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Parameter t = tan K,. The projective parameters of Y- 
equations are then known, but the parameters of X- 
equations are still unknown. They might be freely 
chosen, but to avoid getting into troubles (getting a 
left-handed coordinate system), these must be 
computed analytically. 
According to (Schmid, 1954) it is possible to use two 
constraints to get the reference coordinate system (in 
the new plane) centered in X,=0, Y,=0, and H,=H (H 
is freely chosen). The modified constraints for the first 
image are 
a,b,+d,e,+g,h,H* = 0 
(10) 
a,b, «deg? H'-h?H? - 0 
The second image obtains similar equations. Using 
these constraints, parameters a,, b,, a, b, can be com- 
puted. Unfortunately, it leads to ambiguous signs, 
which have to be determined separately. In the most 
probable case, as mentioned before, a, has the same 
sign as e,, and a, the same sign as e,. Signs of b, and 
b; will then be automatically correct. The remaining 
two parameters, c, and c, can be solved for example 
in this way: 
C, = -g,H%a, or ¢, = -h,H%,, 
(11) 
C = -Z,H"/a, Or C = -h,H”/b,. 
Equations (11) are derived from the properties of an 
orthogonal matrix (Schwidefsky et.al, 1976). 
Equations with the bigger denominators are used. 
2.4 MODEL RECONSTRUCTION 
After the final rectified image coordinates have been 
computed and an arbitrary base length B, has been 
given, the model can be reconstructed using well 
known parallax equations: 
XB, (Y4Y')B, -HB 
=—, Ys———, Z= 
x P; 2p; Bi 
: (12) 
  
  
where the X-parallax is 
p; = XX. 
The 3-D coordinates are linearly deformed, because 
the inner orientation was not treated as an orthogonal 
reference. À 15-parameter projective transformation 
can be used in absolute orientation to transform the 
model to a cartesian ground system. Also knowledge 
of linear object features, e.g. parallel lines, can be used 
to obtain orthogonal references. 
3. MAIN ALGORITHM 
The object reconstruction algorithm is performed in 
two steps: 1) rectification and 2) model digitizing. 
679 
The main idea of the method is to project the images 
to a plane where their epipolar lines become parallel. 
Using at least nine homolog points, 2-D projective 
transformations between the original images and this 
plane, where the rows coincide with epipolar lines, is 
obtained. The images are then digitally resampled on 
that new plane using bilinear interpolation method. 
Âs a result, a normal case of stereophotogrammetry 
is obtained, where the digitizing is easily performed 
along the parallel epipolar rows. 
3.1 RECTIFICATION 
At first, the images are displayed on the graphical 
display instrument. Because the scale and resolution 
of the homolog points at different images may not be 
equal, the identification of the homolog points is not 
necessarily easy. In this case, the first rectification is 
only an approximate one, and new observations can be 
identified from these once rectified images to obtain 
a new rectification. Mostly only one rectification is 
enough. The observations for the possibly needed 
second rectification may be selected completely 
independent on the first step. 
Normally, the operator uses a mouse or cursor to show 
at least 9 homolog points from both images, 15-30 is 
recommended. The observation can be determined 
with the cursor alone, or showing a suitable sized 
window, whose gradient weighted center is the actual 
observation. This method suits best for points and 
corners, whose identification is probably the easiest. 
Measurement accuracy of this step depends on the 
orientation of the images and the measurement 
method. If there is a large convergency angle between 
the images, identification of objects is not easy, and 
the accuracy may be some pixels, therefore a new 
rectification may be necessary. If the images are near 
the normal case, i.e. the convergence angle is less than 
20-30 gons, the measurement accuracy may be up to 
half a pixel, provided that the cursor measurement is 
used. If the method of gradient weighted center is 
used, a subpixel accuracy is possible. 
The images are resampled using bilinear interpolation 
method. Only the stereo overlap area is needed to be 
resampled. If two rectifications are needed, the 
resampling is still performed using the original 
images, not the once rectified images. The final 
rectification parameters (Rj can be computed simply 
by multiplying the first parameters (R,) by the second 
parameters (R,), i.e. R, = R,R,. These are all 3 x 3 
nonsingular matrices. In this way, an unnecessary 
weakening of image quality is avoided. The operator 
can control the result of the rectification visually from 
the display instrument.