ee 
iv 
Steger Carsten 
  
     
  
        
esi 
(a) Bias removed (b) Bias not removed a s (c) Bias removed (d) Bias not removed 
Figure 4: Lines and their width detected in an aerial image of a reduced resolution of 1 m. To assess the accuracy of the 
results, they are shown superimposed onto the original image of resolution 0.25 m. 
r(x,0,1,0.5) —— — r(x,0,1,0.5) —— rÁx,0,1,0.5) ——— 
    
Figure 5: Scale-space behavior of the staircase line f, with w = 1 and a = 0.5 when convolved with the derivatives of 
Gaussian kernels for z € [—3,3] and c € [0.2, 1.5]. 
does not occur frequently for interesting objects in real images. Nevertheless, lines with different polarity (staircase lines 
for short) are interesting from a theoretical point of view. 
In order to extract line points with the profile given by (3), the profile must be convolved with the derivatives of a Gaussian 
in order to get a smooth function. The respective responses are given by 
r,(x,0,w,a) = aGg(x + w) + (1 — a)ós (x — w) (14) 
and its derivatives. Figure 5 shows the scale-space behavior of a staircase line with w — 1 and a — 0.5. As we 
can see, in the one-dimensional case the position of the lines is given by the inflection point of the smoothed line 
profile, where additionally the inflection must be “flat,” i.e., the line position is given by r{(z,0,w,a) = 0 with 
r"(x,o,w,a)r'(x,o,w,a) > 0. This is very similar to the definition of edges in the 1D case given in Section 2.2, 
where the inflection point is required to be non-flat. Considering Figure 5, it is clear that the flat inflection points will turn 
non-flat if o is chosen large enough compared to the line width w, i.e., the line will turn into an edge. Because for a = 0.5 
the line or edge position is at z = 0 for all c, this can be analyzed analytically by requiring r2 (0,0, w,0.5) — 0. With 
this, by a straightforward calculation it can be shown that lines can only be extracted for à < w. For other values of a, 
it can be expected that the line can be extracted for even smaller values of c only. Since for applications c is constant, 
while a and w vary in the image, it is interesting to determine the range of a and w for which lines can be extracted with 
a given c. It can be shown (Steger, 1998c) that lines can only be extracted for 
1 —a,(w,0) <a <as(w,0) , (15) 
where 
1 
w? — 0° +wVw? — 0?  4»y»2-7? 
e a? 
w? — 02 — wv uw? — 0° 
Figure 6 shows the range of w and a for which lines can be extracted with 0 = 1. We can see that the range of asymmetries 
grows for larger line widths. Furthermore, theoretically it is possible to extract lines of arbitrary width with o = 1 in the 
continuous case. 
(16) 
  
as(w,6) = 
| - 
  
Finally, it should be noted that salient staircase lines must be selected based on the magnitude of r^" since the first two 
derivatives will be very small and zero, respectively, at line points. As was the case for the bar-shaped line in Section 2.2, 
the magnitude of r" does not exhibit a unique maximum if c is small compared to w. Since salient lines are selected 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 145