International Archives of the Photo 
correspondences curves from the images with the plane. Those 
intersections must be identical to get the full overall solution. 
When applying this procedure with initial parameters we will 
get two separated curves. Now in order to bring the two curves 
closer ICP algorithm is proposed. 
  
  
  
  
  
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, ur. / = uv 
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s y s 
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/ = C * 
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Figure 1. Projection of correspondences curves 
2.2 ICP — iterative closest point 
The ICP algorithm, first introduced in (Besl and McKay, 1992) 
can be used with several representations of geometric datasets, 
such as point sets, line segments set, implicit curves, surfaces 
etc. The geometric datasets used in this paper are point sets, 
representing the free form curves. The datasets we deal with are 
the projected curve from the first (left) image and the second 
(right) image of the photogrammetric model. As mentioned 
before the data sets are point sequences, Xl and Xr for left and 
right projected curves respectively. 
ICP steps: 
1. Compute the closest points: Yk=C(Pk,X) where C is 
an operator for finding the closest point between P 
and X. 
N 
square distance between the closest points found. 
(gk,dk)=Q(Po, Yk). 
3. Apply the registration: Py479k (Po). 
4. Stop the iteration when the change mean squares error 
small then threshold. 
with X being the model shape, P the data shape and Y 
representing the closest points found. In our case there is no 
model shape and data shape, both shapes are data changing with 
the refinement of the parameters (R.O.P + plane parameters). 
Point sequences are obtained by computing the planar curves as 
ng the perspective center of the 
intersections of cones havi 
images as their origin with image space curves and the plane. 
The plane equation is represented by 3 p 
ax+by+cz=1. Any point from the point sequence could be 
with the 
age plane, by scale 
computed by multiplying the vector, starting 
perspective center through the point from im 
factor. The scale factor can easily found using plane parameters. 
Full description of ICP algorithm can be seen in Besl and 
McKay work(1992). 
grammetry, Remote Sensing and Spatial Informati 
Compute the registration to minimize the sum of 
arameters for instance: 
on Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
X: 
Y, |- IB T O 
Z 
where R = rotation matrix 
T= displacement vector 
f = focal length 
Projecting all the points in both images through equation (1) 
yields the planar curves shown in figure 1. The Euclidean 
distance d(xp,l) between the point xp and the line segment | is 
computed using equation (2) where x1 and x2 are points 
determining the line segment |. 
  
  
  
with the first equation being a 3-D vector equation with 3-D 
point vectors xl, x2 and xp, and with the second equation 
corresponding to 2D image coordinates. 
The closest point xo is the point satisfies the equality d (xp. xo) 
-min d (xp, li) {i=1...n}. With the resultant corresponding point 
computed using least squares 
sets the registration is 
optimization. 
2.3 Experiments with synthetic curves 
Synthetic planar curves were projected from model/object space 
to the images using relative orientation parameters as exterior 
ones. ICP algorithm was performed for the recovery of the 
relative parameters of the photogrammetric model and the plane 
parameters. High sensitivity has been observed to initial values 
of the 3D plane parameters. 
The primary reason for these unsatisfying results and the 
numerical problems confronted is a possible high correlation 
between the plane parameters and the bending angles of the 
model images. The plane representation could lead to numerical 
problems because the multipliers of X and Y are nearly 
negligible compared to Z multiplier. We therefore suggest a 
different representation for the plane. 
n, sin 0 cos @ 
n, |-|sinOsino 
n, cos 0 
n((X — Xo)+n,(Ÿ — Yo)+n,(Z-Zo)=0 (3) 
nX+n,Y +n,Z = D 
where: O-angle from XY plane 
¢ -turning angle around Z axis 
n — unit vector of plane normal 
D - the distance of the plane from origin 
    
  
   
    
    
  
   
  
  
  
  
   
  
  
  
  
  
  
  
  
   
  
    
  
  
   
    
  
   
   
  
  
   
  
   
  
  
  
  
   
  
    
   
   
  
  
   
   
   
  
   
     
    
   
  
  
   
   
International Arch 
The last represent 
and D, but leads to 
0 = 0, the derivati 
not change the 
Therefore, when 
numeric problems 
3. FINDINC 
USING H 
3.1 Homography i 
Homography map| 
second image as if 
(Hartley 2000). As 
to correspondences 
model. In fact this 
8 parameters can 
parameters and the 
Figur 
The homography i 
(1982)), meaning t 
homography. The | 
homogenous coordi 
the other is unique t 
The homography m: 
and the mapping fr 
given by 
where Ur, and Ul ar 
Images respectively. 
One should notice tk 
to the left image an 
the one from right in