ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002 
  
cannot be calculated a disparity at all. To such pixels only with 
prior information or by some kind of interpolation a (often 
inaccurate) disparity can be assigned. With respect to our 
attracting forces that means the following: If there is an edge in 
the left image between (i-1,j) and (ijj) and another one between 
(i -1,j) and (ij) in the right image then there is a force K7(i,i’j) 
originating from (7j) and attracting (i') and another force K,(i- 
l1, -lj) acting from (i-1j) to (i-1j). This is necessary for 
coping with occlusion. 
Let’s consider the external force Kz(i,i’j). Then, first, that force 
depends on the difference |g; (i,j)-gx(i'y)| or, more general, on a 
certain mean value of that difference. That mean value should 
be calculated only over pixels which are in the same image 
regions as pixels (ij) (in left image) and (i) (in right image) in 
order to exclude problems with occlusion. To guarantee this, the 
averaging is performed only over image points (ijj) with 
igi G)-gi)| € threshold and (i^) with |gs(i"J)-ga(i)| € 
threshold, respectively. We denote that mean value as 
Ag(i.i5j)- (g,(.7)-2,6.7])) ^ ae 
Secondly, pure radiometric criteria are not sufficient. Therefore, 
geometric deviations are taken into account too. To do that, we 
consider two region border lines (one in the left image and the 
other in the right image) which contain the points (i,j) and (i), 
respectively. The situation is shown in figure 4. 
  
  
  
  
  
  
  
  
Fig. 4. Borderlines in left image, in right image, and overlaid 
We see that both borderlines are different and do not match. A 
useful quantity for measuring that mismatch is the sum of the 
border point distances along the horizontal lines drawn in figure 
4. Be (ipj+K) a point on the left borderline and (i ’,j+k) a point 
on the right borderline, respectively. For k — 0 the points are 
identical with the points (/) and (ij). Be d, — |i - i^, Then, a 
useful border point distance is 
Ww 
d n = i-a) (17) 
kz-w 
The distance d, accomplishes a certain coupling between 
epipolar image rows which are no longer independent. This 
sometimes can reduce mismatches efficiently. 
With Ag and d, the total distance is 
d(i,i; j)- 0 -d, (i.i; j)- 0 - Ag (i: j) as) 
The smaller that distance between edge points (ijj) and (ij) is, 
the bigger is the force Ky(i,i’,j). Therefore, 
Kp(i,l'; ÿ) exp[-d(i,i'; j)] (19) 
seems to be a good measure for the force Kz. The calculation of 
K, is fulfilled analogously. 
Now, the (external) force K.,(i’,) acting on point (i) can be 
computed as the maximum of all forces Ky(i,i’,j) with different i 
or as a weighted sum of these forces. Here, we take into account 
only points (ij) with | i - i'| € Max disparity. The maximum 
disparity used here is often known a priori. The introduction of 
Max disparity is not necessary. One can also use distance 
depending weighting and calculate the resulting force as 
Kent (i, j) = > Ky Er J): fli —X (i, j))- sign(i —X (i, j)) 
(20) 
with f{| i —x,(i’j)|) being a certain weighting function which 
decreases with increasing distance |i - x,(i’;j)|. Here, we use the 
special function 
  
  
ss ix if lx| € Max _ disparity n 
0 elsewhere 
Up to now we have considered only forces which act only on 
image points (^j) near edges. But we must assign a disparity to 
each point of the right image. Therefore, the disparity 
information from the edges must be transfered into the image 
regions. Within the model presented here, it is useful to do this 
by means of adequate forces, which connect the edge points 
with interior points (i.e. points inside regions). Local forces of 
spring type have been studied for that purpose. Let x(i',j) and 
x(i 1) be two neighboured mass points which we assume to 
be connected by a spring. Then, point x(i’+1,j) exerts the 
following (attracting or repulsive) force on point x,(i’,j): 
K pring E18, 7) = 8 [0,041 J) - x, Gj) -1] 
(22) 
The same force, but with the opposite sign, acts from x,(i’j) on 
x(i 1j) according to Newton's law of action and reaction. 
Experiments with those and other local forces (e.g. internal 
friction) have not been fully satisfying up to now. Of course, the 
stereo information is transferred from the edges into the regions, 
but very slowly. One needs too many recursions until 
convergence. Therefore, one result of these investigations is that 
local forces are not sufficient. We need far-field interaction 
between points x,(i'j) and x(i'^kj) which can easily be 
introduced into our equations of motion. First experiments with 
such forces have given some promising results but that must be 
studied more detailed in future. 
The algorithm is applied here to the standard Pentagon stereo 
pair because that image pair is a big challenge because of the 
many similar structures and the many occlusions. Figure 5 
shows a section of the smoothed image pair (see figure 3 for the 
whole left-hand image). 
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