in 
led 
) is 
(4) 
ed 
ue 
6) 
1S 
rithm /Marr, Poggio 1979/ is one of the most well known 
examples of this group, others can be found in Fôrstner 
/1986/ and Ackermann, Hahn /1991/. The third group 
consists of approaches which, besides the features men- 
tioned above, use relations between these features ("pa- 
rallel to", "to the right of", etc.; for more details see 
Shapiro, Haralick /1987/ and Boyer, Kak / 1988/). 
Following the line of thought of chapter 2, equation (6) 
can be employed to design an object based multi image 
matching algorithm. This algorithm has been developed 
as a generalization of the least squares matching me- 
thods in the last years /Ebner et al. 1987; Ebner, Heipke 
1988/. Similar concepts have been published inde- 
pendently /Wrobel 1987; Helava 1988/. A detailed des- 
cription of this matching algorithm and an evaluation 
using synthetic and real imagery can be found in Heipke 
/1990, 1991/. The outline of this algorithm is shortly 
reviewed here. 
First, a geometric and a radiometric model in object 
space are introduced.The geometric model consists of a 
grid DTM. The grid is defined in the XY-plane of the 
object surface with grid nodes X;, Y; and grid heights 
Z (Xv, Yi) 2 Zu. The mesh size depends on the 
roughness of the terrain. A height Z ( X , Y) at an arbi- 
trary point is interpolated from the neighbouring grid 
heights, e.g. by bilinear interpolation. In the radiometric 
model object surface elements of constant size are defi- 
ned within each grid mesh. The size is chosen approxi- 
mately equal to the pixel size multiplied by the average 
image scale factor. An object intensity value G (X,Y) 
is assigned to each object surface element. The albedo 
p (X , Y) of the object surface is allowed to be variable 
and so is G (X,Y) (see equation (5)). The object sur- 
face elements can be projected into the different images 
using the well known collinearity equations. Sub- 
sequently image intensity values at the corresponding 
locations in pixel space can be resampled from the ori- 
ginal pixel intensity values (see also figure 2). 
In the following, the grid heights Zi), the parameters P 
for the exterior orientation of the images, and the object 
intensity values G ( X , Y ) ofthe object surface elements 
are treated as unknowns. They are estimated directly 
from the observations g(x, y) and control information in 
a least squares adjustment. Thus, g(x,y) depends on 
Zi and on p. The surface normal vector ñ is a function 
of the object surface inclination, and therefore also a 
835 
  
  
  
  
  
Figure 2: Transformation from object to image space 
function of Z,;. The direction of illumination; is allowed 
to vary from image to image. For each object surface 
element, as many values g(x, y) can be computed as there 
are images, and as many equations of the following type 
can be formulated: 
8G (Zu pi). (Zu.p)) » G(X,Y) Rm 
jzl,..,n (7) 
& Gs CZ pi) Xj Cu pi)) 
image intensity value, observation 
from image j 
j image index 
Zi unknown grid heights used to interpo- 
late Z(X,Y) 
Di unknown parameters of exterior 
orientation of image j 
G(X,Y) unknown object intensity value 
n( Ze) vector in the direction of the object 
surface normal 
Sj unit vector in the direction of illumi- 
nation for image j 
n number of available images 
In optical systems the decreasing image irradiance away 
from the principal point due to the term cos/a (see 
equations (1), (3) and (5)) is normally compensated for 
using more than one lens and special coating. Therefore, 
this effect does not need to be modeled here. 
For one object surface element the object intensity value 
G, and thus the product of all values influencing G (see