distortion of the 
; of the points 
its of the beam. 
directly from the 
ZL:- 
equations. 
n from spatial 
and translations). 
far as DLT is 
: of the objective. 
14 ; the system of 
matrix form : 
y) = R(x’,y’) with 
ich is written as : 
| (5) 
e co-ordinates are 
ements are done 
>, it is necessary to 
aking into account 
(6) 
wing parameters: 
ce (0), figure (4). 
jyint of reference. 
reference and 
  
(reference 0) 
Figure 4 : General diagram of the parallax 
1 1 B.c. AH 
APL, =PL, =PL, = Be (— -—) = — — (7) 
H : H HH 0 
0 
By developing (7) above , we could get: 
H,.APL AH. - BcAH 
which gives : AH (Bc+H .APL )-H,.APL =0 
that is to say a function of the type F( AHi,APLi) = 0. 
The total differential (dF=0) leads to the matrix equation AX=0 
where : 
€ t 
zl A B |;X- E ; consequently : X -['A4] i 
Vector X represents the oblique of measurements in distance 
and parallaxes. 
3-2- DLT treatment 
DLT treatment must be made beam by beam. Stages of this 
process include: 
1. Calculation of the beam parameters. 
2. Calculation of the distortion factors. 
3. Following beam. 
4. Calculation of the new points. 
The distortion of the objective concerns the radius and the 
revolution and the parameters vary in a linear way. A linear 
interpolation can be carried out to assess the distortion 
parameters relating to the control points and to the new points. 
Thus for each dr, - (r,, — 1) there is a corresponding increase 
dk,G) - k, G1) -k,G) which involves for a dr = (r-r,) increase 
y, Ln G*D-k D) 
and hence, the value of k, corresponding to the radial distance R 
is: 
p; 7 pi) * dp, 
p» 7 p; (3) * dp, 
the verification constraints being expressed as [8] : 
(15 «D «12 )-q) «D e2)s(C^s BAyDZ)s0 (& 
A-(BC/D)=0 
withy A=LL +L L +L L ; 
3 
, 
BELLE «LL xL. 
C=L.L, +e +L L, > 
, 
X2 2 2 
D=L, +L{0 Lj, 
However, if the twenty eight coefficients of the two 
homologous perspective beams are determined, it is possible 
then to determine the objects co-ordinates of any point common 
to these two beams, considering that for each point, there will 
be the following system : 
P.W-Q=0 (9) 
Where : 
a a a 
11 21 31 bi, 
a a a x b 
P= a a 3 [WS y]: le 
i 2 b, 
32 Z 21 
a a a Ds 
We have here a problem with three unknown (3) factors which 
can be solved via the least-squares method (Cholevski method) 
wherein W indicates the vector of object co-ordinates (X, Y, Z). 
Furthermore, this equation can be adapted through the MGPC 
(Multi Geometrical Photo Constraint) (9) for a better analysis 
(camera / object). 
3-3- Total Compensation 
The objective is to establish an observation relation bringing 
together the point, the beam and external data such as the 
distances measured. 
P,Li - W «0 (10) 
RESI. zA o 0 0.9 0 xX 
0 0 0 0 -X -Y -Z -1 yX 
: point i in beam j; wj : plate co-ordinates of point i in beam j 
* foreach beam there are Nx2 equations; 
* form beam one obtains m xNx 2 equations; 
X,Y, x4; Xn (r^ 2x7), 2yx; 
YiY 
14 
; MA y 2yXi (r^ 4 2y^), 
® the unknown factors will be Li parameters of each beam, i.e. 14x m; 
—113—