Transformations (Blais, 1983). Emphasis is placed on 
those areas of concern in general OLT relating to close- 
range, industrial photogrammetry. These include the 
issues of system response time, generation of 
approximate values, the compensation for systematic 
errors, blunder detection methodology and datum 
establishment. A procedure is suggested for sequentially 
updating the system obtained via a free-net adjustment. 
2.0 SEQUENTIAL ESTIMATION FOR ON-LINE 
TRIANGULATION 
The process of modifying least squares computations by 
updating either the normal equations matrix or its inverse 
has been used in control and signal processing for some 
time in the context of linear sequential filtering. Taking 
into consideration the sequential nature of the 
photogrammetric data collection process, the utilisation 
of sequential algorithms for OLT follows naturally. 
The appropriate simultaneous solution for  photo- 
triangulation is the well known bundle adjustment. All 
data must be available prior to the adjustment. 
Conversely, the sequential procedure builds the object in 
a stepwise fashion, proceeding image by image (or point 
by point) and incorporating data into the system as it is 
collected. 
The primary goal of OLT for aerial triangulation is to 
provide a clean data set for a final, rigorous simultaneous 
adjustment. This is achieved by accommodating blunder 
detection and re-measurement quickly within the 
measurement process. Observations are added to the 
system as they become available and deleted or replaced 
if found to be unacceptable. Sequential algorithms 
enhance this process by updating the system with new 
information without starting from scratch with the entire 
data set. In the aerial case, blunder detection takes 
precedence over the solution vector while in the VM 
application, the monitoring of object point precision 
throughout the measurement process assumes the 
highest priority. 
Notable sequential algorithms which have been examined 
for OLT include the Kalman filter, which updates the 
inverse of the normal equations matrix (Mikhail & 
Helmering, 1973), the "Triangular Factor Update" 
(Gruen, 1982) which updates the factorised normals 
directly, and Givens Transformations which can be used 
to update either the factorised normal equation system or 
its inverse. With respect to general least-squares, 
Givens Transformations possess certain advantages 
over other orthogonalisation techniques such as the 
Householder and Gram-Schmidt methods. (Gentlemen, 
1973; George & Heath, 1980). As applied to OLT, 
several studies have demonstrated the superiority of the 
Givens method over the Kalman filter and "Triangular 
Factor Update" algorithms (Wyatt, 1982; Runge, 1987; 
Holm, 1989). 
Givens Transformations are based on the use of plane 
rotations to annihilate matrix elements. This approach, 
compatible with the Cholesky method, provides a direct 
method for solving linear least-squares problems without 
134 
forming the normal equations. Because all updating is 
done in the factorised normals, numerical instabilities 
associated with forming and solving the normal matrix 
are avoided. Only one row of the design matrix is 
processed at a time, making it ideal for sequentially 
adding or deleting observations in an on-line 
environment. If necessary, the solution vector can be 
obtained at any stage of the process by back 
substitution. The method can easily accomodate 
weighted observations and parameters. A version of 
Givens Transformations presented in Gentleman (1973) 
avoids the computation of square roots, reduces the 
number of required multiplications, and facilitates 
weighted least-squares. This and similar “fast” recursive 
algorithms are gaining favour in the area of parallel 
processing due to the absence of the square root 
operation (Hsieh et al, 1993). Additionally, the ability to 
yield a solution in the absence of a positive definite 
system may prove to be advantageous for the update of 
systems encountered in the free-net adjustment of close- 
range photogrammetric networks. This square root free 
version of Givens is stressed in this study. 
2.1 Least-Squares with Orthogonal Transformations 
First, the use of orthogonal transformations for standard 
least-squares estimation with the familiar Gauss-Markov 
model is illustrated. Given an n x 1 observation vector / 
and an m x n design matrix A such that m 2 n, the goal is 
to determine the n x 1 parameter vector x in such a way 
as to minimise the sum of the squares of the elements of 
the m x 1 residual vector v which is defined by 
y 2 Ax - I. (1) 
Initially considering only unweighted observations, the 
solution is given by 
£=(A"A) ATI @) 
This solution may be obtained with the Cholesky 
factorisation ATA = U"U, where U is an upper triangular 
matrix, or with the related factorisation A"A = U'DU, 
where U is unit upper triangular and D is diagonal. The 
only significant difference between the two is that the 
former uses square roots and the latter does not. 
Applying Cholesky to the normals and to the right hand 
side results in the system 
U“UX=b (3) 
With d = (U M , the system reduces to 
Ux=d (4) 
which is solved by back substitution. 
If the decomposition A - QR is available, where Q is an m 
x n matrix with orthonormal columns and R is an nx n 
upper triangular matrix, the normal equations may be 
written as 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
R"Q" ORÈ = 
Q is orthogo 
nonsingular if . 
Ri=0"1 
Equations 4 ar 
ol. 
R and d are Ol 
transformation 
system may E 
equation matri: 
Extending the : 
P (assuming 
squares solutic 
x= (AT PA) 
Because P is « 
P^ and the | 
modified desig 
Now assume 
represents the 
addition, delet 
orthogonal tra 
closely that c 
observation eq 
unknown para 
stage k 
U [x 
a n7 [9s 
T 
Here, au, rep 
observation, x 
is the right ha 
The total numt 
series of n c 
Givens Transf 
Q = Q, Q, 
to Eq. 8 yields 
and for the rigt