can be rewritten explicitly in terms of the unknown 
initial position po, 
n 
> ipo x ni- sí x nil. (6) 
f=] 
This statement of the sum to be minimised for the 
scanner bundle-fitting problem is the same as that 
required for the frame camera bundle-fitting problem 
assuming scene control points are labelled s;. As far 
as the author can determine, this is the first time that 
the scanner problem has been simplified to that of the 
frame camera. 
Although it seems that conventional frame camera re- 
section solutions (such as those based on the collinear- 
ity condition) may now be applied to solve the prob- 
lem, this is not the case. The ground control points 
have been translated towards one another, their dis- 
tance being a function of the orientation of the scan- 
ner at the time of imaging. Despite the fact that they 
may have been well-distributed across the scene ini- 
tially they have been translated to be spatially clus- 
tered. Tests with SPOT trajectory data have shown 
that ground control points whose image observations 
were initially thousands of pixels apart are translated 
such that the image observations are only tens of pix- 
els apart [Shevlin, 1995] and that conventional resec- 
tion solutions fail to converge to accurate estimates 
when using these clustered points [Shevlin, 1996]. 
The unknown initial orientation of the imaging co- 
ordinate system R has yet to be introduced into the 
problem. Let n; — [|] R;no be the unit direction vec- 
tors of lines in the imaging coordinate system specified 
by known rotations R; of an initial vector no (which 
could be the optical axis, for instance). These vectors 
are transformed into the scene coordinate system by 
the unknown rotation Ro. Using this to rewrite equa- 
tion (6) gives an expression explicitly in terms of all 
unknowns, 
n 
NS Ipo x Ron; — s; x Roni]. (7) 
j=1 
The author has made several attempts to find po and 
Ro which minimise this sum, but without success. 
This lead to the bundle-fitting formulation being put 
aside. However the analysis was in no way a waste of 
effort since it facilitated the simplification of scanner 
problem to that of the frame camera for the first time. 
Since conventional frame camera resection techniques 
are not sufficient for this geometry a new one has been 
derived. In general kinematic analysis is greatly facil- 
itated by the fact that translation and rotation can 
be treated separately therefore aim was to formulate 
two separate resection problems, one for position and 
one for orientation. Since reducing the dimensional- 
ity of the unknown parameter space results in fewer 
parameters being sought together, the probability of 
their optimal determination is increased. 
800 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
5  Coplanarity condition 
The  coangularity condition  (Church's  condi- 
tion [Ghosh, 1988, p. 104]) to constrain the resection 
problem is that the angle 0 between a pair of rays in 
the imaging coordinate system is the same as that 
between the rays in the scene coordinate system. This 
can be written as cosf;; = cosfry =n; -n; = ny -ny 
where the upper and lower case subscripts of angles 
0 and unit vectors n refer to the rays in the image 
and scene coordinate systems respectively. Analytic 
solutions based on this constraint are presented 
in [Ghosh, 1988; Wolf, 1983, pp. 104, 240 resp.], 
however they are not least-squared error solutions. 
This approach inspired the realisation that the prob- 
lem is simplied by specifying the image-forming rays 
with respect to the known scene points instead of the 
unknown focal point. Instead of applying the coan- 
gularity condition through equations written in terms 
of direction vectors it was decided to investigate the 
formulation which results using the scalar product of 
Plucker lines 1; - 1; = 0 = 0 + ed. The dual angle 0 
comprises the length d of a perpendicular joining the 
two lines and a rotation of angle 0 about the perpen- 
dicular. 
Given lines I; 2 Rn;-- es; x Rn; and 1; = Rn; + es; x 
Rn; the scalar product 1; À, is found by distributing 
across the real and dual parts, 
1; À, =0+ed= 
Rn; - Rn; + ¢ (Rn; "Sj X Rn; + Rn; SX Rn). 
(8) 
This gives a new idea for a constraint — that the 
perpendicular distance d between the rays at the focal 
point should be zero, 
d = Rn; - s; x Rn; + Rn; -s; x Rn; 
= s; - Rn; Xx Rn; + s; - Rn; x Rn; 
=S; R(n; x n;) + S; + R(n; x n;) 
= (s; — s;) - R(n; x n;) = 0. (9) 
This expression is one of coplanarity. A vector be- 
tween two scene control points is coplanar with vec- 
tors defining the directions of their image projections, 
see figure 2. The coplanarity constraint is more often 
used for the relative orientation problem. The con- 
straint equation (9) looks interesting for the resection 
problem because the only unknown present is orien- 
tation. 
Letting? g; = ||si — s;|| and c; = ||n; x n;||, the 
squared error function of camera orientation R to be 
minimised is,3 
n 
> (gi, Rey)? (10) 
i=] 
  
?|la|| is used to denote the unit norm of vector a, calculated 
as a/ V |a. 
3(a,b) is used to denote the inner product, calculated as 
a-b 
      
     
    
    
    
    
    
     
  
  
  
  
  
    
    
    
     
    
  
    
   
     
    
    
     
   
     
    
   
   
  
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