Master Thesis - Fachbereich Informatik

4.5. MEASURING POINT DETECTION 87
where ψmax and ψmin indicate the overall maximum and minimum curvature a template
can have. Hence, one has to compute Nψ,total × k templates for each side. This can be
done in a preprocessing step to reduce the computational load. During inspection one has
to determine which templates have to be checked at a given position defined by the center
of the local ROI around a predicted tube edge. For an efficient implementation a look up
table (LUT) is used for this task.
4.5.4. Subpixel Accuracy
The maximum accuracy of the template based edge localization so far is limited by the
discrete pixel grid. The templates are shifted pixelwise within the local ROIs to find the
position that reaches the maximum correlation score. Following the assumptions of tubes
under perspective (see Section 4.2.3) the measuring is performed between the most outer
points of the convex tube edges.
The way the templates are defined the template center corresponds always to the most
outer point of the generated ridge. This is consistent to template rotation, since the
rotation is performed around the template center. In the special case that the template is
not curved, the template center is still the valid measuring point. With the knowledge of
this point within the template and the position where this template matches best in the
underlying image, the position of the measuring point in the image can be easily computed.
However, pixel grid resolution is not accurate enough in this application. For example
one pixels represents about 0.12mm in the measuring plane ΠM in a typical setup for
50mm tubes. The allowed tolerance for 50mm tubes is ±0.7mm. As a rule of thumb for
reliable results, the measuring system should be as accurate as 1/10thofthetolerance,
i.e. 0.07mm in this example. To reach that accuracy one has to apply subpixel techniques
to overcome the pixel limits.
Figure 4.24(a) visualizes the results of the cross-correlation of an image ROI around the
right boundary of a transparent tube with the template that yields maximum score. The
maximum is located at position Mmax =(19, 5). These coordinates refer directly to the
edge position in the image, since the template function is known and therefore the exact
location of the template ridge.
The real maximum that describes the tube edge location most accurate may lie in between
of two grid positions. With respect to the measuring task, the edge has to be
detected as accurate as possible. Interpolation methods have been introduced in Section
2.3.4 to overcome the pixel grid limits in edge detection. The same can be applied at
this stage to the template matching results.
Cubic spline interpolation is used to compute the subpixel maximum within a certain
neighborhood around the discrete maximum. Cubic splines approximate a function based
on a set of sample points using piecewise third-order polynomials. They have the advantage
of being smooth in the first-derivative and continuous in the second derivative, both within
an interval and its boundaries [53].
The interpolation is performed only with respect to the x direction, since this is the
measuring direction. A subpixel location with respect to y has only a marginal effect on
themeasurements.Ideally,themeasuringpointsontheleftandrightsidehavethesame
y value. Assuming the real maximum location is displaced by maximal 0.5 pixels at each