11s presented. 
nd rotational 
mera 1s then 
olutions have 
)8 documents 
| in [Haralick 
as before the 
hotogramme- 
tions exist in 
)] for compre- 
he same con- 
e coordinate 
d coangular- 
constraints in 
nd a wide va- 
n. Currently 
| seem to be 
ified through 
>rimarily due 
1ese solutions 
nevlin, 1995]. 
use an accu- 
> resection of 
hniques. 
hor did not 
hniques since 
f a few well- 
the problem 
ved with the 
on the focal 
ther interior 
form a bun- 
stem. These 
forming light 
1s shown for 
s in figure 1. 
ould then be 
to the scene 
tions of a set 
ordinate sys- 
ontrol points 
| the exterior 
spect to the 
nces between 
ints are min- 
  
(a) Frame camera 
  
(b) Scanner 
Figure 1: Image-forming ray bundles 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
3 Parameterisations 
A computational process is required to find the least- 
squared error solution of this problem. In order to 
determine such a process the problem must be anal- 
ysed using manipulable parameterisations of the prob- 
lem domain elements. The elements are points, lines, 
point-line distance, translation, and rotation (the lat- 
ter two are required to describe both the known re- 
lationship between lines and the unknown relation- 
ship between coordinate systems). The motor alge- 
bra [Brand, 1947] provides convenient parameterisa- 
tions of all these elements. A point can be represented 
by a vector s, a line can be written in Plücker coordi- 
nates as 1 = n + ep x n, and a moment (proportional 
to point-line distance) is1® s = p x n — s x n where 
p is a vector denoting a point on the line and n is a 
unit direction vector. 
4 Scanner problem analysis 
In order to gain some insight into the scanner resec- 
tion problem it will be temporarily assumed that the 
position and orientation of the scanner imaging co- 
ordinate system with respect to the scene is known. 
The squared error between image-forming rays 1; and 
corresponding scene points s; is written,’ 
n n 
Y @si|= > lpi x mi — si x ml. (1) 
i=1 i=l 
The vector difference on the right hand side can be 
rewritten, 
(P: — si) X ni. (2) 
This shows how the moment magnitude and direction 
is a function of the vector between the scene point and 
a point on the line. Note that the vector is unaffected 
by a translation of its end points by t;, 
pi-si-(pi-t)-(s-ti)- 
pi-ti-sitt-pi-si (3) 
The positions p; of points on scanner rays are speci- 
fied by known translations t; with respect to an initial 
unknown position pg, p; = po + ti. Rewriting equa- 
tion (2) gives gives, 
(Po +t; — Si) Xn; (4) 
Making use of observation (3) gives, 
poctti—si-(po-ct;-ti)-(si-ti)- 
po— (si — ti). (5) 
Since both s; and t; are known, artificial scene control 
points s; — s; — t; can be calculated and equation (1) 
  
!|a| is used to denote the square of vector magnitude, calcu- 
lated as a - a.