Description of Magnet Ring Measurement System

The magnet ring measurement system operates as follows. A magnet ring image is corrected and binarized in the usual way. (See [6] and section 3.3.) The resolution of the corrected image is $512 \times 512$ pixels covering area of $100 \times 100$ mm.

Then, the two contours are extracted and rectified to subpixel precision. The subpixel rectification procedure is similar to the one used in [2], with some modifications. The idea is to shift each contour pixel position along the direction of the intensity gradient vector so as to find the precise location where the intensity surface crosses the threshold plane. The gradient vector is orthogonal to the contour, so each contour point may shift inwards or outwards depending on the intenisty surface.

The subpixel rectification procedure along the gradient direction is illustrated in figure 16. In this figure, $T$ is the threshold, $X$ the discrete location of the contour point obtained without rectification, $X_S$ the rectified non-integer location. The latter is computed by linear interpolation of the intensities $I_-$ and $I_+$ of the neighboring pixels.

  

Figure 16: Subpixel contour rectification.

The measures computed are defined in accordance with the definitions given in section 6.1:

1.

$D_{max}$ is computed by projecting the contour to a baseline of changing angle, then selecting the most extending projection. The angular resolution is $1^\circ$.

2.

$D_{min}$ is computed as $D_{max}$, but for the least extending projection.

3.

$d_{max}$ is the same as $D_{max}$, but for the hole.

4.

$d_{min}$ is the same $D_{min}$, but for the hole.

5.

To compute $O_{D}$, we consider the diameter as the function of baseline angle. Then, we take each pair of diameters separated by $90^\circ$ and select the pair for which the absolute difference of the values is the highest.

6.

$O_{d}$ is the same as $O_{D}$, but for the hole.

7.

$N_{cnc}$ is computed by its definition.

8.

$N_{cir}$ is computed as follows. The mean radius of the outer contour is obtained via the mean diameter for all baseline angles. (See Point 1.) Starting from the centroid, we then compute the absolute deviations of the contour points from this mean radius. The corresponding area deviations are approximated by products of the radius deviations and a portion of perimeter spanned at $1^\circ$. Then the sum of the areas is normalized as defined in (3).