nf nf nf +nf,+n-= (8) 
where: 
= HB, 
9h ^ d, y,*b. yYStbhy Yy*2cy Y. 
2B y,g, 
m Way DEAN, 
2o Y,*ey,- B8, 
n-3gy,*8,a-bgJ,*?oXJY. 
n.- g,*a, 
Interestingly, Fishler and Bolles (1981) 
and Zeng and Wang (1992) also arrived at a 
fourth order polynomial function of a 
variable, but the geometric representation 
of this variable ‘is difficult to' express. 
In Eq. (8), clearly the polynomial variable 
is the focal length at point 2. 
An iterative solution can be adopted to 
solve Eq. (8), by using the initial value 
of f; equal to fi; . A closed form solution 
of Eq. (8) can be found in Dehu (1960). The 
solution will yield one to four real roots. 
In most cases ‘this. will ‘lead to. two 
imaginary and :two: real roots. If the 
Solution leads to four real roots, the two 
real roots for f; which are in closest in 
value to fi (without changing sign) will be 
chosen. Such an approach can not be adopted 
if the variable in the fourth order 
polynomial as in Fishler and Bolles (1981) 
and Zeng and Wang (1992) does not represent 
an identifiable geometric entity. The 
availability of a fourth control point, 
will help to find the proper root from the 
two chosen real roots. Substituting the 
roots for f. in Eq. (7), will result in the 
determination of the corresponding roots 
for f3 
The perturbations of the image domain 
Coordinates to enforce a constant scale, 
will result in a three dimensional image 
model. The control point coordinates in 
this model will be i $j: fj) and i=1,2, and 
3. Zi Di are defined in Eq. (13. The 
relationship between the new image model 
and the object space is expressed by the 
following three dimensional conformal 
transformation: 
X: 1 X, X. 
y =-M y |+ y (9) 
S 1 
Z ez 
where: 
Xo, Yo, Zo, Camera Position in object Space. 
M three dimensional orthogonal trans 
formation matrix, such = that>0M will 
represent the camera orientation matrix. 
CLOSED FORM THREE DIMENSIONAL COORDINATE 
TRANSFORMATION 
To obtain the exterior orientation 
parameters for the photograph, we need to 
solve Eq. (9) for X.,Y.,Z. and omega, phi 
and kappa that defines the orientation 
matrix M.» Although- Eq. (9) is’ linear in 
Xo, Yo,Zo , it is non-linear in^ terms ofthe 
scale (s) and omega, phi and kappa 
rotational elements and require initial 
estimates. The initial value for the scale 
can be obtained from “Eq. (3). The 
orthogonal orientation matrix M, can be 
computed using quaternions (Horn, 1987). 
This procedure can be summarized in the 
following steps: 
1. Let the origin be the first point. 
2... Take ‘the line from the’ first to the 
second point to be the direction of the 
new x-axis. 
Place the new y-axis perpendicular to 
the x-axis and assume a right handed 
coordinate system to define the 
direction for z-axis. 
4. Let the coordinates of the three points 
in each system be expressed as a three 
dimensional vector: 
Vni >Vm2 3 VY»3 
in. object space: Voi Vo2»Vo3 
5. Construct 
A ar Venim 
then 
[V9] 
in image model 
; var 
^I] 
is a unit vector in the direction of the 
new x-axis. 
  
6. Now let 
y eG vul 
(Vs Va)? XxX X. 
then RT 
5 ob 
ys — 
Ys 
represent the direction of the new y- 
axis. 
  
  
  
  
7. The Z axis is defined as: 
A YT pee 
Zu Kat» 
Repeat steps 5 to 7 to the object space 
system to find C cwm : 
9. The rotational elements to be computed 
are the one that performs the 
transformation: 
Co 
392 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
  
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