T pl, 
( AG PAG 
T 
SAID AG 
T 
ALP A, | 
APA, - ASP A, ) 
to become a singular or at least an ill- conditioned 
matrix. In that case the image inversion problem cannot be 
solved by the equations (21) alone, unless additional 
observation equations are added well-defined 
procedure of regularization. 
in a 
3. Regularization of FAST Vision by Choice of Appro- 
priate Facets and by Curvature Minimization 
Sufficient regularization may already be obtained by 
choice of the appropriate size of facets. Indeed, for the 
representation of the object grey value function So y» 
in relationship (21) a constant ratio of 2x2 pixels per 
G-facet has been found to be a reasonable compromise 
between resolution and accuracy. This ratio can be used 
everywhere. It is independent of the image signals, as 
long as pixel size itself is in correct relation to image 
signal. In contrast to this Z-facets should have a variable 
size in principle, in one window already, because the 
Z-parameters in (21) depend exclusively on the grey value 
gradient, which is a space function of X, Y. The reali- 
zation of that optimal idea seems to be too complicated. 
We decided for a constant size of Z-facets in combination 
with stabilizing constraints with local weights. This 
approach has many advantages as will come out in this 
paper. 
In this section the limitation of two simple stabilizing 
methods will stabilization only by 
constant, rather large Z-facets and by curvature mini- 
mization with global weights. 
The numerical experiments in this section are performed 
with two stereo image pairs, generated of the same 
object. It can be described as a gable roof: two planes 
with an inclination of 20° meet at a ridge. There is a 
shady plane containing the grey values from O to 12T, 
and a plane exposed to the light containing grey values 
from 128 to 255. The simulated photographs of the object 
were taken with a standard deviation of 4 grey values 
(white noise), pixel size is 20 um x 20 um. This is the 
first image pair used the first experiment. The 
following photogrammetric parameters are the same for both 
image pairs: image scale 112000, base-to-height ratio 
11.6, for comparison with standard accuracy of today's 
photogrammetry: Ol OooD-018 m, with D - distance 
object-image. 
The second image pair has been generated with a 
slightly different texture (see fig. 3.1): In the centre of the 
object, there is a 5x5 Z-facets region containing the grey 
value constant 127. 
The parameters chosen for the FAST Vision process are: 
be demonstrated: 
in 
size of Z-facets 2m x 2m 
G-facets per Z-facet 4x4 
pixel per G-facet 2.083 x 2.083. 
The 12x12 Z-facets, selected for surface 
reconstruction, are located on both sides of the ridge 
(c. fig. 3.1), and the ridge coincides with the boundary of 
Z -facets. 
In all experiments, the iterations of FAST Vision are 
started from a horizontal surface plane through the roof. 
which were 
827 
  
Fig. 31: Generated pictures 
and position of the 
ridge within the 
Z-facets. The hatched 
facets represent the 
region with constant 
grey values. 
  
t 
ridge 
In the first experiment only the grey value equations 
(2D have been evaluated, thereby applying the 
following simple regularization procedure: increase the 
size of Z-facets till sufficient stability has been obtained. 
All the other parameters of FAST Vision have been kept 
fixed. 
Results: 
We started with the above mentioned size of 2mx 2m, 
but convergence was obtained not before a Z-facet size 
of 5m x Sm. Here, stabilization has to be paid by poor 
resolution. However, the accuracy figures, computed from 
least squares, are very good: standard deviation s, of the 
observations G, G': sg- 3.997 grey values (a priori 
S9= 4), mean 5, of the standard deviations of all 
Z:5, = 0.042 m, which is in good agreement with the root 
mean square of true Z-errors: rms (dZ) = 0.068 m. 
The second experiment shows the performance of regula- 
rization by curvature minimization. The second image 
pair (fig. 3.1) and all the other parameters, given above, 
have been introduced. For regularization only global 
weights A have been used. 
Results (see fig. 3.2 and 3.3): 
The Z-resolution has been improved from 5m x 5m to 
2m x 2m, but at the expense of a rather high regulariza- 
tion parameter ^. No convergence is obtained with 
422000. Fig. 3.3 shows, that the ridge (roofline) is flatte- 
ned by a high regularization parameter À. The region 
with constant grey values lowers the ridge at this point. 
Both effects can be seen clearly in the dZ-graph. In the 
region of constant grey values, there are no deterministic 
grey value gradients, only very small stochastic 
gradients, resulting from grey value noise. Nevertheless, 
it is possible to reconstruct the surface in that region, but 
only with high regularization parameter A. High A - on the 
other hand - has a rather far extending impact on the 
surroundings of each Z _,as can be seen from the 
distribution of positive and negative dZ-values, see fig. 3.3. 
Also, there is only a weak agreement of 5, with 
rms (dZ). Regularization by curvature minimization with 
global weights is not satisfying.