)LE
Givens
tion in
ompre-
0852),
ng our
set up
ration.
ird and
resent-
e] most
ed esti-
nsion n
ector x
(1)
> u and
1k-defi-
nk defi-
:nerally
bserva-
regular
'ank de-
19852).
nverses
. 50 ff).
>xpecta-
t
(2a)
(2b)
(2c)
as unbi-
y means
(3a)
(3b)
(3c)
triangu-
hose the
igidity.
n as
(4a)
where
x is the vector of object point coordinates,
t is the vector of orientation elements,
A, and A» are the associated design matrices; and
e, l, and P are the true error vector, constant vector,
and weight matrix for image point observations,
respectively.
x and t are considered here as unconstrained (free) param-
eters. If observations are available for some or all of the
object point coordinates, a second system of observation
equations is added, that is
ve, mIr-4 ; P. (4b)
c
Similarly, observations for the orientation elements would
add
ze, = Il, P, (4c)
The least squares principle, applied to Equations (4a),
(4b), (4c) leads to the combined minimum
vIPy 4 vIP,v t VIP. v, — Min. (5)
e cc
For the purpose of simplicity and without loss of generali-
ty, we will operate in the following derivations only with
the reduced minimum principle
vPy > Min,
that is, we will consider only Equation (4a) as observation
equations.
The resulting normal equations are of the form
dCi
i NE Nt l,
with
N,,- AIPA, ,. 1, - AIPI
N, = A'PA, , L = AIPI
Nu = A,PA, .
N is further assumed to be regular. In an off-line environ-
ment Equation (6) is usually solved by applying Gauss or
Cholesky factorization. The former can formally be de-
scribed as a LU factorization, decomposing N into a prod-
uct of lower and upper triangular matrices L and U, i.e.,
eel t
or, with L = UTD (D is a diagonal matrix), in thc alter-
nate formulation
Hb
> >
d'u)
After the reduction of the right hand side, the solution
vector is computed from
Hit t
by back-substitution.
>
In photogrammetric triangulation the factorization is usu-
ally done as a stepwise procedure, stopping the reduction
of N right before it enters what was originally the N,, ma-
trix. This procedure leads to the pre-reduced normals N,,
ie.
Nat = In, (10)
with N, 2 N,—NLN,IN , and
Ip = NN,
N is finally factorized to an upper triangle N,, and t is
obtained by back-substitution from
The mechanization of this off-line factorization algorithm
takes advantage of the fact that N,, is a block-diagonal
matrix with 3 x 3 submatrices along the diagonal. There-
fore, the reduction of the point coordinates can be done on
a "point by point" basis, leaving the structure of the N,,
and N,, matrices unchanged, i.e., producing no new fill-
ins in those matrices. This particular feature, based on the
structure of N,,, is the key to a successful application of
the Triangular Factor Update technique in on-line triangu-
lation.
Assuming a sequential process and interpreting Equation
(4a) as the status of the measurement system at stage k-1
of the process, we get the following system if one or more
image coordinate observations are added, including new
parameters x,y and fy:
=e =A, Xt Ant ls P
x t (12)
—em =A +A — ar P
() ^ A19 2 (h) à Pa
E E] e
The updated normal equations of the stage k are of the
form
44 = : (13)
[ L
and
