MATHEMATICAL FORMULATION & DIGITAL ANALYSIS 1367 
necessary to perform analyses using error ellipsoids for all points in the photogrammetric 
solution. In many applications, the diagonal elements (02, 02, and 02) in the variance-and- 
covariance matrix (0) should provide sufficient insight into the mapping accuracy. For cases in 
which extremely high accuracy is required, the error ellipsoids for a few carefully selected 
points may be computed for analysis. 
STEREOPHOTOGRAMMETRIC MICROSCOPY 
Largely through the work of A. Boyd at the University College London, significant ad- 
vancements have been made in the use of scanning electron microscope (SEM) photographs 
for stereoscopic measurement. Boyd has developed an instrument which can be used to plot 
contour maps or cross-section profiles from a stereoscopic pair of SEM photographs that have 
been taken with a tilt angle difference of 10 degrees. He has also developed an instrument 
system for real-time stereoscopic viewing of SEM images®. For the purpose of analytical 
computation, Boyd has developed a computer program for computing the three-dimensional 
coordinates (X, Y, and Z) of object points®. The program can also compute height differences 
along cross-sectional profiles which are perpendicular to the axes of tilt, as well as facet slopes, 
area, and volume. The program is designed for use in a Hewlett-Packard HP9830A program- 
mable calculator. In addition to the above numerical parameters, the program can also plot the 
cross-section profiles and the horizontal position of the object points. 
Both the stereoplotting instrument and the computer program developed by Boyd are based 
on the following equation: 
X Xz (31) 
* tano ” sin da 
  
Zi 
where Z, is the height difference between two object points measured in a direction normal to 
the less tilted photo, which is designated as the left photo, X, is the distance between the two 
corresponding image points on the left photo and is measured along a direction perpendicular 
to the tilt axis, X, is the corresponding distance measured on the more tilted photo which is 
designated as the right photo, and Aa is the difference in the tilt angle of the specimen when 
the two photos are taken. 
Equation 31 is derived using the principle of parallel projection. Although the geometry of 
the SEM images is basically a central perspective projection, the error introduced by the 
assumption of a parallel projection is relatively minor when the magnification is 1000X or 
higher. However, equation 31 has one severe limitation that was not fully recognized in 
Boyd's publications. This equation is valid only ifthe “left” photo has a tilt angle of 0 degrees. 
If the tilt angle of the “left” photo is not 0 degrees, then the value of Z;, computed from 
equation 31 would not represent the height difference normal to the plane of the left photo. 
More specifically, Z;, would not be measured in a direction normal to the plane in which X, is 
measured. 
A more general equation which expresses Z; as a function of the tilt angle of the left photo 
can be easily derived from the geometry of parallel projection. The following expressions can 
be derived directly from Figure 8: 
  
  
N 
N 
a 
  
1 £ 
99 
  
  
  
SEC 
XL 
Fic. 8. Parallel projection of SEM images.