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
