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(b) Dr; = 0 on gradient (d) Ar; = 0 on gradient (f) Sr; — 0 on gradient 
Figure 15: Extracted edges for a T-junction with h = 2 by Dr; — 0 mapped onto r; (a) and onto || Vr;|| (b); by Ar; — 0, 
(c) and (d); and by Sr; — 0, (e) and (f). The graphs in the lower row are rotated by 120? with respect to the graphs in the 
upper row. 
third definition, since for this definition the extracted edge often will approach the junction to within one pixel, so that 
the linking algorithm will be able to extract a junction. Furthermore, the direction of the edges extracted by the first two 
definitions in the vicinity of the junction are much worse than by the third definition. As is the case for lines, the edges 
extracted by Sr; — 0 will point directly at the junction, whereas for the other two definitions this is not necessarily true, 
a fact that was also noted in (de Micheli et al., 1989). It is a surprising fact that the edge directions for Sr; — 0 are much 
more stable than the gradient-based directions, since the former ones are based on the second derivatives, while the latter 
ones are based only on first derivatives. 
This discussion gives another good reason to regard edges as bright lines in the gradient image. While the edge positions 
are equally good outside of junction areas, the line-based approach has a clear advantage in junction areas since the edge 
directions still point in the right direction in the vicinity of a junction for the edge that cannot join the junction. Therefore, 
the algorithm proposed above for lines can be used to complete the missing junctions for edges as well. 
Let us conclude this section by giving an example of the proposed approach to complete missing junctions for edges. 
More examples for lines and edges can be found in (Steger, 1998c). Figure 16 displays the results of three different 
extraction schemes to illustrate the results of the analysis of the behavior of the different edge definitions at junctions. In 
Figure 16(c), the results of the edge detection with pixel resolution using the definition in (24) are shown, i.e., the edges 
were computed by a Gaussian gradient operator, non-maximum suppression in the direction of the gradient, a hysteresis 
thresholding operation, computing the skeleton of the resulting region, and linking the edge points into contours. As was 
to be expected by the analysis above, this algorithm fails to detect almost all of the salient junctions of the building. If 
the edges are extracted to subpixel resolution by the definition in (26), some of the junctions are detected because the 
more robust estimation of the edge direction allows the edge to come within one pixel of the junction, as is shown in 
Figure 16(b). From this figure, also the accuracy of the extracted edge positions becomes readily observable. Especially 
the gable and perimeter edges of the roof are almost perfect straight lines. Finally, in Figure 16(a), the results of edge 
extraction with the additional step of completing the junctions is shown. It is apparent that the algorithm was successful 
in extracting all of the junctions in this part of the image, and has therefore produced a complete description of the house 
roof. 
4 LINE AND EDGE EXTRACTION IN COLOR IMAGES 
Sometimes it is useful to extract lines and edges in color images. More specifically, this is true if we cannot find a color- 
space transformation that gives us a gray value image in which the lines or edges have sufficient contrast. Therefore, 
in this section the subpixel edge and line extraction algorithms proposed in this paper will be extended to multispectral 
images of any number of channels. 
We will start by extending the edge detection algorithm. As described above, edges can be regarded as bright lines in 
the gradient magnitude image. Thus, to make use of the subpixel edge detector we need an equivalent of the gradient 
  
152 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.