movements of the specimen stage can be controlled 
precisely. So in order to make the procedure of 
surveying more simple, one usually takes each 
micrograph of a stereopair just with a tilt angle $1, 
v2 respectively. In this case, the Eqs. (1) can be 
simplified as follows: 
cos (X-Xo) - sin (Z-Zo) 
X 
(2) 
k(Y-Yo) 
Y 
According to Eqs. (2), the formula, which can be 
used to calculate the model coordinates from the 
photo point coordinates, can be expressed as: 
X = (sinb 2x1 - sin 3x») /sinC 2-39 1) +XO 
Y = (ya + Ve) / (ka tk.) +Yo (3) 
Z = (cos zxa - cos 1X») /sin (i 2-1) 1)+Z0 
EM3DPS adopts the Eqs. (3) as the intersectional 
equations(similar to the real-time program in an 
analytical plotter). 
Note: In Eqs. (3), the y coordinate axes of the left 
and the right micrographs coordinate system are 
parallel to the tilt axis of the specimen stage. 
2.2 RELATIVE ORIENTATION EQUATIONS 
As in conventional photogrammetry, a stereomodel of 
the object can be formed by means of intersection 
of conjugate rays established by using two 
micrographs of the same specimen. Because EM 
imaging system is regarded as parallel projection, the 
condition of coplanarity can not be used in 
micrographs relative orientation. On analytical 
plotters, relative orientation with EM photographs 
becomes quite simple if, for generating the second 
micrograph of a stereopair, one uses only the 
elements of tilt and the associated x translation. But 
in EM digital image surveying system, the relative 
orientation with the same two micrographs is not 
simple, because during the procedure of digitizing 
micrographs by using a CCD camera or a scanner, 
the direction of sampling can not be ensured to be 
perpendicular to the tilt axis of specimen stage. This 
means that the y axes of the screen coordinate 
systems of digital images(similar to the y axes of 
the comparator coordinate systems on analytical 
plotters) are not parallel to the tilt axis of specimen 
stage. In order to make use of the Eqs. (3)to 
calculate the 3D model coordinates of the specimen, 
the angle between the tilt axis of the specimen stage 
and the y axes of the screen coordinate systems of 
the left and the right digital images must be gotten. 
   
Ya Ys 
  
  
   
  
  
  
  
  
Si XsS2a Xs 
Fig. 1 Relative Orientation Geometry 
This is the task of the relative orientation in EM 
digital image 3D surveying system. According to the 
parallel projective features(the planes formed by 
conjugate rays are all parallel to each other and 
perpendicular to the tilt axis of the specimen 
stage), the ralative orientation equation should be 
expressed as follows(refer to Fig.1) : 
yps 7 yDa (4) 
Where yps,ypa are y coordinates of ps and p's in 
01-X1ÿ1 and 0z-Xaÿz coordinate systems. The y 
axes of these two coordinate systems are all parallel 
to the tilt axis of the specimen stage. o4 and o2 are 
a couple of conjugate points near the center of the 
left and the right images. S,-XsYs and Sz-X' sY" s 
represent the screen coordinate systems of the left 
and the right images. Eq. (4) can also be expressed 
by: 
((Yps-Yo3)00s 0 1- (Xps-Xox)sin 0 1) 
- ( (Yp's-Yoz)cos 0 2- (Xp s-Xo2) sin 0 2) =0 (5) 
where Xps, Yps, X01, Yo, are the screen coordinates 
of the points ps and o, on the left photograph; Xp 
1, Yp a, X02, Yoz are the screen coordinates of the 
points pa and oz on the right photograph; 0 1 isthe 
angle between 01-X1Y1 and S1-XsYs; and 6 2 is the 
angle between 02-x2ÿ2 and Sz- X'sY's . After 
linearization, Eq. (5) becomes: 
af of = 
e doot$ d o7 7-0 (6) 
where Be (Ypa-Yo1) sin 0 °,— (Xp:-X01) cos 6 °, ; 
Sn (Yp's-Yo2)sin 0 ^2- (Xp 4-X02) cos 0 2; 
f? 2 ((Yps-Yog)cos 0 5- (Xps-Xoj)sin 0 ) 
- ((Yp'a-Yoa)cos 0 *2- (Xp 4-Xoz)sin 0 ^2) 
0%, 0°% are the approximate values of 01 and 0 > 
respectively.