iple 
ints 
nal 
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ms 
the 
28) 
9) 
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0) 
9) 
DI 
Ww OU & BP o 
By using all potential unique permutations of photo pairs as 
observation equations, and using one window as the radio- 
metric reference template, as previously described in section 
3.1, we can form up to onl) distinct pairs of conjugate win- 
dows, or nn, n2 corresponding observation equations. 
Each observation equation (equation 15) can be linearized 
with respect to any preselected set of m orientation param- 
eters per photo (o;,..., o},,01,..., o},) as 
) "m? 
of,o , 0 Oz; i 
gi(æi, Yi) — e(z, y) = 95 (23,95) + 9j. 5; 401 T 
1 
  
By; , Oz; , ; 
+ uti a m FI +. 
Öy; ,; 
+ c SS dol, (33) 
The reference template (in photo 1) has to be kept stable, 
therefore the exterior orientation parameters of photo 1 will 
be kept constant during the matching process. Thus, the so- 
lution can be considered the digital equivalent of dependent 
analog orientation. Since the original model is non-linear, 
the final solution is obtained through iterations. The design 
matrix for this case will have similar sparsity pattern to the 
one shown in Fig. 1, but the dimensions of each block of 
nonzero elements will be (n; -n;) x m. After each iteration, 
the image coordinates of point P in photo j are updated 
due to changes in orientation parameters 
a 2; . Oz. . 
T; =z; + doi 7 +... + Doi, dom (34) 
and 5 8 
o y; . y; . 
y; - y del tot de, (35) 
By solving the above system we inherently ensure that con- 
jugate image rays intersect at a point in the object space. 
While plain least squares matching is solely a radiometric 
adjustment, the use of object space constraints to express 
the relationship of two or more conjugate windows allows 
the combination of the radiometric and geometric solutions 
in a single adjustment procedure. The model can be ex- 
panded to include the object space coordinates of point P 
which can be introduced into the adjustment by properly ex- 
pressing the scale factors as functions of them. In addition, 
the technique can be expanded to simultaneously adjust ob- 
servations of more than one point in the object space. The 
images of all points in each photo will be related through 
a common set of exterior orientation parameters and the 
adjustment can thus proceed in a global manner. 
3.3 Matching in the Object Space 
By examining the image formation process we can extract 
some rules which can later be used in the matching pro- 
cess not only as constraints but also to expand the problem 
into the radiometric and/or geometric reconstruction of the 
object space itself. 
Fig. 2 shows four image windows wy, ...w, displaying ap- 
proximately the same surface patch S in four overlapping 
images. The surface is described by two continuous func- 
tions, one geometric Z(X,Y) (elevations) and another ra- 
diometric G(X,Y) (gray values). Assuming a local tessela- 
tion, whereby the surface patch S is represented as a Digital 
Elevation and Radiometry Model (DERM, a term analo- 
gous to DEM) with a resolution of n; x nz grid points, the 
805 
patch is defined by n, - n; elevations and by an equal num- 
ber of gray values. The reconstruction of the patch would 
therefore involve the determination of these 2 14:713 param- 
eters. These parameters can be determined by defining the 
geometric and radiometric transformations which relate S 
to its images w, ... wy. Each image window w; corresponds 
to a gray value function g;(x;,y;), related to S through a 
geometric transformation 
(zi, y) = T(X,Y,Z) (36) 
and a radiometric one 
gi(zi yi) 7 T;[G(X, Y)] (37) 
Assuming the object space patch S to be a Lambertian sur- 
face, the recorded image irradiance g(z, y) (image gray val- 
ues) is directly related to the surface radiance G( X, Y ) (sur- 
face patch gray levels). Furthermore, taking into account 
the relatively small size of the surface patch, the rather 
complex radiometric relationship between image and object 
space can be effectively approximated by a linear transfor- 
mation 
gí(z,y) = 75 * riG(X, Y) (38) 
Assuming a Lambertian light source, the values of the ra- 
diometric shift (r$) and scale (ri) parameters are unique for 
each image and they are functions of the surface albedo as 
well as of the angles formed between the image window w; 
and the normal to the surface [Horn, 1986]. 
The radiometric adjustment is typically performed prior to 
the matching process, by forcing each window w; to have 
the same average and standard deviation of gray values as 
the reference template w;. Thus, we actually force 
i ta ia 
rizrl-re and rcr (39) 
Subsequently, the gray values g;(z;,y;) of w; are assigned 
to the surface patch S. The assignment can be performed 
either directly: 
G(X, Y) = gi(z21,yı) (40) 
or through an inverse linear transformation 
G(X,Y) - g(z, y) — ro (41) 
Ty 
In order for an inverse linear transformation to be used, the 
parameters ro and r4 have to be determined using a priori 
  
Figure 2: Overlapping image windows in the object space