348 
poor contrast. (C) Schematic illustration of how a 3D point distribution is generated. The vertical shift is measured by a laser 
interferometer with an accuracy better than 10nm. 
This approach has two main advantages. 1.) A point distribution with homogeneous precision in all three 
dimensions is provided even for submicron error level. 2.) The construction of the third dimension by moving 
the specimen table in the vertical direction makes it possible to flexibly adapt the vertical range of the calibration 
standard to the actual depth of field. Therefore, a single grating can be used for calibrating quite a wide zoom 
range, though the depth of field is dependent on the magnification. 
4 MATHEMATICAL FRAMEWORK 
A. Weak perspective equations 
Most of the descriptions of imaging processes involve somehow the well known perspective equations. ! 
C 
Dir and MEY I zs (1) 
  
€— &o — ZR 
The vector zg — [rn, yn zn]. 
R(x ie Z9) 2 
An alternative formulation for (1) can be found. Defining C/D as the object to micrograph magnification M 
and substituting ¥ = zg/D which is always smaller than 1, the perspective equations can be converted into a 
Taylor series of the form: 
can be related to any reference frame with a rigid body transformation zg — 
oo oo 
E-&=M-zpr) 0* and n-m=M-yr) PP (2) 
k=0 k=0 
According to their functionality we classify the terms M - xr - 9% as k order terms of perspective distortion. The 
zero order term simply describes an orthographic projection, for obvious reasons often called weak perspective. 
As in the microscopic case ? «& 1, the high order terms are rapidly converging to zero. A rough estimation for 
the stereo light microscope shows that the first and second order terms of perspective distortion can maximally 
be 11pm and 4pm, respectively. Thus, the second order term is already smaller than the pixel spacing (8um) 
of the video cameras mounted on the microscope. 
   
  
  
  
  
Perspective 
Center 
z 
^ Reference 
Frame 
Object Point X 
cl 1! Xr 
Ç Rotated Object 
PA Coordinate Frame 
= Figure 4: The left and right image of the 
05076 stereo view can be combined to a stereo-rig. 
mage Point * Micrograph 6 Trigonometric function values of rotation an- 
: : : gles qi/7, 01/7, K1/r define the rotation matri- 
Figure 3: Illustration of the perspective ces R,,. © is the rotation from the calibra- 
mapping function tion grating frame into the reference frame. 
B. Combining two mapping functions to a stereo-rig and defining a common reference frame 
The stereo view of the microscope can be described with a combination of two weakly perspective imaging 
functions. In light microscopy two different optical systems are involved in stereo imaging (Figure 1). Therefore, 
two separate sets of mapping parameters have to be estimated. However, we expect very similar characteristics 
for the left and right zoom lens systems. In addition, in the case of CMO optics, the rays partially run 
  
1Scalar values are written as simple symbols, vectors and matrices are symbolized with lower and upper case bold face letters, 
respectively. 
2? The various vector symbols and their significance are explained in Figure 3 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995