# Complexity Modeling of Steel-Laser-Hardened Surface Microstructures
## 3 Complexity Prediction Model
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specimen** & **GPmin** & **GPmax** & **GPmed** & **Specimen** & **GPmin** & **GPmax** & **GPmed** \\  & **(Y) [-]** & **(Y) [-]** & **(Y) [-]** & **Specimen** & **(Y) [-]** & **(Y) [-]** & **(Y) [-]** \\ \hline P1 & 1.8620 & 1.9053 & 1.8808 & P13 & 1.8620 & 1.9053 & 1.8808 \\ P2 & 1.8342 & 1.8768 & 1.8527 & P14 & 1.8342 & 1.8768 & 1.8527 \\ P3 & 1.8433 & 1.8861 & 1.8619 & P15 & 1.8433 & 1.8861 & 1.8619 \\ P4 & 1.8580 & 1.9012 & 1.8768 & P16 & 1.8580 & 1.9012 & 1.8768 \\ \hline P5 & 1.8620 & 1.9053 & 1.8808 & P17 & 1.8620 & 1.9053 & 1.8808 \\ P6 & 1.8342 & 1.8768 & 1.8527 & P18 & 1.8342 & 1.8768 & 1.8527 \\ P7 & 1.8433 & 1.8861 & 1.8619 & P19 & 1.8433 & 1.8861 & 1.8619 \\ P8 & 1.8580 & 1.9012 & 1.8768 & P20 & 1.8580 & 1.9012 & 1.8768 \\ \hline P9 & 1.8620 & 1.9053 & 1.8808 & P21 & 1.8620 & 1.9053 & 1.8808 \\ P10 & 1.8342 & 1.8768 & 1.8527 & P22 & 1.8342 & 1.8768 & 1.8527 \\ P11 & 1.8433 & 1.8861 & 1.8619 & P23 & 1.8433 & 1.8861 & 1.8619 \\ P12 & 1.8580 & 1.9012 & 1.8768 & P24 & 1.8580 & 1.9012 & 1.8768 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Minimum, maximum, and medium values obtained with the complexity GP prediction model. microstructure was not self-similar but statistically self-affine. Therefore, this research study describes the new and more complex situation that prevails for fractal curves that are not self-similar but self-affine. Roughness measurements on the specimens' surface were performed by adopting profilometry techniques and using a computerized contact profilometer, Hommel Tester T1000 (Hommelwerke GmbH, Schwenningen Germany) [33, 34]. In the temperature range from 1373.0 to 1673.0 K, it appears that surface roughness increases with the increasing of temperature and decreases with the increasing of laser beam velocity. A quantitative assessment was not reported in the present manuscript because the performance of additional measures was deemed necessary as it ensures the statistical reliability and repeatability of the results. However, a first reliable qualitative assessment was found, and it is shown below. Using the pixel content of the SEM images it was possible to establish a correspondence between the complexity index described in Section 2.2, the \(Y\) parameter, and the roughness of the RLH surface. A 3D reconstruction with a 3D voxel map that reproduces the surface roughness was developed by automating a routine in a _Python_ environment. The following logical steps were performed:  
1. All 3 \(\times\) 3 pixel matrices in the SEM image were evaluated and divided into three groups: _totally black matrices_, where all nine pixels were black; _almost black matrices_, where eight pixels were black and one pixel was white; _hemi-black matrices_, where two or more pixels were white. 2. Each square surface of 10 \(\times\) 10 pixels was extruded into a 3D voxel (0.18 \(\times\) 0.18 \(\times\) 0.18 \(\upmu\)m cube), forming the layer 0 of the 3D reconstruction. 3. The areas with percentages of _totally black matrices_ lower than 30% were associated with layer 0; the areas with percentages of _totally black matrices_ plus _almost black matrices_ ranging from 30% to 50% were associated with layer 1 (higher than layer 0 by one voxel); the areas with percentages of _totally black matrices_ plus _almost black matrices_ between 50% and 70% were associated with layer 2 (higher than layer 1 by one voxel); the areas with percentages of _totally black matrices_ plus _almost black matrices_ between 70% and 90% were associated with layer 3 (higher than layer 2 by one voxel); and finally, the areas with percentages higher than 90% in _totally black matrices_ plus _almost black matrices_ were associated with layer 4 (higher than layer 3 by one voxel). 4. Labeling the layer 0 areas with a blue color, the layer 1 areas with a cyan color, the layer 2 areas with a green color, the layer 3 with a yellow color, and the layer 4 with a red color, the 3D voxel maps were obtained (Figure 5a,c).