À 
last dps position 
Figure 1: A Strip of Stereo-pairs Formed by Sequential 
Triangulation to Bridge Over Areas Without 
Satellite Signals. 
This solution is normally obtained by first 
factorizing the normals(ATA) into the product of a 
lower triangular(L) and upper triangular matrix(U) 
resulting in the system 
LU. =b. (3) 
By letting d - L^" b, the system becomes 
UR=d (4) 
and can be solved by a simple back substitution. 
If, however, the decomposition 
À = OR (5) 
is available, where Q is am x n matrix whose columns 
are orthonormal and R is a n x n upper triangular 
matrix, the normal equations can be written as 
RTQTQR£-RTQTI. (6) 
Since Q is orthogonal, Q"Q - I and additionally, 
since R is nonsingular if A"A is nonsingular, we get 
R2-97]1. (7) 
One can see that (4) and (7) are equivalent where 
U = R and d = Q'1l. The solution to this system can 
be obtained without forming the normal equation 
matrix(A"A), thus avoiding the instabilities 
associated with its formation. 
As only d is needed for the solution, Q is not 
explicitly required [Gentleman,1973]. Obtaining R 
and d is a matter of applying a series of Givens 
Transformations to A and 1. 
If the design matrix A is associated with a weight 
matrix P, the solution is given by 
2e (ATPA ATP]. (8) 
For uncorrelated observations P is a diagonal matrix 
and the design matrix A can be premultiplied by P*. 
The OR decomposition is then applied to this modified 
design matrix. If the observations are correlated, P 
is fully populated and positive definite, and can be 
factorized by the Cholesky method into the product of 
a lower and upper triangular matrix, 
P=LLT=UTU. (9) 
In this case matrix A is premultiplied by U before 
the QR decomposition is performed. 
Sequential Estimation 
In many photogrammetric applications, new 
measurements must be added to a system once a 
solution is computed. In such a case it is of 
advantage to directly update the reduced normal 
equation matrix R and avoid a full solution of the 
new system. Therefore, operations are needed to add, 
delete, or replace observations or to impose 
constraints. All of these operations can be based on 
Givens transformations, as explained below. The 
development below was proposed by [Gruen, 1985]. 
At stage k-1 the reduced system takes the form of 
(4). The addition of one observation equation 
including a set of new unknown parameters leads to 
the following form(stage k) 
R x d 
iz --- (10) 
T 
a‘wl LY lu 
where a",, is the new coefficient vector, y is the 
new parameter vector of length p, and l,, is the 
right hand side of the new observation equations. 
Applying a series of n (number of total system 
parameters) Givens Transformations 
Qs On Oi e Qi (11) 
to (10) gives 
R Oln-p  |R; in 
gl-9-—1p- — (12) 
a’ 11 0 11 
dlin-p d | }n 
ol 0 |lp = . (13) 
The updated solution vector is found by 
backsubstituting into