Full text: XVIIth ISPRS Congress (Part B3)

  
  
In both cases of collinearity equations (l-a,l-b) 
these projective equations may be linearized by 
first-order Taylor expansion to the general form 
(Ackermann, 1975,1981; Barbalata, 1979; Fraser, 
1980): 
  
  
Ÿ+BÂ=E 
where: 
v B B E 
AS 
$2151, &—]|8. 0l, 5714] , E -lc (2) 
v. 0 -1 E 
V is the vector of residuals (v from collinearity 
equations,V from constraints equations and V from 
observation equations arising from a priori know- 
ledge regarding the object ground coordinates), 
B are the matrices defined by Jacobians, 
Ais the correction vector of projective  parame- 
ters (^) and object ground coordinates (^). 
A o — 
ZE = Diag. (>, >, 3 ) is the covariance matrix 
associated with the merged obseryation equations 
(= for collinearity equations, 2. for constraint 
equations and for suplementary equations concer- 
ning the object ground coordinates, where their 
independence is assumed). 
The corresponding normal equations will be: 
= Oo 
C+C 
-—i. (3) 
In actual computation the vector/A^in (3) is first 
solved from a set of reduced normal equations and 
then the vectors 4. are solved for one at a time, 
(Helmering, 1977; Grün, 1982), respectively : 
A= [v Rc RE] [c - 8 Gin (GE) 
oO ^ 
N+N N 
  
  
= (4) 
dx GU 7 Coa SES) - (NH) TNA 
An iterative procedure is used; the iteration_is 
stopped when the corrections in matrices / and ^ 
become negligible small. 
A computer program called PROJECT 5 was develop- 
ped by the author and its formulation is based on 
the principles of the above paragraph (Barbalata, 
1980b). 
3. ERROR PROPAGATION 
Since the unknown coordinates of control points 
were carried in the present solution, the error 
propagation associated with these also emerges as 
a by-product of the solution. 
From the partition employed in (3) it follows 
that the inversion of the general coefficient ma- 
trix N of the normal equations can be accompli- 
shed by the method of submatrices, i.e.: 
Oo ^ —1 A 
N + N N Q 
^ d. Le (5) 
NT NEM T $4 
e» x 
According to the fundamental definition of the 
variance-covariance matrix, the covariance matrix 
of all the triangulated points may be simply ex- 
pressed as: 
x -02à (6) 
640 
where:G 2 is the unit variance estimated from the 
adjustment and Q is the cofactor matrix of all 
the triangulated coordinates. 
From (5) it may be shown easily that the matrix Q 
is given by: 
Q = —1 IST 
Zi 
=> 
(7) 
+N N'TQNN! 
man. land 
+8 FRY] 
nz" 
(8) 
where Q = 
It is seen from (7) that the cofactor matrix Q 
may be expressed as: 
a- 4 u (9) 
in which : U » N- RT QR N-! (10) 
The evaluation of the entire cofactor matrix Q 
presents some difficulties. However if one is not 
interested in the correlations existing between 
coordinates of different triangulated points, at- 
tention may be confined to the evaluation of the 
cofactor matrix Qj of the coordinates of the in- 
dividual point j. 
For the j-th control point, the cofactor matrix Q 
would be: 
Qj 7» Np le Ns NT o ss! (11) 
and the covariance matrix : 
vx Ixy Óxz 
Si=0z Ÿ Óvz| =; 
7x (zy G7 
(12) 
4. ERROR ELLIPSOIDS 
Error ellipsoids are used to evaluate the accura- 
cy of photogrammetric determination of position 
in three-dimensional space (Mikhail, 1976; Wong, 
1975,71976). 
Although least squares theory of adjustment does 
not require a specified distribution, most of the 
statistical testing following the adjustment is 
concerned with multinormal distribution of random 
vectors with density function: 
1 (Xp ) TST (X=py) 
f(x X aa eX )= ———— Tin DM Zn exp - ————————-—-———-- 
1° 2° x. 
"^ Qn" 2 
with mean vector p, and covariance matrix 2. 
T 
Referring to equation (13), the function (X-p,) 
Sl(X-p,), which is a positiv definite quadratic 
form, represents a family of hyperellipsoids of 
constant probability. 
In the case of three dimensions, the density func 
tion becomes: 
  
! (x1x2X3"02) TS! (x1x9X3-5) 
f(xi(x9x4)" ---z295 exp -- = 
OS 2 
(14) 
and the ellipsoid equation for the j-th point, 
assuming for simplicity n.70, becomes: 
  
  
X 
1% 
Hip (X,Y,Z);“ XY z); E | Ko (15) 
213 
PAE 2 Sled mb 1 
m^ rr bd AA dM
	        
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