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Steger Carsten 
  
minimum second directional derivative. However, as this line approaches the other two lines it rapidly turns away from 
the junction, and ends up running parallel to the line on the y-axis. Therefore, allowing lines to be restricted extrema in the 
direction of the minimum second directional derivative is definitely a wrong approach. What is striking, though, is that the 
direction of the line on the y-axis at the point where it changes its type points exactly at the junction. More examples with 
different configurations seem to indicate that this is true in general (Steger, 1998c). Therefore, a good algorithm to extract 
the missed junctions is to search for other lines along the last extracted line direction. As can be seen from the examples, 
the lines not continuing to the junction will in general be the lines with the smaller contrast. Hence, while following the 
line, the gray value in the image should increase monotonically for bright lines and decrease monotonically for dark lines, 
respectively. Since it is desirable to avoid using the image intensity directly, this condition can be reformulated to the 
condition that the dot product of the gradient of the image and the line direction must always be positive for bright lines 
and always negative for dark lines. Of course, the length of the line along which the search is performed should not be 
arbitrarily long. A good restriction is to use the same length 2.50 as for the determination of the line width. 
Let us now analyze the behavior of edges in the vicinity of junctions. Consider a T-junction where three areas of different 
intensity meet. A suitable model for this type of junction is 
0, y<0 
He = 1, y20AT«0 (22) 
hiy20AZz20 
It is obvious that this models any possible configuration of intensities, since arbitrary intensities can always be scaled 
and translated such that one is zero, while another one is one. Hence the only free parameter is the third intensity. For a 
Gaussian smoothing filter, the scale-space description of this junction is given by 
r;(x,y,0) = óc(y)((h — 1) (x) +1) (23) 
and its partial derivatives of first and second order. 
As described in Section 2.2, the edge points in the one-dimensional case are given by the maxima of |r. (z,0)|, or 
equivalently by the zero crossings of r(x, 0), where additionally 7." (x, 0)r,(z,0) < 0 is required. As we have seen, 
the first interpretation quite naturally leads to the conclusion that edges can be modeled as lines in the gradient image. In 
the two-dimensional case, these two edge definitions have been translated to the requirement that an edge is either given 
by the maxima of the first directional derivative in the direction of the gradient (Haralick, 1984) or by the zero crossings 
of the Laplacian (Marr and Hildreth, 1980). The first condition can be expressed as 
2 an 2, 
TUT zz T 2r;TyTzy + Ty yy 
Dr, = 
  
ur, (24) 
T2 72 
while the second condition can be expressed as 
Are = 75: +70 = 0. . (25) 
As is well known, these two definitions are equivalent if and only if the mean curvature of the image vanishes (Berzins, 
1984). This difference is especially noticeable for corners, where the first definition will move the edge positions to the 
inside of the corner, while the second definition will broaden the corner. In contrast, the interpretation of edges as lines in 
the gradient image leads again to the use of the De Saint-Venant condition to model the behavior of edges in 2D. In this 
case, edges are given by 
€z€y(€za — €yy) — (et — eder) 
€2 shes 
  
Sry = ez, (26) 
where e;,...,€,, are calculated according to (9)-(13). We will use these three definitions to explain the often counterin- 
tuitive failures of edge detectors to extract junctions. 
In order to analyze the behavior of the different edge definitions at junctions, the response r; can be substituted into 
(24)-(26). Figure 15 shows the edge positions extracted with the different edge definitions for a junction with h — 2 
mapped onto the response r; and onto the absolute value of the gradient || Vr;||. More examples can be found in (Steger, 
1998c). As was the case for the analysis of the line junctions, the spurious edges for the three definitions have to be 
disregarded. For Dr; — 0 these are the points where the absolute value of the gradient has a minimum in the direction 
of the gradient. For Ar; — 0 they are the flat inflection points, while for Sr; = 0 they are the points that are restricted 
minima in the direction of the maximum second directional derivative and the minima and maxima in the direction of 
the minimum second directional derivative. If this is kept in mind, the first interesting fact noticeable from these figures 
is that the edge positions extracted by Sr; — 0 approach the junction much closer than the edges extracted by the other 
two definitions. Therefore, it can expected that the first two definitions will miss junctions much more frequently than the 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 151