Andrew Bibitchev 
  
2 ELEMENTS OF THEORY 
So, let there exist two images of the one and the same 3d scene. And let shooting geometry be known. We need to 
extract candidates to 3d straight edges of man-made objects presented in the scene. 
2.1 Integral Intensity Step Functional 
To begin with, consider construction and usage of integral intensity step functional, i.e. functional, which describes 
intensity behaviour along straight line. 
2.1.1 Heuristic. Let i(x, y) be gray scale discrete image of 3d scene, x —0,...s, — 1, y 2 0,...s, —1. We need to find 
  
(0,0) — 3 straight intensity steps of function i(x,y), ie. 2d edges. For convenience, 
parametrization € = (d,œ) of straight line L is used: 
| (Xo. Yo) L(d,o) 2 | (x. y) | (x—-xg)cosa 4 (y- yo)sina « d], (1) 
y € (d,a)e E = [-dax : max 1<[0; ), 
  
  
  
de where (xo.Vo )= gs ts, ) is center point, dax = +1|52 +s? (see Figure 1). 
Figure 1. Parametrization °°, f ; 
We propose the following heuristic. Suppose g = (e... 2, )= le] ; (cos a, .sina,, ) is gradient of intensity i(x, y), then the 
probability P(d, a; x, y) that point (x, y) belongs to 2d edge L(d,a) is proportional to 
nf enh min’ joa rt +l] = 
260% 
  
  
Here le] = A gi T 2; is gradient length; a, is angle between vector g and horizontal direction, a, € [0:27) ; 0, is 
constant, which has sense of angle dispersion; q » O is constant, which defines contribution of high gradient values. 
For example, if we are interested only in strongly marked 2d edges we should choose q «« 1 and vise versa if we are 
interested also in watery steps we should choose q »» 1. As to gradient g(x,y) of intensity i(x. y) its discrete analog 
can be calculated using convolution with Sobel masks: 
-1 0 1 -] -2 -1 
2,0 »2|-2 0 25i». 2,0 »2| 0 0 0 [rio y. (3) 
-1 0 1 1 2 1 
The main premise for the proposed heuristic is that in ideal case (no noise, no sampling and no quantization) probability 
P(d,œ;x, y) is proportional to 
0(g) 5(Ac), (4) 
where Ao = mina - a.a. - a, +7 
  
Lx>0 
} oo, 20 is step function and ó(x) 20 (x) is Dirak generalized 
XS 
function. Thus, the first multiplier in expression (2) is weight and corresponds to 6(e|) and the second one is 
regularized function 8(Aa). 
2.1.2 Integral Intensity Step Transform. Using heuristic (2) we can construct the following functional: 
  
72 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
del 
In 
Na 
the 
qo T T 
ow» 
us 
In 
co 
ca 
th 
de