uces the 
seem to 
ken into 
(11) 
features 
lied in a 
gorithm 
ented in 
re many 
, the so- 
54(1,]),) 
(12) 
(13) 
ground) 
> clouds 
ng. One 
es good 
AS" (i,j) 
ion. To 
D, f) 
(14) 
1e order 
id other 
ns may 
he right 
Herbert Jahn 
  
One can see from (14) that it is not useful to use too big values of x; such that K : is much bigger than 
  
v 
d,/2. Therefore, at each pyramid level x; can be chosen adaptively according to some average values of V.J | and d,/2, 
e.g. from K, (V. 
  
  
) — 1/2 or similar relations. 
Further measures to reduce ambiguity are edge preserving smoothing of both images as a pre-processing step and a 
smoothing procedure applied to the disparity increments As") (i,j) (before (14) is applied) in order to reduce noise in 
the original images and in the disparity increments. For edge preserving smoothing of the original images the algorithm 
described in (Jahn, 1998) was applied. A modified algorithm for disparity smoothing is described now: 
1 
As (i, j) e As (i, j)-e« V w "(i jk.) [As Qj 1) As "(I 7) (15) 
kl=-1 
1 
(c8 ——————— ). 
Vw", jk.) 
k,l=—1 
With the special weights w = 1 the algorithm (15) computes the ordinary mean value recursively which blurs the 
disparities. But, the hypothesis to be used is that the disparity is constant or smoothly changing inside homogeneous 
image regions. Therefore, as for edge preserving smoothing of the original images (Jahn, 1998) the weights are chosen 
according to 
(n) 
wi, j:k,1)= u 
uo * [g, (i k, j -1)- e, (i. j)] 
  
(16) 
2 
These weights favor contributions to (15) where g;(i+k,j+/) = g;(i,j). u” is a parameter which can be kept constant or 
which can be chosen to tend to zero with increasing n in order to accelerate convergence. (15) generates a coupling of 
image lines because there is smoothing in y-direction. This diminishes errors in single image lines which sometimes can 
be seen as stripes in disparity images generated with pure epipolar algorithms. 
Some general words concerning the recursive algorithm (4). If one looks at anaglyph images with big parallaxes then 
one observes that it takes some time until the final 3D impression is obtained and that this time increases with 
increasing disparity. That behavior can be explained with recursive algorithms of type (4) where the disparity increment 
is limited by some constraint as e.g. (14). Furthermore, the iterative determination of the disparities also reduces 
trapping in wrong maxima (as compared with “one-step” algorithms). Another argument for algorithms of type (4) is 
that the disparities in every image point (i,j) of the left image can be calculated in parallel (this holds also for the order 
constraint (14) and the smoothing procedure (15)). Therefore, when appropriate parallel processing hardware will be 
available then real-time stereo matching becomes feasible. Of course, this does not mean that the algorithm which is 
used here is an ultimate one. It only shows to a certain direction of further research. 
When big disparities in both directions are present in the image pair, then the x- and y- disparities must be estimated 
together as proposed by (4). If the geometry is strictly epipolar then an one-dimensional version of the algorithm can be 
used (x; — 0). But, if the geometry is near-epipolar (only small y- disparities are present) then the algorithm can be first 
applied in epipolar direction (or another epipolar algorithm can be used) to estimate s, coarsely with subsequent s, 
estimation with the same algorithm applied in y- direction (with recursive application if necessary). This seems to be a 
good processing scheme for pushbroom stereo image pairs. 
3 RESULTS 
A few results to be presented now show the capabilities of the method and the difficulties which should be overcome by 
future research. First, a stereo pair of a rural region with a small village generated with the airborne camera WAAC is 
considered. Figure 1 shows the image pair. 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 441