m the 
f all 
rix Q 
(7) 
(8) 
rix Q 
(9) 
(10) 
rix Q 
not 
tween 
, at- 
f the 
e in- 
rix Q 
(11) 
(12) 
cura- 
ition 
Wong, 
does 
f the 
ce is 
andom 
(13) 
T 
Lou 
ratic 
3 Of 
func- 
ix) 
(14) 
int, 
15) 
The expression Hip(X, Y,2) j-K? from equation (15), 
for a specific value of K, is an ellipsoid boun- 
ded by a parallelipiped with the dimensions 2KÜy 
the direction of X, 2K(0y in the direction of Y 
and 2K0, in the direction of Z. Such an ellipsoid 
is known as an ellipsoid of constant probability 
and the value of such probability depends on the 
selected value of K. 
In the shifted coordinate system X',Y' ;2', consi- 
dering the ellipsoid with K=1 as a representativ 
one, the standard ellipsoid is obtained: 
x'2 Y'2 712 
  
+ = + = | (16) 
dx ox gà 
the axes of the ellipsoid coincide with the axes 
of a rotated coordinate system. 
The system X',Y',Z' is obtained from the original 
system X,Y,Z by three rotations w, f, X about the 
X',Y',Z' axes respectively. With respect to this 
new system, it becomes clear that Gy:7 a, Gyr = b, 
(7v= c and the new random variables X' Y' D are 
uncorrelated, such thatOy:y: 7Oyi7 7 0,1 7:70. 
Consequently, through the transformation from X; 
Y,Z to X',Y',Z' it was possible to replace a set 
o three dimensions of correlated random varia- 
bles by another set of the same dimensions of un- 
correlated random variables. 
The transformation matrix can be constructed by 
determining the normalized eigenvectors of the 
covariance matrix and using them as columns. This 
is equivalent to diagonalizing the covariance ma- 
trix 5., since the resulting covariance matrix of 
the uncorrelated set of random variables X',Y',Z' 
is always in diagonal form. The elements of this 
diagonal matrix are the eigenvalues A 15 92, À3 of 
the original covariance matrix S and the values 
of the semiaxes a, 5b, c, of the ellipsoid can be 
readily shown to be the square roots of the ei- 
genvalues Ar» A»; Az 
In actual computations, the semiaxes a, b, c of 
error ellipsoid are determined by diagonalizing 
the covariance matrix =; ; respectively: 
as 
"RIS 
(17) 
pc 0.20.1020 0,1 A. 9.0 
0 Cy 0 0 »2 0 = 0 A2 5 | 
0 
0 0 Gy o 9 2 d 
where R is an orthogonal matrix whose columns are 
the normalized eigenvectors of the matrix >-; M 
à», À3 are the eigenvalues of the matrix s and 
the X',Y',Z' is a rotated coordinate system uch 
that the random variables in the directions of 
its axes are uncorrelated. 
The computational flow itself, used to evaluate 
the accuracy of photogrammetric determination of 
position in threedimensional space, is actually 
quite short and consists in the following major 
steps : 
Let 5; be a covariance matrix of the coordinates 
of the triangulated j point obtained from the 
least squares adjustment (12). Furthermore, let 
the eigenvalue equation be defined by the charac- 
teristic matrix equation: 
(5 - AI)x = 0 (18) 
which represents a set of homogeneous linear equ- 
ations, 
641 
The characteristics equation of the covariance 
matrix 2;is evaluated as: 
2 2 2 
eod C Oxv| Py Gyz| |x Gxzp _ 
(=A)z +Tr (= NS - 2|* + 2 =0 
3 yx 714 Poy & | |%x 67 3 
(19) 
where : 
Tr( 2) 04 qz- trace of the covariance matrix 
[Ej toni of the =; covariance matrix 
Since the matrix S; 02) is symmetric, then the 
eigenvalues are real and the eigenvectors are all 
mutually yt that is: 
XIX. -X X-0 (20) 
Let the three eigenvectors of the covariance ma- 
trix 2 be denoted as XA,,; XA2> XA3,respectively: 
X4 Xi X 1 
Xa at X2 bh. Fags 1% of XA, = |X 5 (21) 
X X 
3 3 3 
A Az ^s 
For an eigenvalue A;, (i=1,2,3), we solve the set 
of homogeneous linear equations: 
gx À; Üxy Üxz XI 0 
Ex Ay Oy; lie] «lo (22) 
Üzx Üzy 027 =A; X, A 0 
The eigenvalues and eigenvectors for the matrix Z; 
are evaluated by an iterative procedure which con 
sists in the following steps: 
A. From Eq.(22), for i71 it results: 
=6Gû 
a) Aykı 76. ox? CO S +02 5 
c) ÀX3 "Üzy X4 *GzyX) + 0x3 
At the first iteration, for X) x yx. 
from Eq.(23c) we obtain: 
A= Try + Po (24) 
At the second iteration the computed value of 
can then be substituted into Eqs.(23a,b) to solve 
for Xy <> X9 and Aq as below: 
1 
X 9T RA, Oso r1 D 
1 
X= SA Trust 6 + 0 yz) (25) 
=0 x + X +02 
M zx | zy 2 07 
The iterations are stopped after the solutions 
have been checked to be convergent. At the end of 
the k-th iteration, the solutions are: 
1 
1,k 1,k=1 (1,%~1) 79 
1 
1 
xl ‚K LA Gu ne fux M 02 
(26)