in group 1 and s?" in analogy for group 2. For FAST 
Vision s characterizes the goodness of fit for the grev 
values and s$}” the still inherent amount of smoothing of 
the surface. These quantaties may be used for quality 
control of the reconstruction results during iteration. 
If convergence exists, then in the limit, for i — o, we 
obtain (irrespective of PD: 
DC. X 
e use 
(33) 
SUED us NM 
d Q 
l 
The results (33) are not influenced by P5, the weights of 
curvature equations, but speed of convergence and the 
condition number of the main matrix in (26) are depen- 
ding on P, (WROBEL, 1974). 
For quality control, the standard errors of x*?, derived 
from the inverse of (AT PA), may be determined 
according the same iteration scheme (WROBEL, 1974). 
However, the success and complete characteristics of the 
iteration process, described so far, will depend upon the 
iteration matrix G and the matrices behind it: 
G= (AJP A+AJP,A) ATP A,. As has been shown in 
detail in WROBEL et al. 1991 their properties are formed 
in essence by the rank and condition number of matrix 
ATP A, originating from the grey value equations. 
There are two cases, and they give rise to two variants, 
of which adaptive regularization is composed of: self 
adaptive regularization and pyramid assisted regularization. 
4.2 Adaptive Regularization 
FAST Vision is started with self adaptive regularization, 
applying a global weight P,=\l the 
constraints. À should be chosen as low as possible. This, 
  
matrix for 
already, can produce the final result, if matrix CAÏP,A,) 
has full rank and an acceptable stability. The solution 
vector x“i converges to the correct solution, irrespective 
of P,. If the iteration tends to i—>, there is no smoothing, 
as can be seen from relationships (33). As long as i«o, 
the qualitiy of surface reconstruction can be controlled 
by statistical tests: 
eThe standard error of unit weight, SD see (31), may 
be compared with s, a priori, in order to check the 
goodness of fit of the reconstructed grey value function 
G(X.Y), see (16), to the given image grey values G, 
G… 
e Additionally, the amount of smoothing, which is 
inherent in the surface at iteration step (i), may be checked 
by comparison of the mean curvature residuals sS, (32), 
with zero or with an alternative figure greater than zero. 
With respect to the statistical errors of the reconstructed 
surface smoothing can be tolerated to some extent. 
However, the conditions of self adaptive regularization 
will hardly exist in an entire object window. As discus- 
sed earlier, weak or zero grey value gradients give rise 
to the that the parameters Z,.. 
somewhere in the window, are very badly or even 
not at all determinable. Then, matrix CATPJAD is 
ill-conditioned or even not of full rank. The solution 
Cid 
situation, some of 
vector x"!” still converges, irrespective of P^, but it will 
depend upon the start vectors x(?? and c°9) This has 
829 
been shown in WROBEL et al. 1991. Therefore, this solution 
cannot be accepted in general. So, all parameters Z 
have to be examined, whether they can be accepted 
(= step 1) or whether they deserve more stabilization 
(= step 2). We sketch a procedure in short lines, with 
which the problem can be overcome. 
Step 1: Localization of unstable Z 
Compute the standard errors of Z s from the stabilized 
equations (26), and compare them with corresponding 
standard errors from constraining equations alone. Z 
with statistically significant differences in those errors, are 
supposed to be stable. They will be accordingly 
processed in step 2. However, often the matrix of constraints 
(AJP,A)) is not invertible. In that case a relative 
comparison of the standard errors, computed from the 
stabilized system is possible, in any way: All standard 
errors are compared with the smallest standard error 
S(Z, Qa Of With another reasonable threshold from 
experience. Z,,, with a standard error greater than the 
threshold by a constant of about 5, say, are regarded 
as unstable. The appropriate definition of suitable thresholds 
is already a matter of application. It will be discussed 
elsewhere. 
Step 2: Pyramid assisted regularization of unstable Z,.. 
By that procedure, the curvature values c of unstable Z 
will not any more be estimated in the least squares 
procedure as unknown parameters. By contrast, they are 
given ‘reasonable’ curvature values as ‘observations’, zero 
values, say, and are processed accordingly in FAST 
Vision. We prefer curvature values, derived from the 
surroundings of unstable Z _ by interpolation. For FAST 
Vision, this simply means to use curvature values from 
the next higher level of surface pyramid. This variant of 
adaptive regularization, therefore, is called pyramid 
assisted regularization. 
To be more precise, those ' reasonable observations of 
curvature, of course, are arbitrary information and may 
produce not very reliable facets. In view of photo- 
grammetric practice, these facets could be marked by the 
computer and be inspected by an operator. 
To bring the overall procedure to an end, after step 2, the 
computations of FAST Vision are repeated with the now 
two classes of facets. 
It may be argued, the final result will be influenced by 
the choice of P,- Al at the beginnning. This objection is 
not true for the solution vector, but the standard errors 
are perturbed by the constraints: they are always too 
low. The numerical example in fig. 3.2 has shown this 
already. However, if reliable quality assessment for each 
parameter Z,, is a strict demand, the following step = 
may be performed. 
Step 3: Refinement of standard errors of parameters Z 
The amount of A in P,=Al at the beginning has had to 
stabilize the weakest parameters Z _. After step 2, these 
are made very stable, so A may be substantially lowered. 
Again, this may be pursued globally, but could be done 
also locally in relation to the individual standard errors of 
Z,, which have been computed already in step 2. 
So, finally, the above mentioned rule - regularization as little 
as possible - can be realized.