sion sf a 
jon- linear, 
A closed 
he spatial 
osed space 
the scale 
proach has 
convergent 
which are 
resection 
known fact 
photograph 
.ane. One 
s the fact 
in a frame 
(z ^ image 
is being 
cal length 
point. 
the image 
mera focal 
the photo. 
f observed 
ject space 
to remain 
ve in the 
This will 
Coordinates 
(z image 
The new 
owed as a 
id | rotated 
he object 
sed three 
tween the 
4ly formed 
ill result 
ition : and 
. describes 
etween the 
sition and 
-ation, and 
nt, can be 
| conformal 
vation the 
dimensional 
d constant 
eflect the 
image is a 
constant Z 
  
(focal length) and a variable scale. With 
our proposed solution we: regain the 
constant scale of the model by varying the 
image coordinates and the focal length of 
the points. .One-point.needs- to be fixed in 
its measured image coordinates and focal 
length to eliminate singularity problems. 
INVARIENT PHOTO SCALE 
The proposed concept of invariant photo 
scale is analyzed mathematically for the 
case of photo with three control points, 
which is the minimum control required for 
space resection. 
By fixing the focal length at point 1 (fi), 
^ ^ 
changed image coordinates i-2,3 
the g g Xi |, ( ) 
due to focal length perturbations at points 
2 and 3 can be expressed as: 
^ X 
NF i 
fA 
i=2,3 (1) 
jp 
The horizontal distances between the three 
points after these changes are given as: 
7p (Fx Fux) pv) 
Where i=1,2 and j=i+1,3 
(2) 
The scale of the newly formed three 
dimensional image model can be expressed 
as: 
  
scale = 
25 
dir ff) 
D? 
ij 
i=1,2 and j=i+1,3 
where: 
  
D-G-X) «QGi-Y) «Z-Z) 
and X,Y,Z are the object space coordinate 
in a right handed cartesian coordinate 
System. 
Expanding Eq. (3) and eliminating the scale 
variable will result in equations; 
fraf. bf ref .=B z 
df. ref, +8 hf (5) 
where: 
= Da! Di 
a:= Dia! Dis 
r-xtyf, 
iT f. 
a7 3 xxt Y yy f1)f ao 
bi-2axystY st ff ilao 
c=- arr 
B=(a-Dr./ a, 
d=0-ar! f, 
&7 0 YES f, 
g2g xS Y, MI, 
h.-- ari! f, 
B=-r 
From Eq. (4), fs; may be derived as: 
fio ra rf (6) 
Substituting Eq. (6) in Eq.(5), will result 
ina? 
f o e. 
frog um. -. 
  
  
  
es s 
V- BA B 
n aod 
Ad s 
P ah, 
a Or 
Ci 
bi, 
* € 
Substituting Eq. (7) in Eg.| (4d) ‘will result 
in the following fourth order polynomial 
equation: 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996