In the past, also functions of higher order differences of 
Z,, have been proposed for stabilization. Twomey (see 
HUANG, 1975, pp. 187) found, that any constraint, which 
is quadratic in Z,,, may be used to produce a solution 
resembling (4) with (6). This finding has been fairly 
well confirmed by RAUHALA et al, 1989, when compu- 
ting a Digital Terrain Model (= DTM) from scattered 
Z-data and comparing 15 different constraining functions. 
So, one has to use one type of constraining function, but 
in general it is not decisive, which one. 
model of 
object surface 7 X>Y 
facets of 
   
     
   
orthophoto 
opt. density D(X,Y 
  
facets of 
object opt. density 
Fig. 1.1: FAST Vision: Simultaneous reconstruction of object 
surface Z(XY) and optical object density 
D(XY) or object grey value function G(X.Y) 
resp. 
However, the presence of breaklines in the surface or of 
other non-smooth surface elements and their reconstruc- 
tion is the weak point for the application of such a 
functional. Applying the functional (6) with a large 
weight A, can lead to errors, if the terrain to be recon- 
structed really is "rough". Surface edges may degenerate 
to arcs. 
Since breaklines of topography or edges of workpieces 
play a fundamental role for morphologically correct 
reconstruction or for object recognition and for other 
postprocessing tools of object surface data, it has to be an 
ultimate goal to preserve these object characteristics as 
well as possible. Therefore, the following rule has to be 
pursued: As much regularization as necessary - as little 
regularization as possible. This rule emphazises the priority 
of data consistency in (4), and consequently, the 
necessity for optimal or near optimal local weights A, 
instead of one global parameter A. In this context, there 
are many proposals (see HUANG, 1975, pp. 184, 
WEIDNER, 1991). 
Now, the new regularization principle, given here, also 
relies on a curvature functional, but with a substantial 
difference to all approaches discussed so far: We do not 
regard the expectation of surface curvature to be zero. In 
our opinion, this assumption is true very rarely, both 
globally and locally. Therefore, the approximation to 
reality only by proper weighting is very crucial. In 
825 
contrast to these approaches, we are introducing estimates of 
local surface curvature c with non-zero expectation. We 
are minimizing only their residuals together with the 
error functional of the image grey values, ie. 
find z, minimizing llAz - bl? « X IK Bz - o)IP? CD 
Here, the adaption to locally quickly changing surface 
curvatures will be obtained primarily by the estimated 
curvature values c themselves, as will be shown later. 
A. now, is of minor importance. 
The remainder of this paper is organized as follows: In 
section 2 a brief presentation of surface reconstruction by 
facets stereo vision (= FAST Vision) is given. Section 3 
reports on numerical results of FAST Vision, stabilized 
only by proper choice of facets or by standard curvature 
minimization. The theory of the new method will be 
derived in section 4, section 5 numerical 
examples in comparison with the former methods will be 
demonstrated. Finally, section 6 contains some remarks 
on open questions. 
and in 
2. Object Surface Reconstruction by Facets Stereo Vision 
(FAST Vision): The Basic Equations 
Facets Stereo Vision is a method developed by Wrobel 
(WROBEL, 1987, 1991), which fulfills the task of simul- 
taneous reconstruction of object surface Z(X,Y) and object 
grey value function G(X,Y). The relationship of a point 
on a surface and its images in the pictures P,P … can be 
described with regard to radiometric and geometric 
characteristics. If the sensor, with which the picture was 
taken, is a metric camera, the geometric relation between 
the object coordinates X.Y.Z and the image coordinates 
x.y in P is given by the well-known perspective 
equations: 
n Oxo OY rZ 
  
  
x=xa- "e (8) 
O Hai X Kol ing C l-lg 2-29) TK 
1,4 X-X9) «t, 4 CY-Yo) r4 Z-Z0) : 
Y= Ya- :e (9) 
O rngOCXorl-Yg 42-29) TK 
Xo9.yo9 and c, are the interior orientation parameters, 
Xo lo Zg and n; are the exterior orientation parameters. 
In this paper all these are assumed to be known. 
The radiometric relation between an object grey value 
G(X,Y) and image grey values is modelled by linear 
transfer functions T, T', . . ., which are invertable: 
GO) =T(G(xy))=T(G (x y))=... (10) 
with G, G' the grey values of pictures P, P,. . . . 
Of course, sensors with a different geometry can be chosen 
instead of (8), (9) and the transfer function does not 
have to be linear. Now, an image ray defined by the pixel 
coordinates x,y' and the perspective center Xo. Yo.Zo of 
image P' may intersect with an approximation of the 
surface at X9.Y9,Z9. Then the expansion of G(X.Y) into a 
Taylor series leads to: