## 2 Materials and Methods
### Deflectometry  
To extract three-dimensional information from the surface of the rolling elements, the method presented in [27] is to be applied in a modified form. This method can be ideally adapted for use with a line scan camera by using a sufficiently fast exposure device. The illumination patterns presented in work [27] are used in a modified form. The formulas  
Figure 1: Schematic structure of the recording mechanics.  
Figure 2: Illumination switching logic.  
were adapted for the laboratory setup for rolling element testing. In this case, \(\sigma_{w}\) stands for the screen width in meters, and \(\sigma_{h}\) stands for the screen height in meters. The vector \(\vec{\omega}\) describes the position on the screen. The intensity \(P_{x/y}(\vec{\omega})\) in the respective coordinate direction results as a linear brightness curve according to Formulas (1) and (2). Formula (3) was not adjusted and is adopted for this work.  
\[P_{x}(\vec{\omega})=\frac{1}{2}\cdot(\frac{\omega_{x}}{\sigma_{w}}+1), \tag{1}\]  
\[P_{y}(\vec{\omega})=\frac{1}{2}\cdot(\frac{\omega_{y}}{\sigma_{h}}+1), \tag{2}\]  
\[P_{c}(\vec{\omega})=1 \tag{3}\]  
According to Formula (1), a horizontal brightness gradient is created for \(P_{x}\). This curve shows a linear brightness increase in the horizontal direction. Ideally, this goes from the brightness value of zero to the maximum possible brightness of the LEDs used. \(P_{y}\) represented by Formula (2) is a vertical brightness curve that goes from the value 0 to the maximum possible brightness value of the LEDs. \(P_{y}\) corresponds to \(P_{x}\) rotated by 90\({}^{\circ}\). \(P_{c}\) described by Formula (3) is the full exposure of the object, which means that all LEDs are driven at full power. The resulting brightness gradients are generated by selectively controlling the brightness of the LEDs. The LEDs then illuminate a focusing screen that further homogenizes the pattern. In Figure 3, these brightness curves can be seen both on the real matrix and as a schematic.  
The images recorded with the three illumination patterns, shown in Figure 3, contain the three images \(L_{x}\), \(L_{y}\), \(L_{c}\). From these three images, the tilt information is extracted, as given in Formulas (4) and (5).  
\[R_{x}=\frac{L_{x}}{L_{c}}, \tag{4}\]  
\[R_{y}=\frac{L_{y}}{L_{c}}, \tag{5}\]  
With the help of these two gradient images, the reflection vector \(\vec{r}\) can now be calculated, according to Formula (6), for each pixel, considering the screen width \(\sigma_{w}\), screen height \(\sigma_{h}\), and the distance between the object and the screen d.  
\[\vec{r}=\left[\begin{array}{c}\frac{\sigma_{w}}{\sqrt{\sigma_{h}^{2}+d^{2}}} \cdot(2\cdot R_{x}-1)\\ \frac{\partial_{h}}{\sqrt{\sigma_{h}^{2}+d^{2}}}\cdot(2\cdot R_{y}-1)\\ \sqrt{1-{{{\vec{r_{x}}}}^{2}}-{{{\vec{r_{y}}}}^{2}}}\end{array}\right] \tag{6}\]  
As the last step, the surface normal \(\vec{n}\) is calculated pixel by pixel as the angle bisector between the reflection vector and the fixed camera vector \(\vec{v}\), according to Formula (7).  
\[\vec{n}=\frac{\vec{r}+\vec{v}}{||\vec{r}+\vec{v}||} \tag{7}\]  
Figure 3: Illumination pattern (\(P_{x}\), \(P_{y}\), \(P_{c}\)).  
The result represents the gradient field of the surface and will be reconstructed as a virtual surface afterwards (cf. [27]).