## 2 Materials and Methods
### Considerations about the Types of Distributions of the Parameters of the Vision Subsystem
#### 2.2.1 Procedure for Measuring a Glass Line Scale  
A NPL (National Physical Laboratory) glass line scale (Figure 2a), with a nominal length of 100 nm and a distance of 0.1 mm between graduations, was used. Its main purpose was to serve as a high accuracy standard for performing calibrations in the industrial area. The graduation width was about 10 \(\upmu\)m. The scale contained two parallel horizontal lines, about 50 \(\upmu\)m apart (Figure 2a), at the start and the end of the scale, used as an alignment axis. The coefficient of linear expansion of the scale, as indicated by the manufacturer, is: \(\alpha=(0.5\pm 0.1)\times 10^{-6}\,\mathrm{K}^{-1}\).  
The glass line scale was calibrated by the CEM using a microscope-CCD camera assembly and laser interferometric system. The results up to a nominal position of 0.4 mm were as follows (Table 2).  
The measurands are the distances \(d_{k}\) between the reference centreline of the zero graduation and the centreline of the \(k\)-th graduation. The measurements were made on the section of the scale to fall between the two horizontal lines used as an alignment axis. The field of vision of the Ultra-QV350 equipment for the measurement conditions used above (magnification \(10\times\)) was \(0.62\) mm \(\times\)\(0.47\) mm. Therefore, the distances between graduations of the glass line scale are measured up to the nominal position \(0.4\) mm so that it is possible to compare the results obtained in different positions.  
This phase was divided into the following steps:  
* acquisition of multiple images of the glass line scale. Ten images at intervals of \(30\) s were captured, with the standard in the same position, and the information obtained from the images was used in the subsequent calculation of uncertainties;
* correction and normalization of the luminance of the image [19];
* detection of the pixels that formed the edge of the glass line scale graduations (Figure 2b), employing the Canny edge detection method with \"global thresholding\". Each \(k\)-th graduation of the glass line scale, with \(k=0,\dots,n_{t}\), has two edges, called an odd edge and even, the edges of a \(k\)-th graduation are characterized by a several number of points (\(p_{k,att}\), \(p_{k,rem}\)), and, for each of the \(k\) graduations, the pixels of the odd edge \(\left.\left(u_{i}^{k},v_{i}^{k}\right)\right|_{i=1}^{p_{k,att}}\) and the even edge \(\left.\left(u_{j}^{k},v_{j}^{k}\right)\right|_{j=1}^{p_{k,rem}}\) are obtained;
* conversion of the previous coordinates (in pixels) of the edge in units of length. The above relationship is determined by the expression: \[\mathbf{X}_{m}=\mathbf{K}^{-1}\cdot\mathbf{U}_{m},\] (3) where \(\mathbf{U}_{m}=\left(u_{m}^{\prime}-\delta_{u_{r}}^{\prime},v_{m}^{\prime}- \delta_{v}^{\prime}\right)\) and \(\mathbf{X}_{m}=(x_{m},y_{m},1)^{T}\) are the coordinates in units of length. The components of the matrix \(\mathbf{K}^{-1}\), the terms of geometric distortion (distortion coefficients \(k_{1},k_{2},p_{1},p_{2},s_{1},s_{2}\), and the geometric distortion centre \(u_{r},v_{r}\)), were obtained in the previous calibration of the vision subsystem (Section 2.1).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nominal Position** & **Measured Valued** & **Deviation from** & **Expanded Uncertainty** \\
**(mm)** & **(mm)** & **Nominal Position (\(\mu\)m)** & **(\(k\) = 2) (\(\mu\)m)** \\ \hline
0 & 0.00000 & 0.00 & 0.08 \\
0.1 & 0.10003 & 0.03 & 0.08 \\
0.2 & 0.20006 & 0.06 & 0.08 \\
0.3 & 0.30001 & 0.01 & 0.08 \\
0.4 & 0.39999 & \(-\)0.01 & 0.08 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calibration data of the glass line scale.  
Figure 2: (a) Glass line scale. (b) Edges detected by processing the image of the line scale.  
determination of the distance between graduations. Once the coordinates of the edge pixels have been processed, the distances \(d_{k}\) between the zero graduation and the \(k\)-th [16] are determined).