}\]  
\[H(i,j)=e^{-\frac{\beta^{2}+\beta^{2}}{2\sigma^{2}}} \tag{4}\]  
Subsequently, for edge detection, a gradient value \(G\) was calculated as a directional filter response in the horizontal and vertical directions. Accordingly, all maxima were white, i.e., the edge \(M\) and all zero values were black. \[\
abla G(u,v,c)= \begin{pmatrix}G_{x}(u,v,c)\\ G_{y}(u,v,c)\end{pmatrix} = \begin{pmatrix}\frac{\partial bw^{\prime}}{\partial x }(u,v,c)\\ \frac{\partial bw^{\prime}}{\partial y}(u,v,c)\end{pmatrix}  \tag{5}\]  
Figure 2: Illustration of the image processing steps.