Steger Carsten 
  
  
  
; Range of asymmetries and widths 
08r £ 
06 - / 
: ; 
04 - x 
0 0.5 1 15 2 2.5 3 3.5 4 
  
  
  
Figure 6: Range of widths w and asymmetries a for which staircase lines can be extracted with c = 1. The extractable 
range is contained between the two curves. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
4 1 
35 + True line position + True line width d—— Rm of line detectability 
x 0.8 
3} Line position 4 Edge position MÀ 
as | i 
2.5 Edge position 06 + 
w 2 a 
LSE A J 04} 
/ Line position 
LE 4 
Range of line detectability 0.2 
05 Ft € : True line width 4 True line position —#i 
Edge position X 
0 A À. À , À k A 0 A A - n 
4 -3 2 -1 0 1 2 3 4 3 -2 -1 0 1 2 3 
X x 
(a) (b) 
Figure 7: (a) Location of a staircase line with a = 0.75,w € [0, 4] and its edges for o = 1. (b) Location of a staircase 
line with w = 1.5, a € [0, 1] and its edges for 0 = 1. 
based on this value it is necessary to derive a restriction on o similar to the restriction 0 > w/ v3 for bar-shaped lines. 
This analysis can only be carried out for symmetrical staircase lines with a = 0.5 since only then the line position can be 
determined analytically. With these parameters, we obtain 
o>w (17) 
  
Combined with the above restriction 0 < w, the value of c should be chosen in the range 
wA (3— V6)/3 «a « w (18) 
if lines of width w are to be extracted. 
If staircase lines are extracted with the above definition and their edges with the definition of Section 2.2, the results will 
again generally be biased. Figure 7(a) shows the line and edge positions for a staircase line with a — 0.75 for w € [0, 4] 
extracted with 0 = 1. As can be seen, lines can only be extracted for w > 1.424 for this choice of a. Furthermore, the line 
position moves to the weak side of the line, i.e., the side with the smaller gradient, for small w. For large w the extracted 
line widths correspond closely to the true line widths, while the extracted line position asymptotically approaches the true 
line position. However, this convergence is very slow. It is also instructive to examine the behavior of the line positions 
and widths if c and w are kept constant, while the asymmetry a is varied. An example of this behavior for o. — 1 and 
w = 1.5 is displayed in Figure 7(b). As is to be expected, the behavior of the line and edge position is symmetrical with 
respect to a — 0.5. The line position is biased by a large amount for small and large a. The edge position is also biased, 
albeit by a smaller amount, for these choices of a. 
From these examples it can be seen that the extracted line and edge positions of staircase lines generally exhibit a bias, 
similar to the bar-shaped lines. The major difference is that for staircase lines the extracted line position always remains 
within the true line. Furthermore, the extracted line width is always too small, while it is always too large for bar-shaped 
lines. Therefore, for staircase lines a bias removal algorithm is also essential to obtain meaningful results. 
In order to model the bias of lines with different polarity the technique of Section 2.2 can be used. As was the case for 
bar-shaped lines, the line positions and widths of staircase lines form a scale-invariant system. Therefore, we can work 
  
146 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.