Master Thesis - Fachbereich Informatik

4.6. MEASURING 89
side of the tube, the worst-case displacement is 0.5 at one side and −0.5 at the other side
leading to a total displacement of 1. A straight line connecting the two measuring points
in an Euclidean plane is slightly longer than the distance in x. Following Pythagoras’
theorem the maximum expectable error due to a vertical inaccuracy is:
errory = � l 2 +1− l (4.18)
where l is the pixel length between the left and right measuring point. With respect to
the definition of the camera’s field of view and the image resolution, the length of a tube
is about 415 pixels in an image. In this case, the worst-case error is about 0.0012 pixel.
Assuming one pixel represents 0.12mm (a typical value for 50mm tubes) this corresponds
to an acceptable error of 0.14µm which is far beyond the imaging capabilities of the camera
used (each sensor element has a size of about 8.3 × 8.3µm).
Other than in the vertical direction, a subpixel shift of the best matching template
position in horizontal direction has a significant influence on the length measurement
results. Again, assuming a maximum error of 0.5 pixels if discrete pixel grid resolution is
used, the total error at both sides sums up to 1 in worst-case. If one pixel corresponds to
0.12mm as in the example above, this means the measuring system has an inaccuracy of
the same length purely depending on the edge localization. Obviously, this error depends
on the resolution of the camera and can become even worse if one pixels represents a larger
distance.
The interpolation considers five discrete points: The maximum matching position Mmax
and the two nearest neighbors left and right to Mmax in x-direction respectively. In
Figure 4.24(b), the interpolation results of the local neighborhood around the discrete
maximum of Figure 4.24(a) are drawn into the plot of the match profile at y =5. It
shows the interpolated values describe the sampled values quite well. In this example, the
interpolated subpixel maximum equals the discrete maximum. This does not always have
to be the case as can be seen in Figure 4.24(c). Here, the discrete maximum is located at
x = 12, whereas the subpixel maximum lies at x =12.2. In the first case, the neighbor
pixels of the maximum yield almost equal results at both sides. On the other hand in the
second example, the right neighbor of the maximum is significantly larger than the left
one. This explains the shift of the subpixel maximum toward the right. The precision of
the subpixel match localization is 1/10 pixel. Mathematically, much higher precision is
possible,butthesignificanceofsuchresultsisquestionablewithrespecttotheimaging
system and noise, and increases the computational costs unnecessary.
4.6. Measuring
The result of the template matching are two subpixel positions indicating the left and right
measuring point of a tube. This section introduces how a pixel distance is transformed
into a real world length and how the measurements of one tube are combined. Therefore,
a tracking mechanism is required that assures the correct assignment of a measurement to
a particular tube. This means, one has to detect when a tube enters or leaves the visual
field of the camera.