Master Thesis - Fachbereich Informatik

80 CHAPTER 4. LENGTH MEASUREMENT APPROACH
important to have a closer look on the response of the edge detection results with respect
to the input data. Consistent characteristics can used for the design of the right template.
Figure 4.17 shows examples of the SOBELX operator applied to test images. In this
case, the response corresponds to the right ROI of three transparent tubes (Figure 4.17(a)-
(c)) and one black tube (Figure 4.17(d)) at different positions in the image with respect
to the x-axis. The tube boundary can be detected intuitively by humans even under the
presence of background clutter. However, one can find the edge response differs between
the different plots due to image contrast or perspective.
Figure 4.17(a) shows an almost straight edge (close to the optical center of the camera)
with a quite uniform region left of the ridge belonging to the tube, and a more varying area
on the right due to the background structure. It can be observed that the edge response
is stronger at the ends of the ridge than in the center, which is due to the transmittance
characteristic of transparent tubes (see Section 4.2). More light is transmitted at the
center leading to brighter intensity values and a poorer contrast, while the corners (‘L’corners
between horizontal and vertical boundary of a tube) are darker and yield a better
contrast. This effect can be seen also very clearly in Figure 4.17(b). In addition, the tube
boundary looks convex due to perspective since it is further away from the camera center.
Vertical edges of printings on a tube’s surface are also extracted by the edge detection step
as can be seen in Figure 4.17(c). In this case, the straight line of an upsight-down capital
‘D’ falls into the right local ROI, causing the smaller ancillary ridge on the left of the tube
boundary. Figure 4.17(d) includes the boundary of a black tube. Due to the strong image
contrast the edge response is about three times stronger compared to transparent tubes.
The influence of the background clutter reduces to a minimum and since printings are not
visible on black tubes at back light, this problem vanishes completely. The edge response
does not differ in intensity at the ends like with transparent tubes.
4.5.3. Template Design
The goal is to design a universal, minimum set of templates that covers all potential edge
responses of both transparent and black tube boundaries. The templates must model
different curvatures to be able to handle perspective effects. Assuming a constant horizontal
orientation and a constant size, the curvature is the only varying parameter between
templates. The following two-dimensional function has been developed that can be parametrized
to approximate the expected edge responses:
� �
y
Tψ(x, y) =aexpb
HT
� 2
− (x − (ψy2 )) 2
2σ 2
�
(4.14)
It is based on a Gaussian with standard deviation σ in x-direction extended with respect
to y. The curvature is denoted by ψ. A value of ψ = 0 represents no curvature, while
the curvature increases with increasing values of ψ (ψ ≤ 1). The first summand in the
exponent of the exponential function can be used to emphasize the ends of the template
in y-direction which is motivated in the characteristic response of transparent tubes. The
edge detector results in higher values at the ends than at the center. b controls the amount
of height displacement. If b = 0, the template is equally weighted. HT corresponds to the
template height. a determines the sign of the template values. For bright-dark edges like at
the left boundary the edge response is negative, thus a