a=1 for a non homogeneous estimation 
or prediction (inhomBLUUE/P) 
TrTn_—! 
= uso for an homogeneous estimation or 
Hs B Qu HB 
prediction (homBLUUE/P) 
In the case pg = 0 formulae (4) and (5) can be simplified as: 
x=N"ATQ,™ 
(7) 
$- Q,B'Q, !HI 
In order to make the estimation of vector x more robust, espe- 
cially for the part relating to the coordinates of points to be 
updated by photogrammetric measurements, it is necessary to 
introduce some conditional equations in system (3): 
1+v=Ax+Bs 
(8) 
c=Dx 
where: 
D is the coefficient matrix of the conditional equations; 
c is the known vector of conditional equations. 
System (8) can be considered as a mixed linear model with 
constraints and the BLUU estimate of vector x can be obtained 
by (Crosilla and Visintini, 1996): 
&=N"[ATQ,"4-aBu,)+D'n] (9) 
where: 
Sina IT. nn lat n. 
n- [DN D'| [c- DN? ATQ," 1 oBy,)] 
Again in case pg = 0, formulae (9) can be simplified as: 
{= N^ ATQ, 1 + Dx] (10) 
where: 
A 
n=[DN"D"]" [e- DN^!ATQyl| 
3. CONSIDERATIONS ABOUT 
THE METHOD PROPOSED 
3.1. Photogrammetic conditional equations 
As was reported previously to make the estimation of vector X 
more robust, system (8) can take into account some conditional 
equations relating to the sub-system of the DLT. 
v2 [L2 L2 «145-04 «Lg «vos nonu) 
- (LiLg EE L5Ljo + LaL y + (LsLg + L6L40 + LyL;1 y = 0 
(11) 
v? Z (LL * LjLg € L4L7)(L9? - Lig? * L4?) 
— (L41Lo + LoL + L3Ly1 XLsLo + LeL10 + LyL;1 ) =0 
These two first conditional equations follow from the fact that 
among the eleven L; (j 2 1, ..., 11) DLT parameters just nine 
are independent, that is those relating to the three internal 
orientation and the six external orientation parameters for each 
image (Bopp and Krauss, 1978). 
196 
LiLgtLoLip*L4Li — 
2 2 = 
Lo? - Lyg? Lg? 
_ LsLo + L6L10 +L7L141 _ 
2 2 2 
Lo‘ +Lio“ +L411 
V3=X0- 0 
vV4=Yo 0 
2 2 2 
Vg =~C 2 x 2, LH E _9 
5 x 0 2 2 2 
Lo" +L" +L; 
2 2 2 
L 
2 .latLg tls <0 
2 
Vg —-—c, —yo + = 
à L9? - Li? «Ly? 
These four more conditional equations (Crosilla, Guerra and 
Visintini, 1993) report relationships joining Xo , yo , c, and 0 
to some functions of the L; parameters; they can be applied if 
the internal orientation parameters are exactly known. On the 
contrary, if the internal orientation parameters are not exactly 
known, they can be considered as sthocastic values 
characterized by a certain dispersion. 
Conditional equations such as (12) are particularly useful for 
the solution of the numerical problem, since they make it more 
robust also in case the values of xq, yo, c, and c, are vay 
approximate. As a value of x; and y, the coordinates of the 
image centers can be used with enough precision, while c, and 
c, can be substituted by a unique value given by the nomini 
focal lenght of the objective multiplied by the photo 
enlargement. In this way a sort of probability ellipses is 
identified in correspondence to the image center: the coefficient 
of radial distortion K, can be better estimated since it refers to 
a point of central simmetry defined with enough precision. 
32. Correct weighting of the observational equations 
The problem of the correct weighting of the observational 
equations, both cartographic and photogrammetric, represents à 
very important aspect within the practical application of modd 
(8). 
The method used is the so-called "danish method" (Krarup, 
Juhl and Kubik, 1980) which assigns a variable weight p; a 
each iteration, function of the correspondent standardized resi 
dual vs;, according to the following expression: 
.k 
hA (vs; ) = e 005 VS; (13) 
where: 
k is a coefficient equal to 4.4 in the first three iterations and 
equal to 3 in the following ones. 
The values so computed are used to find the diagonal matrix Q 
of the expression Qy = BQ,B" +Q. 
This method, however, should be used with a particural cat 
since it can cause divergent solutions. The analitycal model (8) 
is in fact correct only if the functions are linear or linearizable 
around a very approximative value. In this regard the approxi 
mative values of the unknowns are in general very far from the 
true values, for the reason described in the next paragraph 33 
The increments from such value estimated in the first iterations 
are therefore significantly different from zero, and for this 
reason the coefficient matrices A and B should contain also tie 
partial derivates of heigher order. 
Since the program implements only terms of first order, the 
estimation of the residual v by the relationship (6 
ÿ=1-AX-BS is wrong, because v is also function of ti 
heigher order terms not considered at all. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
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