So rms( dZ) So n 
  
6000 4.260 0.245 0.0304 | 13 
  
  
  
  
  
2000 no convergence 
X global regularization parameter 
So standard deviation of unit weight 
(in grey values) 
rms(dZ) root mean square of true Z-errors 
[metres ] 
$2 mean of standard deviations of 
Z-unknowns [metres] 
n, number of iterations on zero level 
  
  
of image pyramid 
  
Fig. 3.2: Experimental results for regularization 
with curvature minimization 
  
  
  
  
Fig. 3.3: Regularization with curvature minimization: 
Reconstructed roofline with A= 6000 (left) and 
dZ-graph (right). 
Zmax = «1118 m and dZmin=- 0.335 m; the 
dark facets represent the region with constant 
grey values (left) and facets with dominant 
negative differences (right) 
4. Adaptive Regularization of FAST Vision 
This new method refers to relationship (7). Actually, it is 
based on a proposal given earlier by Wrobel (WROBEL, 
1973, 1974) for stabilization of ill-conditioned linear 
systems. Here, use is made of the curvature constraints 
(6) for every Z-facet as in section 3, however, with 
curvature values c, that are estimated from grey value 
observables during the reconstruction process. 
41 Derivation of an Iterative Method for Regularization 
with Constraints 
There are several computational procedures (WROBEL, 
1914) to solve the basic approach. Here, we present a 
very simple one. 
To fulfill functional (7), a system of two groups of obser- 
vation equations is set up: 
[n FARM Siig (25) 
(V2 VAT C J 9, (OR 
/ / 
/ 
For later use we set el ) with I = unit matrix. 
1 
A I / 
The first group originates from the ng observation 
equations of the FAST Vision method, derived from the 
image grey values, see (2D. The second group 
comprises the n, additional from discrete curvature 
equalions at the facets, with c the unknown vector of 
curvatures. A, is formed by D D, and D y 
obtained with (6) from the Z-facets at the start of each 
FAST Vision iteration step, P, is a diagonal matrix, 
representing local regularization parameters A. 
The minimization of vTPv leads to: 
T T T ; T 
A BA+AP A, MP x | [AB i =0 (26) 
Pa As P, ] Vc Os, 
From the second line of (25) and of (26) follows: 
This leads to: 
vIPv= vIPv = (27) 
Ki la 
ie. the residuals of the original observation equations 
v,=A,x -l, are also the residuals of the extended system 
(25). Similarly, it can be shown that 
e x from system (26) is equal to 
x, =(ATpAD™ ATR], 
e Q,,. the cofactor matrix of x from the inversion of 
(26) is equal to (ATP, AD”, 
e and, finally, the redundancy of system | is the same 
as that of the enlarged system (26). 
Now, the numerical solution of system (26) can be 
performed to advantage by the block GauB-Seidel iterative 
method (WROBEL, 1974). The first iteration step (o) is 
started with an arbitrary vector x(O? . The second line of 
(26) is solved for c2? , while x“? is kept constant: 
c(O? -—-A,x(O?. Then the first line of (26) with c‘° as a 
constant, is solved for x“!? : 
£1), T T zl. T T CO) 
KU (CAP A, + ALP A GAP ATP, c. 
The full rank of AIp, A, cannot be guaranteed, but the 
sum CAIP, A," AIP,AD has been made invertable 
through regularization. 
E : ; ; a T T 
{ var = 4 A.B A 
On the i-th iteration step, Sen B= A, P A,+A,P,A,, the 
solution for the unknowns x*i+1? cf? reads: 
gi Az (28) 
Ci+l)_p-f 1 T -T C, EE 
X zB CA; Pil; - A2P^c y. (29) 
Substituting cf? according to (28) in (29), results in a 
blockwise GauB-Seidel iteration method for computing x: 
xD. pl. alp A, x) ,BrlaTp 1=GxP Blaïp 1. (30) 
G is called the iteration matrix of the method and the 
method itself is called adaptive regularization. In every 
iteration step (i), we also may compute the residual 
e and ver, by inserting x? and c(? in (25). 
From these vectors the following statistical functions are 
derived: 
vectors v 
= (AD) (3D 
nj-n X 1 7 
HOT VD 
[v P.v ) / HE NZ 
Eur vp » \ ai 
2 = 55} : (32) 
n . - « 
a 
«b is the standard error of unit weight of an observation 
V 
tl 
Q 
If 
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