International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
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Figure 1 Sequence, stereo double matching restraint 
4. THREE-DIMENSIONAL MOTION 
PARAMETER DETERMINATION 
4.1The basic methods of three-dimensional moving 
parameter estimation in solid sequential images 
According to dynamics and the space analytic geometry theory, 
rigid body movement in the three-dimensional space can be 
decomposed into rotation and translation. Supposes that the 
three dimensional coordinates of random feature points p at 
time t on moving rigid body is (X; isi jr after time Af it 
moves to feature P whose three-dimensional coordinate is 
(x.y 2 Y 0 =1,2,-+-,n) , and the corresponding 
relation of P and D is as the following rigid body moving 
equation: 
Ps RE al G=12,-n) (10) 
Where R is a 3 X 3 rotational matrix, T is translation vector 
named 7 — (Ax, Ay, Az)” .Thus the rotational matrix R 
and translation. vector T of the above equation can be 
determined from sequential images. According to the equation 
(10), suppose there are three feature corresponding points at 
time t and time t+ AZ on the object, then has: 
dn Bo pee n (11) 
P,-P, P,-P 
The equation (11) is a linear equation about the R, the condition 
that has the unique solution is that the rank of coefficient matrix 
is less than 2, in a nother word P,-P, and P,-P; must be not co- 
line .It means that the three-dimensional moving parameter can 
be determined uniquely so long as more than three uncoline 
three-dimensional feature points on the object are obtained. In 
order to guarantee the precision and the computation speed, the 
There into: 
actual computation adopts the method of center of gravity to 
the three dimensional feature point coordinates. Supposes the 
gravity we T the three dimensional feature points 
is 
set {P, } and P. 
i 
ne pe! (12) 
Thus: PB A RP E=12 0): 043) 
HR 
After Obtaining rotational matrix R, translational vector can be 
determined as follow: P 
T=P —RP (14) 
4.2 The solution of rotational matrix R 
As revolved above matrix R is a 3x3 orthogonal matrix, 9 
components of R are calculated directly, but the algorithm is 
complex, the solution is not orthogonal, the reliability is poor. 
Because there are only 3 independent variables in the rotational 
matrix, by selecting these 3 independent variable for 
computation, no doubt it will reduce computation quantity, and 
enhance the reliability of algorithm. By applying this linear 
algorithm based on skew-symmetry matrix decomposition to 
determine three-dimensional rotational matrix parameters, not 
only computation. quantity is reduced, but also linear 
computation is realized. 
As Cayley theorem, a three-dimensional orthogonal matrix Rif 
satisfy L-R is full rank, it is can be decomposed skew- 
symmetry matrix S and unit matrix I uniquely, that is : 
Rz xS Sw (15) 
    
  
  
  
   
  
   
  
   
   
    
   
   
  
  
   
  
  
  
    
    
      
     
   
   
   
  
     
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