Steger Carsten 
  
  
  
T-Junction 
no 
  
  
  
  
  
4 2 0 2 4 
(a) T-junction (b) Extracted lines (c) Lines in the (x, y)-plane 
Figure 13: (a) A T-junction with lines of equal width and contrast. Extracted lines by the De Saint-Venant condition for a 
T-junction with lines of equal width and contrast: (b) mapped onto the response r;(z, y, 1), and (c) in the (x, y)-plane. 
  
  
  
   
  
-4 2 0 2 4 
(a) Extracted lines (b) Lines in the (x, y)-plane 
Figure 14: Extracted lines by the De Saint-Venant condition for a T-junction with lines of equal width and different 
contrasts of h; = h3 — 2 and h» = 1: (a) mapped onto the response 7; (x, y, 1), and (b) in the (x, y)-plane. 
The scale-space description of this junction is given by 
ri(z,y,0) = hi (és (y + wı) Is $c (y m. w1))6c (x) + 
ha (do (y + wa) — Do (y — W2))(1 — do (x)) + 
h3(Do(T + W3) — Po (x — w3))(1 — do (y + w1)) (20) 
and its partial derivatives of first and second order. Figures 13(a) and (b) show a T-junction with lines of equal widths 
wı = wW — ws — 1 and equal contrasts hy = hy = h3 — 1, and the corresponding response r;(z, y, 1) for (z, y) € 
[-5,5] x [-5,5]. 
In order to analyze the scale-space behavior of the line positions the zero crossings of the first directional derivative in the 
direction of the maximum second directional derivative must be obtained. To analyze the behavior of lines in the 2D case, 
we can use De Saint-Venant's condition S — 0 (Koenderink and van Doorn, 1994). This condition is given by 
Vr? HVri 1 TaTy (Tas — Ty) — (r2 — 2 ey 
T … 2 2 
Vr; Vr; Tz tr, 
  
  
Sz =0. (21) 
This condition will extract a superset of the lines the proposed algorithm extracts because it will also extract the points 
where there is a restricted extremum in the direction of minimum second directional derivative, and because it does not 
discern between restricted maxima and minima. Nevertheless, because the extracted lines form a superset of the lines 
returned by the proposed algorithm, the reasons for missed junctions can be studied. 
Figures 13(b) and (c) show the lines extracted by the condition S = 0 for the T-junction of Figure 13(a). One thing 
immediately strikes the eye: the “hump” in the middle of the Gaussian smoothed junction. As can be seen, the three 
extracted lines seem to intersect in the middle of the junction, at the maximum of the hump. However, closer inspection 
reveals that the line of the y-axis actually is a restricted maximum in the direction of the minimum second directional 
derivative in the vicinity of the junction. Therefore, failures to extract junctions, even for lines with identical contrast can 
be explained. From this it might seem that one should allow lines to be restricted extrema in the direction of the minimum 
second directional derivative as well, in order to extract all junctions. The next example shows that this is not the case. 
Figure 14 shows a junction with h; = hs = 2 and hy = 1, i.e., the left line on the y-axis is now darker than the other two 
lines. As can be seen, the line on the left part of the y-axis again changes to a restricted maximum in the direction of the 
  
150 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.