Application of Image Pyramid for Surface Reconstruction with FAST Vision (=Facets Stereo Vision) 
B.Kaiser, M.Schmolla, B.P.Wrobel 
Institute of Photogrammetry and Cartography 
Technical University Darmstadt 
Petersenstr. 13 
W-6100 Darmstadt 
Federal Republic of Germany 
Washington 1992, Comm. III 
The performance of FAST Vision, which is an iterative method for the reconstruction of both object 
surface and orthophoto from two or more images, is determined by two measures: radius and rate of 
convergence. On one hand the radius of convergence should be as large as possible in order to make 
it possible to start FAST Vision with coarse approximate values for surface heights, which can be de- 
termined without much effort. On the other hand, a high rate of convergence is desired to reduce the 
time needed for reconstruction. Both aims can be achieved by the application of an image pyramid 
within the FAST Vision method. Experiments show, that by this way the rate of convergence can be 
increased by the factor of about 10. Furthermore, combining the image pyramid with the technique of 
object lifting delivers a method for obtaining very reliable starting values under very general circum- 
stances. 
Keywords: DTM, image matching, orthophoto, rectification, 3-D 
1. Introduction 
The non-linear basic equation of surface reconstructi- 
on by FAST Vision (Wrobel, 1987. Wrobel et al, 
1992) has to be solved iteratively after carrying out 
a Taylor-linearization. The performance of iterative 
procedures is characterized by two measures: 
a) radius of convergence, i.e. the maximum differen- 
ce of the unknowns from their true values at the 
start of the iterative process. 
b) rate of convergence, i.e. the number of iterations 
necessary to meet a specific break-off criterion. 
The radius of convergence depends on the smallest 
wavelength A... of the grey value signal transmit- 
ted from object surface into image space (Korten et. 
al., 1988): 
rs 08 Amin 
The relation between pixel size Ap and wavelength 
À min Can be obtained by Shannons theorem: A in 2 
2Ap. If the image contains the lowest possible wave- 
length according to the sampling frequency, the 
radius of convergence will be only one pixel. Expe- 
rience with real image material has shown, that this 
pessimistic assumption often proves to be true. 
The rate of convergence decreases with the number 
of height facets in FAST Vision, provided the size of 
the facets remains constant. This fact makes the cal- 
culation of larger DTM's a time-consuming task. In 
order to reduce the number of necessary iterations, 
the start values for the unknowns have to be as close 
as possible to their real values. 
On the other hand, the computation of start values 
for the heights should not be too complicated in or- 
der not to waste much time before the iterations can 
start. A simple method e.g. consists in the choice of 
equal values for all grid-points of the DTM to be re- 
constructed. This is a suitable possibility for small 
windows with a not too complicated surface geome- 
341 
try. In order to guarantee convergence for such a 
simple choice of start values, a larger radius of con- 
vergence may be necessary. This can be achieved 
by enlarging pixel size Ap - in other words: À coar- 
ser image resolution is chosen. In order to meet 
Shannon's theorem, a low-pass filtering has to take 
place before. Otherwise, aliasing could be the result. 
The hierarchical ordering of digital images of succes- 
sively lowered resolution is called image pyramid. In 
general, one starts at the resolution level of the origi- 
nal digitized image (level O of the image pyramid) 
and reduces the resoltution of the image in each of 
the two image coordinate system axes by the factor 
2 (reduction of the image size to a quarter) after 
preceding low-pass filtering to get from level i to le- 
vel (i+l) of the image pyramid. This procedure is 
continued, until the lowest required resolution (level 
n) is reached. A corresponding pyramidal structure is 
build up for the unknowns. Their number in each of 
the two directions of the object coordinate system is 
halved to get from level i to level (i+l). When 
applying the image pyramid to FAST Vision this 
means, that the number of height facets in each 
direction of the coordinate system has to be a poten- 
cy of 2 on level O of the image pyramid (at least 
2^1). After meeting the break-off criterion on level i 
of the image pyramid, the resulting height values 
will be prolonged by bilinear interpolation and be 
used as start values for level (i-1) of the image py- 
ramid. 
The basic theory of image pyramid in FAST Vision 
has been given in (Müller, 1990). In this paper we 
report an experimental application with that ap- 
proach.