corresponding elements of the coefficient vector as 
follows: 
We Ha cn (16) 
Each observation vector is rotated through U, row by row 
until each of its elements is transformed to zero. The 
additional element Q of the right hand side d vector 
maintains the root residual sum of squares and is 
updated with Givens Transformations along with U and d. 
The alternative square root free implementation of 
Givens Transformations used in this study involves 
finding a diagonal matrix D and a unit upper triangular 
matrix U such that 
U= DU (17) 
1 
A row of the product D?U is rotated with a scaled row of 
A (Gentleman, 1973), 
gea — das 
0-- 03a, = YBa, + 
where d is the diagonal element of the matrix D and ô is 
the scale factor for the coefficient vector, initially set to 
one. After one rotation, the newly transformed rows are 
(18) 
Q3, . s au 
099. ond JET a9) 
where 
d’ =d+0a’ 
3’ =dB/d’ 
€ » d/d' 
$26, /d. 
^ — 
RS —— 
Weighted least-squares is simplified with this method by 
initialising the scale factor 5 to the weight instead of to 
one. Introducing an observation several times with 
various positive and negative weights is equivalent to 
introducing it once with the sum of the weights. Thus an 
observation can be removed by reintroducing it with the 
negative of its original weight. 
3.0 OLT FOR SINGLE-SENSOR VISION METROLOGY 
In this section important concerns in OLT with respect to 
close-range, convergent photogrammetry are highlighted. 
These include system response time, approximate 
values, compensation for systematic errors, blunder 
detection, and appropriate datum. 
3.1 System Response Time 
Response time is critical in on-line VM applications and 
particularly so in the industrial environment where 
inspection costs are directly influenced by the extent of 
site disruption. Although significant, improvements in 
computer hardware should not curb the search for 
efficient algorithmic solutions. Sequential techniques 
such as Givens Transformations improve response time 
but efficiency is also affected by the size of the system 
which is in turn dependent on the number of active 
parameters. 
Ignoring self-calibration, phototriangulation involves six 
exterior orientation parameters for each photo and three 
coordinate parameters for each object point. Consider a 
system involving m photos and n object points. In the 
standard formulation of the bundle adjustment, object 
point parameters are eliminated, leaving a 6m x 6m 
system of orientation parameters. In aerial 
photogrammetry, this system is still too large to yield 
permissible OLT response times. The normal case 
geometry of the aerial network permits the use of sub- 
blocks of photos in the on-line procedure. The sub-block 
must be of sufficient size to provide reliability and yet be 
small enough to yield adequate response times. Gruen 
(1981) recommended the use of a 3 x 3 sub-block of 
photos with both 60% overlap and sidelap. 
The irregularity of convergent, close-range networks does 
not offer such a straightforward answer to effectively deal 
with system size. While image sensor parameters are 
less than point parameters (6m < 3n), the elimination of 
point parameters is the optimum solution. The typical 
inspection for a single-sensor VM system will likely 
involve only 50-100 points. As previously mentioned 
however, a sizeable number of exposures may be needed 
to achieve a desired level of accuracy. Using a 50 point 
inspection as an example, as we collect in excess of 25 
exposures, the number of sensor parameters begins to 
exceed the number of point parameters. From this point 
on, a reversal of the standard bundle in which photo 
parameters are eliminated will certainly provide a faster 
response. This approach would also be useful for the 
standard simultaneous adjustment. The incorporation of 
inner constraints for a free-net adjustment and the use of 
additional parameters for self-calibration may also be 
simplified. The most efficient solution is to incorporate 
both elimination techniques into the OLT procedure. It is 
little effort to compare the current number of sensor and 
point parameters to determine which to eliminate. 
Efficiency is also enhanced through the exploitation of 
the sparsity patterns of the reduced normal equation 
system. A special matrix storage technique described in 
Gruen (1982) for the Triangular Factor Update and also 
utilised by Runge (1987) for standard Givens 
Transformations is modified here to accommodate the 
elimination of image sensor parameters as discussed 
above. This technique, when combined with Givens, 
facilitates the direct updating of the reduced normal 
equations. A representative example for six object points 
is shown in Figure 2. 
136 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
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