1. BASIC RELATIONS 
Composing the matrix of projection by its row-vectors 
  
mj - (mj; mz mj; mg), an image point results from 
T. 
m. 
uu - M y or uj ^ TY (1.1) 
0 
after eliminating the tiresome factor u. From M result 
the coordinates of the center of projection by means 
of 
My,-0 or m/y,-0 (1.2) 
(Brandstátter 1996). Referring to the interior 
coordinate system mentioned above, M itself reads 
simply 
1 4-1 45-1 i-1 
M=10 u 0 U34H3 (1.3) 
0 0 Ho Uh 
(Brandstátter 1993). It depends on the four projective 
parameters ju, =1, iu, y, H,, and on the image 
coordinates u,,, u,, of a point u,, that is the image of 
the third affine unit point of the object space (Figure 
1). By means of these elements the image 
coordinates result from 
d y; Hj 5WS Us us ni 
- ;4j-712 (1.4) 
tx. y. 
ki 
representing the pure projective transformation 
whereas (1.1) also contains elements of affine 
transformations. 
Stereoscopic image correlation between two 
images P'(u) and P"(u") is defined by the 
coplanarity condition 
UCU =c "Zu" =0 (z 6/6) (1.5) 
(Thompson 1968, Fuchs 1988) with a matrix Z con- 
taining besides of z,,=1 the significant eight com- 
ponents z; of correlation. Using now, out of the set of 
eight points of correlation, three non-collinear points 
for the definition of the interior affine coordinate 
system, their homogeneous point vectors will contain 
the simple components 
8-5) (8 
and the corresponding system Ag will arise with co- 
efficients as listed in Tab. 1. Therefrom follows 
directly that z,,=0 and hence det(A,)=det(A7). 
Furthermore, from the equations 1 and 2 of A; follow 
two additional simple relations: 
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€. 
By their means, projective correlation is reduced to a 
ox5-system Asz=a consisting of five rows according 
to 
UT(Uj — 1)Z44 * uju2z,? * (u5 — u2)zog * uui, * 
4.5 
*ujup-fz,- u-u OD 
with a vector of unknowns zl = (Z41 Z42 Z920 Zo Z22) : 
  
det(Ag)=de 
1 0 
@ 4 
d d 
e; 9G, and its ex 
terminant: 
Figure 1: Definition of the affine coordinate systems ponents.e 
det(A7)= 
2. THE CONDITION OF CRITICAL > 
CONFIGURATIONS 
Critical configurations referring to projective image = 
correlation will be indicated by vanishing of the 
system determinant, that is . 
det(Ag)=det(A,)=0. (2.1) 
Hence the coefficients of matrix Ag (Tab. 1) will be = 
responsible for critical situations caused by all point 
distributions satisfying this condition. In order to sé 
recognize their spatial distribution, the components of 
Ag in Tab. 1, which are composed originally by plane 
image coordinates, must be transformed to the object = 
space by means of (1.1) or (1.4). Using (1.1), the 
transformation will contain the typical dyadic product The sub 
(m[Ty)(m;"y) = y" (mim;")y = y"Qy y (2.2) FO din) 
coordinat 
But expressing at first the components of Ag by the geon 
means of the abbreviated terms according to (1.4), derived. 
the condition (2.1) will read 
78 
Remote Sensing. Vol. XXXI, Part B3. Vienna 1996