Plate system
Ground
System
Figure 3 : Diagram of the DLT Principle.
x+(x-x )rk +(r +2(x-x,) )p, +
LX+E V+EZ+E 3)
TXy- X—X =
(y, X JP; LX+L Y+L Z+1
yt(y-y,)k t. €Xy-y,) Jp. *
2 X LX+L Y+L Z+L
+ = X—X m5 sta pm mA RUE
( Yo p, L'X+L Y+L Z+1
with r 7 (x^ry?)'?
where:
e X and Y are the co-ordinates comparator of the image
points.
. xo and y, define the position of the center of the plate in the
comparator system.
X -Y 7]. 0g 0 0 0 x
0 0 0.0 -X.:Y -Z =I vx
Each object point known in ground co-ordinates lends itself to
two similar equations. A minimum of seven points is necessary
to solve the problem correctly, but to have a good determination
of the unknown parameters of the transformation, the number of
fulcrums must be higher than that which would be necessary,
which leads us to a resolution by the method of the least-
squares. It is worth pointing out that it is possible to use this
method together with the two conditions (colinearity and
coplanarity) for a more refined treatment [6].
2-4 Adjustment of blocks
In this type of work, one cannot genuinely speak of block
compensation, since the nature of the object is different from its
geographic space. However, there is a great similarity with the
aero triangulation considering the principle of the functional
model used, namely the beams method. In this situation where
one is concerned with in an exact definition of the object, the
total compensation principle (block adjustment) is based on an
installation through DLT calculation and a final compensation
using the external parameters. The observation relations used
must take into account all the perspectives representing the
subject.
Each beam gives place to fourteen parameters and each object-
point known gives two equations. It is thus possible to pose an
observation relations system between co-ordinates, parameters
and distances.
The compensation principle is based on the rigorous method [7]
which deals with the adjustment of the plate observations.
* K, ,p; and p, are the parameters of distortion of the
objective;
e X,Y, Z are the ground co-ordinates of the points
photographed
e L,through L,, are the unknown coefficients of the beam.
The following expressions can be deduced directly from the
system (3):
L,X + L,Y + L32 + L, - xAL, — XYL,0 -xZL,, —
-xr Ak sit +2x )Ap, -2VKAp ~X=1
LX LY #1. Z+ Lr yXL, =yYL y= Y71=
2 2 2 4
er Ak —(r 2x )JAD, -2yxAp —y-r «
where :
e A=L9X + LuY+LuZt+I1
® r,andr, y are residue errors in condition equations.
e IL, .., Lj are parameters drawn from spatial
transformation (scale factor, rotations and translations).
They are considered independent as far as DLT is
concerned.
e L;,, L,3, Ly4 are the distortion coefficients of the objective.
with Ak,=K", =L;2, Ap,=P'=L;3, Ap,=P'7=L;4 ; the system of
equations (4) can be reduced to the following matrix form :
xZ x (r +2x ) 2yx ZI I
YY V7 quac ya ac (ey) pe: I
3 - Treatment Processes
3.1 Preliminary treatments
These treatments include:
- The orientation of the photographies : (x,y) = R(x’,y’) with
[] rotation matrix of the photography which is written as :
x! cosa —sin a || X
8l : (5)
y sina cosa || y
- The observation treatment :
The observations in space image of the plate co-ordinates are
sullied with errors in dots since measurements are done
separately (non stereoscopic dots). In this case, it is necessary to
check the reliability of the observations by taking into account
the parallax equations (6).
PL. =x; —X; i=l+n (6)
The calculation reference is based on the following parameters:
* H, H, distance between i and the reference (0), figure (4).
e PL longitudinal linear parallax of the point of reference.
e AH, =H, — H,, difference in distance.
e AP], difference in parallaxes between the reference and
point I.
—112—
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