Full text: XVIIth ISPRS Congress (Part B5)

   
    
   
   
   
   
  
    
  
  
  
    
   
   
   
  
   
   
     
    
  
  
   
   
   
   
     
   
   
   
  
  
  
    
  
  
  
   
   
  
   
   
  
  
  
  
    
   
   
  
   
  
    
    
  
  
   
      
   
  
     
   
   
  
   
   
    
     
: projector coordinates from the grid 
codes, with yO0z-d2 for all the 
points, where d2 is the projector 
focal distance 
If the coordinates x2,y2,z2 are substituted in eqs. 
(6) by their expressions functions of x0, y0O, z0 and 
of the angles of rotation around the axis, 
epsl,eps2,eps3 , it can be written: 
gaml = N*(y1*z0-z1*y0-eps1* (y1*y0+z1*z0)- 
eps2*y1*x0+eps3*z1*x0) 
gam2 = N*(z1*x0-x1*z0+eps1* 
x1*y0+eps2* (z1*z0+x1*x0) +eps3*z1*y0) 
gam3 = N*(x1*y0-y1*x0+eps1*x1*z0-eps2*y1*z0- 
eps3* (x1*x0*y1*y0)) 
(7) 
In the eqs. (7) it is assumed that the angles 
eps1/2/3 are small enough to allow a 1 st order 
approximation in the transformation equations of the 
X2 coordinates by a rotation around the origin. The 
variables  eps1/2/3 are then small variations of the 
3 angolar variables omega, k, fi: 
omega: rotation around x axis 
k : rotation around y axis 
fi : rotation around z axis 
Let be: 
GAMMA = GAMMO+delta GAMMA 
GAMMO = (gami0,gam20,gam30) 
XP = XPO +delta XP 
then: 
XPO = (1,-f2y,h2) 
delta XP - (0,delta y,delta z) (9) 
GAMMO = N*((y1*z0-z1*y0), (z1*x0- 
x1*z0), (x1*y0-y1*x0)) 
delta GAMMA = N*((-eps1i* (y1*y0+z1*z0) - 
eps2* (y1*x0)+eps3*z1*x0), 
(eps1*x1*y0+eps2* (z21*z0+x1*x0) 
+eps3*z1*y0), 
(eps1*x1*z0- 
eps2*y1*z0-eps3*(x1*x0+y1*y0))) 
For each point -i- : 
d (i) ABS (GAMMA*XP) 
ABS (GAMMO*XPO+delta GAMMA*XPO+GAMMO* 
delta XP) (10) 
Products in equation (10) can be written explicitly: 
GAMMO*XPO = 1*gam10-f2y*gam20+hz*gam30 
delta GAMMA*XPO=(eps1* (-1* (y1*y0+z1*z0)- 
f2y*x1*y0+hz*x1#z0)+ 
eps2* (-1*y1*x0- 
f2y*(z1*z0+x1*x0)-hz**y1*z0)+ 
eps3*(1*z1*x0-f2y*z1*y0- 
hz* (x1*x0+y1*y0))) *N 
{11) 
The scalar product: delta GAMMA*XPO , can be written 
in a concise form: 
  
delta GAMMA*XPO=eps1*A(1,i) 
+eps2*A(2,i)+eps3*A(3,i) 
where i: point index 
The last term in equation (10) can be written: 
GAMMO*delta XP=gam20*delta y + gam30*delta z 
Let 
A(4,i) = gam20 
A(5,i) = gam30 
eps (1) = epsl 
eps (2) = eps2 
eps (3) = eps3 
eps (4) = delta y 
eps (5) = delta z 
then 
f = SUMMAT(i) d(i)**2 
= SUMMAT (i) (GAMMO(i)*XPO(i)* 
SUMMAT (k) A(k,1i)*eps(k))**2 (12) 
and 
d f/ d eps(k) = 2*(SUMMAT(i) A(k,i)*d(i)) (13) 
where d f/d eps(k) means the partial derivative 
The unknowns eps(k) can be derived by setting to 
zero the partial derivatives (13), which are a 
system of 5 linear equations in 5 unknowns. 
In the CCP program the equations described here are 
inserted in an iterative Scheme, where the 
rototranslations eps(k) are treated as perturbations 
to the the projector position. At each iteration the 
new projector coordinates X2 become the initial 
projector coordinates X0 for the new iteration, 
until convergence is reached. 
REFERENCES. 
BOOKS. 
American Society for Photogrammetry and Remote 
Sensing 1989. Non-Topographic Photogrammetry. 2 nd 
edition. 
GREY LITERATURE. 
American Petroleum Institute, 1991. Recommended 
practice for planning, designing, and constructing 
fixed offshore platforms RP2A. 19th Edition. 
L. Azzarelli, E. Baj, M Chimenti, 1984. 
Preprocessing procedures for automatic plotting. In: 
Int. Arch. Photogramm., Vol XXV Part A2 Commis. II. 
E. Baj, 1988. Prototype of a metric projector. In: 
Int. Arch. Photogramm. Remote Sensing, Vol.27 Part. 
B5 Commis. V, pp 24-31. 
E. Baj, G. Bozzolato, 1986. On line restitution in 
biostereometrics using one photogram and a metric 
projector. In: Int. Arch. Photogramm. Remote 
sensing, Ottawa-Canada, Vol.26 Part 5, pp. 271-278. 
     
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