l 
  
where g,,g, are the gradients in z— and y— directions, a; 
the partial derivatives of the collinearity equations. The 
initial gray levels g^(T2, T?) of the image patch are deter- 
mined by transforming the initial elevations Z?( X, Y) to the 
image with the initial collinearity equations 72, Ty. The pa- 
rameters Aa; include the 6 exterior orientation elements and 
AZ, the unknown elevations of the surface patch. Finally, 
G(X, Y) are the unknown gray levels of the surface. 
3.2 Parallel Projection 
If the exterior orientation in the collinearity model is deter- 
mined from one image patch only the normal equation sys- 
tem will be ill-conditioned. The perspective center is only 
weakly determined by the small image patch since the in- 
tersecting bundle rays form very acute angles. Of course, if 
the same image is involved in several well distributed surface 
patches, the situation improves, but only if a simultaneous 
adjustment is performed. 
If the surface patches are determined individually then a 
parallel projection should be used instead of the central 
projection. The direction of the projection is defined by 
the bundle ray through the center of the surface patch (see 
Fig. 2). This direction can easily be determined in image 
space. Let [z,,y,, —c|T be the center of the image patch in 
the traditional photocoordinate system (origin at perspec- 
tive center). The bundle ray through the center of the patch 
is then defined by the following equations: 
Zz = ng y —mz (7) 
c yc 
n = — m= 
Ze Ze 
The angles a, 8, determine the spatial direction of the bun- 
dle ray that represents the image patch. 
1 
Vv1+ m2 4 n? 
TT e 
cos(v) = 
V1 + m? + n° 
cos(a) = 
Il 
With these three independent angles about the axis of the 
photo coordinate system the rotation matrix R is formed 
in the usual fashion. The corresponding rotation matrix 
R, in the object space coordinate system is obtained by 
multiplying R with the rotation matrix from the exterior 
orientation Re 
Rs = ReR (9) 
If we rotate the surface patch S by Rg then the projection 
becomes parallel to z in the photo coordinate system. Since 
we deal with a parallel projection the object/image space 
relationship is trivial, that is, z — X',y — Y' where z,y 
are the photo coordinates and X',Y' the rotated surface 
coordinates. 
Now we complete the geometric transformation by adding a 
translation vector [z;, y,]" and a scale factor s. This compen- 
sates for not including the perspective center in the trans- 
formation. Finally, the following transformation equations 
describe the parallel projection 
257 
   
   
     
Ay 
_ 
S 
    
    
    
. 
SN 
Figure 2: Direction of parallel projection defined by bundle 
ray through center of surface patch S 
= (ruX +T12Y + T13Z)8 + Xe (10) 
= (rnX +r2Y +7237)8 + uy, 
Linearizing the general observation equations 3 with respect 
to the geometric transformation of equation 10 we obtain 
equation 11 which corresponds to equation 6 
  
7 2T, 2T, 
r-G(X, Y)-9*(T2,1,)- Y, ge + gy—— | Aa; (11) 
i=1 Va; Va; 
with the partial derivatives 
X 
ER = a? =s{0,0,0]| Y 
a Z 
X 
2T, 
y = al = [rs1, 732, 733] Y 
e Z 
9T, = a3 = —scos(4) [rai, Taz, Tas) X 
26 Z