Herbert Jahn 
  
Of course, (1) or (2) does not hold in regions which are occluded in one of the images. Therefore, in images with many 
occluded areas (e.g. in cities) each method which relies on a assumption like (1) in every pixel will not work very good 
in those regions. Future investigations are necessary to alleviate that problem. The smoothing procedure to be 
considered at the end of this chapter is one (preliminary) approach. Here, the problem is not considered in detail. But it 
must be stressed that the problem of occlusion generally cannot be solved exactly because of missing information. The 
same is true in homogeneous regions with constant gray value where no disparity information is available. A priori 
assumptions and interpolation processes are necessary preventing exact measurements in all image pixels. Precise 
measurements are possible only in image points which can be identified non-ambiguously in both images. Even the 
human visual system with its huge stereo processing capability is not able to measure precisely! Of course, using more 
than two images (as can be done with the above mentioned digital stereo cameras), the number of occluded image 
points can be reduced and the precision can be enhanced. Here, that possibility is not considered because here the 
development of the new parallel method is in the center of interest but in the future the method should be generalized to 
the multi-viewing case. 
Furthermore, (1) does not hold exactly because of illumination changes, non-lambertian reflection properties of object 
surfaces etc.. That can be taken into account very roughly by adapting the mean gray values and standard deviations of 
both images. A better performance can be obtained if the image statistics is adapted locally using the local median and 
the local mean absolute difference instead of mean gray value and standard deviation. This must be studied more 
carefully in the future (see also Wei et al. 1998). 
To determine the disparities s, = s,(i,j) and s, = s,(i,j) in each point (i,j) first the least squares measure 
107s 5) lg. Gi. j)- geli+s.j+s)f (3) 
is considered. For every (i,j) J, is a function of two variables s,, s,. To look for the minimum of J, here the method of 
steepest descent is considered. Of course, there are many other minimization methods (Himmelblau, 1972) but here 
only the principle is essential and therefore comparisons of the existing methods are not performed. 
(0) (0) 
According to the method of steepest descent starting from an initial point (s 5 ) the disparities are changed in the 
X y 
direction of the negative gradient of J: 
s*»(i, j) 2 s? (i, j) -w- VJ, (i, j:s) . (4) 
(r) 
$ Ko 
In (4) s?) = (r) is the disparity vector at recursion level (or discrete time) 7. The matrix K = is 
S 
y y 
determined later. 
To compute the gradient of J, first the derivative in x-direction is considered: 
2 - le, j)- e, s, js, jud ios 5) 
s. ox 
x=i+s, 
  
S : : : 
R = () even if there is a shift between 
  
; ; J 
This representation has the drawback that ——- vanishes in points (i,j) where 
S X 
X 
81 and gr in that point. Therefore, according to (2) it is better to use 
sS 5 5, 
4e sd 
- 
PESE july 6) 
Sn 2 , J 7 ( 
EL can be used too. 
X 
  
instead of (3) because then the derivative 
  
438 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.