ADAPTIVE REGULARIZATION - A NEW METHOD FOR STABILIZATION 
OF SURFACE RECONSTRUCTION FROM IMAGES 
Prof.Dr.-Ing. Bernhard P. Wrobel, Boris Kaiser, Julia Hausladen 
Institute of Photogrammetry and Cartography 
Technical University Darmstadt (Germany) 
Washington 1992 - Comm. III 
Abstract 
Regularization of ill-posed image inversion problems using a stabilizing smoothing 
functional has one weak point: Breaklines (edges, creases, cusps, . . ) will not be 
properly reconstructed, if the parameters for smoothing are not chosen in an almost 
optimal way. Often curvature minimization is applied with global or local weighting. 
Global weighting tends to smoothen too much, whereas optimal local weighting is a 
crucial and time consuming operation. In this paper a smoothing functional is intro- 
duced using locally estimated curvatures and minimizing only their residuals together 
with a functional of image grey value residuals. The amount of object surface 
smoothing can be controlled by statistical tests. This procedure is called adaptive regulari- 
zation. The impact of weights is of less importance than before. The basic equations are 
presented related to the object surface reconstruction approach called facets stereo 
vision (= FAST Vision). A series of experiments is presented at this congress in another 
Qon 
paper from KAISER et al. 1992. 
Key Words: DTM, Image Matching. Orthophoto, Rectification, 3-D 
l. Introduction: Ill-Posed Problems and Regularization 
Surface reconstruction as a problem of inverse optics 
belongs to the class of problems, which are ill-posed in the 
sense of Hadamard (TIKHONOV et al. 1977), ie. at least 
one of the following conditions is not met by these 
problems: 
existence of a solution, (D 
uniqueness of the solution (2) 
stability: the solution depends continuously on the 
initial data. (3) 
Problems not satisfying condition (1) may be called 
over-constrained, which is rarely the case in inverse 
optics. Problems, which do not fulfill one of the other 
two conditions (or both) can be regarded as under- 
constrained (BOULT, 1987). Meeting condition (3) does 
not ensure the robustness against noise in practice. Not 
meeting (3) means, that small changes in the initial data 
cause large ones in the results. 
In order to provide numerical stability, the problem does 
not only have to be well-posed, but also be well- 
conditioned (POGGIO et al, 1985). Additional assumptions 
can turn ill-posed problems into well-posed ones. The use 
of supplementary information of a qualitative nature 
(e.g. smoothness of the solution) yields the regularization 
method (TIKHONOV et al, 1977). Taken more generally, 
the term regularization refers to any procedure turning 
ill-posed problems into well-posed ones. In computational 
optics ill-posedness is closely related to the occurrence of 
noise. Surface reconstruction requires regularization even 
in the absence of noise in order to bridge areas, in which 
the gradients of grey value signal are too low. 
In order to restrict the space of solutions of a problem 
Az = b, a stabilizing functional |[Bz|| is introduced: 
find z, minimizing IlAz- bll2 + À - IIBzll2. (4) 
824 
with A, the regularization parameter. À controls the 
compromise between regularization and data consistency. 
The qualitative assumption, expressed in the choice of a 
specific functional |/Bzil, has to show physical plausibility. 
It is a very common approach to assume the reconstruc- 
ted surface to be smooth. An oftenly used functional 
expressing this assumption is the quadratic variation 
CGRIMSON, 1981; TERZOPOULOS, 1988) 
IIBzI? - ff(z2,«222. «22. .) dx dy. (5) 
The surface reconstruction approach FAST Vision, used 
here, deals with discrete surface heights Z,, às parameters 
in an XY-coordinate system. Thus, the stabilizing func- 
tional (5) has to be approximated by second differences, 
related to a set of facets (or grid) of the surface 
(figure 1.1): 
  
2 z 2 «T32 2 
\Bzlliser. = » (re s +2 De (2,8 € D (2 2) 
Z 227 +2 
: r-l,s IS r-l,s 
with D x eg A S UI : 
ZZ -Z +Z 
rs r+l,s r,s+l r+l,s+l 
Day Fret * 5 (6) 
2 -22.. «ZZ 
Rel IS r.s-1 
am 3 : 
h=X 2X10" Trai nies 
Is 
For some of the Z _ (those situated in corners and edges) 
it is impossible to form all of the above mentioned 
equations, because some of the adjacent grid points are 
outside the window to be reconstructed. Such equations are 
simply omitted. 
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