: 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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