1368 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING, 1975 
  
Xz = hcos(a, + 4a) — k sin (a, + 4a) (32) 
X, = hcosa, — ksina, (33) 
and 
Z, = htana, + kcosa, (34) 
Solving for h and k from equations 32 and 33 yields 
kon Xgcosa, — X,cos(a, + Aa) [| (35) 
— sina,cos(a, + Aa)—sin(a, + Aa)cosa, 
and 
he X, + ksina, (36) 
= cosa, : 
Thus, when the tilt angle (a,) of the “left” photo and tilt angle difference (4a) between the two 
photos are known, the height difference above the plane of the “left” photo can be computed 
from the measured distances X; and X, using equations 34, 35, and 36. Furthermore, let 
C, - cosa, 
C3, = cos(a, + Aa) 
C3 = sina, 
C,=C,C, — C,sin(o, t 4a) 
_ CyC, — C,C3 - CC, 
  
Cs CIC, (37) 
and c: 
la €i 
Then, it can be easily derived from equations 34, 35, and 36 that 
Z, 7 X,C; * XgCe (38) 
For a given stereoscopic pair of SEM photos, C4 and C, are fixed constants. Thus, the use of 
equation 38 results in only a small increase in the amount of computation as compared to 
Boyd's special-case equation. When a large number of points are to be measured from a 
stereoscopic pair, the percentage increase in the total computational effort is negligible. 
By setting a, = 0°, equations 34, 35, and 36 are reduced to the following form: 
Z, =k (39) 
_ X,cosda — Xr (40) 
sinda 
and h = X,. (41) 
That is, 
Z, » X,cosda — Xg 
sinda 
which is the equation used by Boyd5>4. 
One ofthe major problems in using SEM images for stereoscopic measurements is the need 
for accurate measurement of the tilt angle of the specimen stage and the magnification. Boyd 
has developed an otpical lever method which can determine the tilt angle difference between 
two settings of the specimen to within £0.05 degrees. According to Boyd, the magnification 
can be determined to *4.2 per cent. The problem of geometric distortions can be minimized 
by the superimposition of an electronically generated grid onto the SEM picture. 
In an approach which is completely different from that employed by Boyd, Ghosh!? and 
Maune?8 modelled the geometry of the SEM images with the collinearity equations. Included 
in the mathematical model are terms for symmetric and asymmetric radial lens distortions, 
electronic distortions, film distortions, magnification in the x and y directions of the final SEM 
photograph, and the tilt angle ofthe specimen stage. This model was used to calibrate an SEM 
at three different magnifications (2000 x, 5000 x, and 10,000 x). An SEM resolution and 
calibration standard which was made from a diffraction grating with 2160 lines-per-millimeter 
in both the x and y directions was used as subject. The CRT display of the subject was