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step(d,œ)= S P(d,a:x,y) = V lg. yy? expl - m laa; rte en ; (5) 
(x,y)eL(d,œ) (x—xo )cos a-t( y—- y )sin ed 265 
  
This functional has sense of integral intensity step along straight line L(d,o) . Note that expression (5) for (d.a) e € 
defines transform of image i(x, y). 
In practice to compute values of sfep(d,o) we can use voting method applied for Hough transform calculation. 
Namely, first of all discrete grid with steps Ad =1 and Aa = H^ 4, i$ introduced in area E of parameters values; then, 
the following algorithm is performed: 
l. put step(d,o) «0; 
2. for each point (x, v) of source image do steps 3 + 6: 
3. calculate g(x,y) using Sobel operator (3); calculate ls ^. Oy; 
A for each discrete value of o such that mine AN -6. «nj 360, do steps 5 + 6: 
S. assign the nearest discrete value of (x — x,)cosa -- (y — yo)sino to d ; 
6 add P(d,a;x,y) to step(d,a) . 
Note that to improve the algorithm performance fixed point mathematics and precalculated tables of functions can be 
used. Moreover, in section 3 application of Gauss pyramids, that result in speed and stability gain, will be discussed. 
In Figure 2 you can see the results of the integral transform (5) for fragment of aerial image. To obtain this results, 
constants q and o,, were assigned to 4 and 0.1 respectively. 
  
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Figure 2. Example of integral intensity step transform: source image (at the left), result of transform (at the center) and 
extracted edges (at the right) 
2.1.3 Finding of Candidates to 2d Edges. Due to form of functional (5) its local maximums have strongly marked 
form (see Figure 2). These local maximums should be treated as candidates to 2d edges. The following procedure can 
be applied for candidates choosing. First of all we find global maximum of step(d,œ) , let me denote it as (d, ,, ) , ie. 
candidate number 1. Then we exclude vicinity (e.g. 5x5) of point (d,,œ,) and find new global maximum in the rest of 
the grid, as a result (d,,Œ&,) will be obtained. And so on. There is one question: when should we stop it? The answer 
depends. For example, if we need to find N 2d edges, it is reasonable to test first kN candidates, where k >1. 
Note that when we define vicinity of point (d;,0/;) we must remember that & =(d,a) is toroidal coordinates, that is 
(d 0) is equivalent to (—d, x) for any d . 
2.1.4 Contour Ends Extraction. Let us remark that detection of contour ends remains serious problem. To resolve 
this problem, one can use Forstner operator [W. Eckstein, 1996] or various corner detectors [A. Singh, M. Shneier, 
1989]. But all known operators of such type have low robustness. For this reason we prefer to solve problem of end 
detection via intersection of contours. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 73