|^ and 
pose 
ts to 
uring 
e can 
inces 
n 2D 
ents. 
ent. 
e can 
man- 
Ss can 
ne et 
three 
| that 
ather 
met- 
r and 
s dis- 
and a 
ween 
ojec- 
rding 
d ap- 
. Till 
ts for 
high 
ing : 
vhich 
sr the 
leter- 
n the 
g the 
times 
esent 
ise of 
ction. 
lane, 
goth- 
other 
ance, 
voids 
S, We 
oper- 
| then 
nents 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
2 DATA SETS 
On the one side, we are dealing with a point cloud coming from 
one station scan. Scan is performed by constant angle ray tilt- 
ing in a vertical plane, followed by rotation around vertical axis. 
Range measures present noise that can be reduced by multiple- 
shots. Points coordinates are defined in the scanner reference 
system. 
On the other side, we have got a digital image and its optical 
model, in an image coordinate system centered on camera point 
of view. It is a conic projection where distortion is corrected. 
Camera devices record 5 Mpixels color image, coded on 12 bits. 
     
A 
  
Figure 1: Point cloud, digital image (details). 
3 SYSTEM FORMALIZATION 
In this section, we develop the explicit system where orientation 
and location unknowns are considered. It is a bundle adjustment 
frame, where we look for translation and rotation between a ter- 
rain reference system (the scanner one) and an image system. 
3.1 Distance between segments. 
Let us project the ends of a 3D segment into the image plane. pi 
and p? are the image projections of the segments ends. Expressed 
in polar coordinates, in the image coordinate system, the straight 
line passing through these points is defined by : 
  
x COS 0 + 7. sino p (1) 
with : 
cos 0 = (pa — pi) Q) 
lip] 
sin (P1 — P2)e 3) 
2122| 
Iprpa| 
The projection of a 3D point P into image is given by : 
AG — 1. RO -—T 
Dr — f Ri Dy = f ( = Du (5) 
R(P — T) R(P — T) 
where : 
e R rotation between world system and image system 
e T translation between world system and image system 
e fthe focal length. 
Note P, P» the 3D segment ends in world coordinates. Con- 
sidering a vector 7i lying into the plane which contains the 3D 
1131 
3D segment 
Distances to minimize 
Point ^: c nad 
cloud: Pr À 
  
2D matched segment — T "€ ds 
Figure 2: Distance between segments. 
segment and the image projection center, and orthogonal to the 
image plane, such as : 
ii 2 R() — T) AR(P, — T) (6) 
Replacing equation (5) into relations (2), (3) and (4) gives : 
Tis ; n 75; 
cos = —;  sin0=— p=-f— 
[172] liri] 
A 2D segments end named p — (1;, L,, f) lies on the projected 
line ; this yields : 
Bolen 
|i 
Left term in equation (8) is the distance between the ends of the 
segment detected in images and the line supported by the 3D seg- 
ment projection. This distance is associated as residual for each 
segment. 
0 (8) 
  
3.2 Distance between points 
On the same scheme, one can define distance between two 
matched points : reaching least-square distance between the im- 
age point and the 3D point projection amounts to minimize the 
sum of squared distances on the X-axis and on the Y-axis. 
Expected distance annulation along X-axis gives : 
PT. : 
R(P — T); 
Along Y-axis : 
x 
pn RO TO d (10) 
R(P — T) 
3.3 Global energy function 
For segments, energy function is derived from (4) : 
  
Ny San) 
= Mu - PI 
E = = (11) 
: (32)