The condition of relative orientation is that the new 
Y-coordinates are equal (Y';Z Y"). Thus: 
dx; e;y' f, dx" ey" f, 
= " " (3) 
gx ;+ hy'1 
  
gx h,y;+ 1 
The original images will be projected on the same 
plane, and the epipolar rays become parallel with each 
other. 
2.2 SINGULAR CORRELATION 
From equation (3) a bilinear equation is obtained 
(d,g2-g,d2)*"x";, + (d,h,-g,e2)x"y 6 
(di-gif)x';  (e,g,-hidoy'x", 4 
(e,h,-h,e2)y";y"; + (e,-h,f,)y”, + 
(fg2-da)x"; + (fhz-e2)y"; + 
(fcf x 0. (4) 
By'renaming the coefficients we obtain 
mx" mx y+ myx +m, yx + 
». " " 
ID5y,y j-m,y -m;x"--mgy ;+m, = 0, (5) 
which is in matrix form 
m, m, m,|[,/ 
[x y' 1] m, m, m. y" =0, (6) 
m, mg my | 1 
where 
m, = d,g,-g,d, 
m, = d h,-g,e, 
my = d;-g,f, 
m, = e;gy,-h,d, 
m, = e,h,-h,e, (7) 
me; = e-h,f, 
m, = f,g,-d, 
mg = f,h,-e, 
M, = f-£- 
Equation (6) is also known as a correlation equation 
(Thompson, 1968). This correlation is singular, so the 
determinant of the correlation matrix M must be zero. 
According to (Jordan et.al., 1972) the determinant of 
the correlation matrix in the case of central 
perspectivity is always zero. However, the elements of 
the matrix M should be solved using det(M)=0 as a 
constraint to ensure correct solution. Conventional 
least squares solution is used. 
2.3 EPIPOLE COORDINATES 
From the matrix M the epipole coordinates of both 
images can be obtained using the following equations: 
/ 
m, m, m, X. [o 
m, m, m,|y, = 0 (8a) 
m,m,mjj/ O0 
678 
” 
m, m, m,'X.| lo 
m, m, mely/| - 0 (8b) 
m, m, my yl 
where x’, Y eZ", ANA X" y" ,z", are homogeneous epipole 
coordinates of the images. Their rectangular 
coordinates are x'-x'/z,, y'=y’/z, and x" Zo 
y y /Z'. An approximate kappa angle of an image 
can be computed from the relative position of the 
epipole and the center of the image, assuming that the 
principal point is approximately at the center of the 
image. This property has been used later when 
computing the projective transformation parameters. 
The projective transformation parameters for both 
images can be solved from the correlation matrix M, 
if the determinant of M is zero. There is an infinite 
number of solutions for the projective transformation 
parameters, which all lead to the same correlation 
matrix. Any one of the solutions can be chosen and 
used in further computations, but some of them may 
seem to be impractical. 
The relations d,/e, and h,/g, can be fixed to a chosen 
value which is approximately the same as the tangent 
of K, (the kappa rotation along the original z-axis of 
image one). Then a practical solution is obtained. The 
above mentioned ratios are found from the comparison 
of the collinearity and projective transformation 
equations. 
Any two ofthe elements m;, m,, m, and m, are always 
nonzero, these being usually m, and m, when the 
situation is near the normal case of photogrammetry. 
If | m, | »| m;| and |m,|»|m;|, we obtain tz-y Jx., 
where t=tanK,, and y’, x’, are the coordinates of the 
epipole point of image one. These can be directly 
computed from the correlation matrix M using 
equations (8). 
Choosing f,-0, the rest of the projective transformation 
parameters can be solved using equations (7), and 
equalities d,/e,=t and h,/g,=-t. When f, gets the value 
of zero, it corresponds to the situation where the 
omega rotation of the first image is zero in the 
conventional collinearity equations, thus preventing 
the arbitrary projection plane to rotate around the eye 
base. 
If | m,|»| m,| and |m,|»|m,| (the most probable 
case), the following expressions for the projective 
parameters have been derived: 
d, = -m, 
€3 7 -Inlg 
f, = -my 
e, = (m,+tm,)/(t?+1) 
g, = (m,-tm,)/(mp(t?+1)) (9) 
d, z te, 
h, = -tg, 
g; = -(m,+g,tm,)/m, 
h, = -(my-g,tm,)/m, 
tv