Steger Carsten
(a) Bias not removed (b) Bias removed
Figure 11: Result of extracting lines and their width from a test image containing a line with varying asymmetry (a) with-
out and (b) with bias removal.
(a) Aerial image (b) Bias not removed (c) Bias removed
Figure 12: Aerial image (a) containing a line with different polarity in the part marked by the square. Lines extracted
from the marked part in (a) without (c) and with (d) bias removal.
can be seen as a dark line. Therefore, one way to extract lines with different polarity is to use the facet model line detector
used to extract the line width in Section 2.2 to extract dark lines from the gradient image. However, as Figure 10(b) shows,
lines with equal polarity also result in dark lines in the gradient image. Since only lines with different polarity should be
detected, this simple approach is not sufficient.
Fortunately, there is a simple criterion by which lines of different and equal polarity can be discerned. From (14) it is
obvious that the gradients point in the same direction on both sides of a line of different polarity, while they point in
opposite directions for lines with equal polarity. Thus, to discern lines of different and equal polarity, the gradients at the
edge positions in both directions perpendicular to the dark line in the gradient image are evaluated, and their dot product
is calculated. Only if the dot product is positive, a line of different polarity has been found. However, it is useful to be
more restrictive in the selection of lines. For example, junctions of three edges are also extracted as dark lines from the
gradient image in the vicinity of the junction. Therefore, in the implementation a dark line point is only accepted as a
line point of different polarity if the directions of the gradients are sufficiently close. Here, sufficiently close is defined
by the gradient directions forming an angle of less than 30?. With these criteria, lines points of different polarity can be
extracted fairly reliably.
To link the individual line points into lines the same linking algorithm as for bar-shaped lines is used (Steger, 1998b,
Steger, 1998c). The two hysteresis thresholds are applied to the eigenvalues of the dark lines in the gradient image, which
are equivalent to the third directional derivative in the direction perpendicular to the original line. Figure 10(c) displays
the results of extracting lines of different polarity with ¢ = 1.5 from the image in Figure 10(a). As can be seen, with the
approach discussed above the line of equal polarity is not extracted.
Figure 11(a) exhibits the results of extracting the line position and width with o = 1.7 from a test image containing a line
with varying asymmetry. As can be expected, the line position is strongly biased. In contrast, the line width corresponds
to the true line width for most of the line. The result of applying the bias correction to the extraction results is shown in
Figure 11(b). As can be seen, the extracted line position corresponds to the true line position, while the extracted line
width is very close to the true line width.
To conclude this section, Figure 12(a) displays an aerial image containing a line of different polarity in the part marked by
the square. This image was obtained from (Fritsch and Sester, 1994). The results of extracting lines of different polarity
with o — 1.4 and thresholds of 2 and 7 without and with applying the bias removal are displayed in Figures 12(b) and (c),
respectively. As can be seen, only the road corresponding to the staircase line has been extracted. Since the line has a fairly
148 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
