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3. PHOTOGRAMMETRIC RESTITUTION 
3.1 Parallel Projection 
As we have mentioned, we will now consider parallel projection 
to be the normal case in microscopic imagery. Several works 
have shown, that errors due to the projection model are smaller 
than errors in image point measurement at magnifications of 
approximately 1000x and more. 
The mathematics have to be adapted to the different model. This 
will include the bundle block adjustment and any subsequent 
section 1n space. 
First we will examine how the interior and exterior parameters 
of the sensor and the images change. 
Since the ray of sights are parallel they will not merge in one 
central point, hence there is no more projection centre. The 
missing projection centre has effects on both, the sensor and the 
image. On the sensor we can no longer construct a 
perpendicular on the image plane, known as the principal point. 
Also there cannot be a calibrated focal length anymore, i.e. the 
distance from the principal point to the projection centre. 
Instead the image is described by a global scaling factor. So the 
number of unknowns per sensor has dropped from three to one. 
However, additional parameters like radial and tangential lens 
distortion, or affine transformation parameters may still be 
applied (Elghazali, 1984). The global scaling factor, which 
substitutes the calibrated focal length is also the mathematical 
explanation for the ambiguities described in figure 3, in contrast 
to central projection, where each point has an individual scaling 
factor, depending on its spatial distance from the projection 
centre. Since in case of parallel projection each image point has 
the same global scaling factor, the object points distance has no 
influence on the images geometry anymore. Hence, the tilting 
direction of planar shapes is not distinguishable. 
Due to the missing of a projection centre, we cannot clearly 
define the position of the image plane in object space. While the 
parameters of the rotation remain unchanged, it is sufficient to 
give a two dimensional offset along the x-y-plane to represent 
the imaging situation exactly. The z-component may be dropped 
since any image along the line of sight will give the same result. 
So the number of unknowns per image is now five instead of 
six. In figure 6 the geometry of parallel projection with its 
consequences is illustrated. 
A. 
» 
  
  
os. 
Figure 6. Parallelprojective geometry 
3.2 Parallel Block Adjustment 
After making clear why and how the geometry changes, we 
would like to describe the differences mathematically. 
Previous works often used the samples tilting angle as the only 
parameter for the three dimensional object reconstruction. The 
forward section was simplified to this assumption and the use of 
two images was enough (Hemmleb, 1996; Burkhardt 1981). In 
this work we want to extend the mathematical model to full 
flexibility for several reasons. First, the parallel block 
adjustment provides us with highly accurate results in image 
orientation and sensor properties. This way small errors in the 
sample tilting process are modelled by the rotation angles as 
well. These results will be applicable to any arbitrary image 
setup. So this flexibility takes into consideration any future 
hardware development, which may allow specimen movement 
along more than one axis. 
Additionally, any accurate subsequent section in space requires 
full knowledge of the image orientations. The formulas which 
are introduced later, will show an approach to derive accurate 
object points from an arbitrary number of images. So not only 
the accuracy is enhanced by taking advantage of observations in 
several images, but we are also able to check for blunders, as 
will be explained later in this chapter. 
As mentioned above we now have to consider 5 unknowns for 
each image and only one unknown for each sensor. 
In the following, the image coordinates will be given in pixels, 
related to the digital images centre. Hence the scale factor has 
the unit pixel/um, or pixel/nm respectively. 
The collinearity equation for parallel perspective will be as 
follows: 
x X- X. 
0 Z 
x = mR; (X = Xo) + Ry, (Y — Y ) + Ry Z] 
y 2m|R; (x — Xo)+ Ry (Y — Yp) + Rs, Z] 
where | m scaling factor 
X, y = image coordinates 
X, Y, Z = object coordinates 
X0 Yo = projection centre (see Fig. 6) 
R = Rotation matrix 
The following model will be used as a Rotation matrix: 
cop 0 sing) (1 0 0 cosx —sinx 0 (2) 
Box 9 1 0 [0 coso -sino||sinx cosx 0 
-sing 0 cose) (0 sino coso 0 0 1 
When entering the parallel block adjustment, with observed 
image points x, y, given control point coordinates X, Y and the 
unknowns ©, c, K, X», Yo and m, we have to apply the following 
observation equations: 
—213— 
SIRES. 
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