Steger Carsten 
  
  
    
(a) Extracted edges (b) Junctions not completed (c) Edges at pixel resolution 
Figure 16: Edges detected in an aerial image of resolution 0.25m. (a) Edges extracted with the proposed algorithm. 
(b) Edges extracted without junction completion. (c) Edges extracted with pixel resolution by Dr, — 0. 
magnitude. Let us regard the multispectral image as a function f : IR? — IR^, where n is the number of channels in the 
image. For multivalued functions, there is a natural extension of the gradient: the metric tensor G, which gives us for 
every vector v € IR? the rate of change of f in the direction v. For notational convenience, we will regard G as a matrix. 
Thus, the rate of change of the function f in the direction v is given by v" Gv, where (Di Zenzo, 1986, Cumani, 1991) 
T T n  Ofi Ofi n» 21: 9h 
G fe fo Îa Jy RE Y Or Oz 3 ed ox 3 (27) 
files Si) \ CMS TES 1 
i=1 ox 0y 
  
i—1l Oy Oy 
Of course, for edge extraction the partial derivatives are computed by convolving the image with the appropriate deriva- 
tives of Gaussian masks. Some approaches (Sochen et al., 1998) add the identity matrix to G. This added term will not 
cause major differences, especially in noisy images, where it is hardly discernible from noise. As in the ID case, we define 
the gradient direction by the vector v in which the rate of change is maximum. The vector v can be obtained by computing 
the eigenvectors of G. The largest eigenvalue of G is the equivalent of the gradient magnitude, i.e., the maximum rate of 
change, in the 1D case. Contrary to the 1D case, the rate of change perpendicular to v is usually not 0. For n = 1, the 
above definition is equal to the squared gradient magnitude. The only difference to the usual gradient is that we can only 
determine the orientation of the edge within the interval [0 : x) from G. But since we regard edges as lines in the gradient 
image, this information is unnecessary anyway. 
Figure 17 compares the results of extracting edges from a color image of the Olympic Stadium in Munich with the results 
of extracting edges from an equivalent gray value image. As we can see from Figure 17(b) the borders of the soccer 
field are clearly visible in the color gradient image. In contrast, in the corresponding gray value gradient image the 
borders cannot be distinguished. Consequently, the borders of the soccer field have been extracted from the color image 
(Figure 17(c)), while they have been missed in the gray value image (Figure 17(f)). 
To extend the line extraction algorithm to color images, we can regard lines as dark lines in the color gradient magnitude 
image, as described in Section 2.3. Note that this is reasonable because in a multispectral image lines may have a bright 
bar-shaped profile in one channel, a dark bar-shaped profile in another channel, and a staircase profile in a third channel. 
Therefore, the distinction between these types of lines is not meaningful in color images. With this definition of lines, the 
color line detector is straightforward to implement. Unfortunately, what cannot be implemented is the line position and 
width correction since. as mentioned above, the different channels may have different line profiles, which have different 
biases, but are combined into a single image by (27). Therefore, line position and width correction in multispectral images, 
unfortunately, seems to be impossible. 
Figure 18 displays the lines extracted from a color image of a set of cables with the proposed approach with o. = 3.5. 
Note that the line positions and widths are very good for most of the image. Only the bottom line shows a significant bias. 
5 CONCLUSIONS 
In this paper, the subpixel accurate line extraction algorithm proposed in (Steger, 1998b), which extracts bright or dark 
lines lines with different polarity, has been extended to handle lines with equal polarity. A scale-space analysis similar to 
the analysis in (Steger, 1998b) has been carried out to explicitly model the bias of the line positions and widths of lines 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 153