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.
iz