It is well known that increasing the number of photographs at 
each camera station will increase the accuracy of 3-D co- 
ordinates of the object points measured in photogrammetry. 
Table 4.3 illustrates the results of the simulation test with six 
camera stations and 200 targets. When the number of 
photographs increase, the standard errors for x, y and z decrease 
and they are inversely proportional to the square root of the 
number of photographs as reported by Fraser (1992). 
  
  
  
  
  
  
Number of c, (mm) Gy (mm) c, (mm) 
photographs 
1 0.04686 0.04686 0.05752 
2 0.03313 0.03313 0.04067 
4 0.02343 0.02343 0.02876 
6 0.01913 0.01913 0.02348 
8 0.01657 0.01657 0.02034 
k 0.04686k ^ 0.04686k 0.05752k 
  
  
  
  
  
  
  
Table 4.3 Number of targets = 200 co, = 0.001 (mm) a = 90° 
Changing the network geometry gives different accuracy for 
estimated 3-D co-ordinates. Table 4.4 and Fig. 4.2 illustrates 
the influence of network geometry on the accuracy of 3-D co- 
ordinates by changing the convergent angle a. A large angle 
will cause the accuracy to worsen in x and y, and get better in z. 
It can be seen approximately 110? will give the best accuracy 
for x, y and z (RMS values) and that angles between 100° and 
120? are reasonable. The q-value is equal to 0.5 in this situation 
as reported by Fraser (1984). 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
aO c, (mm) Sy (mm) o; (mm) -Sj,(mm) 
60 0.04351 0.04351 0.08145 0.05893 
80 0.04558 0.04559 0.06330 0.05217 
100 0.04826 0.04827 0.05307 0.04992 
108 0.04943 0.04944 0.05023 0.04970 
110 0.04973 0.04974 0.04960 0.04969 
112 0.05004 0.05005 0.04901 0.04970 
120 0.05127 0.05128 0.04690 0.04986 
140 0.05420 0.05421 0.04317 0.05080 
160 0.05642 0.05643 0.04117 0.05184 
  
  
Table 4.4 Number of targets = 200 Number of cameras = 6 
o, = 0.001 (mm) 
œ 
o 
  
x 
o 
a 
o 
| 
  
Standard errors (Hm) 
8 
dm FL 
a 
o 
T T T T T T 
90 100 110 120 130 140 150 160 
o 
o 
al 
o 
œ 
[s] 
The convergent angle o. 
Fig. 4.2 3-D co-ordinate accuracy of different 
network geometry 
5. CONCLUSIONS 
In this paper an iterative separated least squares estimation 
method is introduced and compared with the simultaneous least 
squares estimation method using a simple example. This 
method has been applied to the solution of collinearity 
equations as a two step separated adjustment method. 
Simulation tests showed that this method gave the same result 
as the traditional bundle adjustment. The advantages of this 
method are: (i) it is much faster than the traditional bundle 
adjustment. The bundle adjustment shows an exponential 
increase with the number of target, while this iterative method 
is linear; (ii) less memory is required than the traditional bundle 
adjustment. With the bundle adjustment, the inversion of the 
large matrix requires considerable memory space as the number 
of unknowns increases. With the iterative method, the sizes of 
the matrices to be inverted are 3x3 and 6x6 no matter how 
many cameras and targets involved; (iii) it is reliable and 
robust. Simulation tests show that the convergent property of 
the separated solution is as good as that of the bundle 
adjustment; and (iv) it is more flexible than the direct linear 
transform method, as camera orientations are continually 
updated and a full functional model of all camera parameters 
can be included. Further work is undeway to implement this 
method in a real-time system and to consider other aspects such 
as: datum problems; further mathematical analysis; and real- 
time specific issues. 
6. ACKNOWLEDGEMENTS 
The assistance of Prof. M.AR. Cooper is gratefully 
acknowledged in discussions concerning least squares methods. 
The support of Dr. T.A. Clarke is also acknowledged in 
supervising the research and assisting in the preparation of this 
paper. 
7. REFERENCES 
Cooper, M.A.R. 1987 Control Surveys in Civil Engineering. 
Pub. Collins, London. 381 pages. 
Fraser, C.S. 1984. Network design considerations for non- 
topographic photogrammetry. Photogrammetric Engineering 
and Remote Sensing, Vol. 50 No 8. pp. 1115-1126. 
Fraser, C.S. 1992 Photogrammetric Measurement to one part in 
a million, Photogrammetric Engineering & Remote Sensing, 
Vol 58. No 3. pp. 305-310. 
Granshaw, S.I. 1980 Bundle adjustment methods in engineering 
photogrammetry, Photogrammetric Record, 10(56), pp. 181- 
207. 
Gruen, A. 1985. Algorithm aspects of in-line triangulation. 
Photogrammetric Engineering and Remote Sensing. pp. 419- 
436. 
Hill, A. Cootes, T.F. & Taylor, C.J. 1995. Active shape models 
and the shape approximation problem. British Machine Vision 
Conference, 1995. pp. 157-166. 
Karara, HM. 1980 Non-metric cameras, Developments in 
closer range photogrammetry - 1. Ed. K.B. Atkinson. Applied 
Science publishers, London. 222 pages. 
Marzan, G.T. & Karara, H.M. A computer program for direct 
linear transformation solution of the collinearity condition and 
some applications of it. Proc. Symposium on Close-range 
photogrammetric systems, Champaign, Illinois. pp. 420-476. 
(670 pages) 
Mikhail, E.M. 1981 Analysis and Adjustment of Survey 
Measurements. Pub. Van Nostrand Reinhold Company, New 
York. 340 pages. 
Miles, M.J. 1963. Methods of solution of the adjustment of a 
block of aerial triangulation. Photogrammetric Record. Vol. IV. 
No 22. pp. 287-298. 
Shmutter, B, & Perlmuter, A. 1974. Spatial intersection. 
Photogrammetric Record, Vol. 8(43): 94-100. 
Shortis, M.R. 1980. Sequential adjustments of photogrammetric 
models. PhD Thesis, City University, 248 pages. 
592 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996 
KEY V 
ABSTR 
A know 
a calibr 
maps w 
image c 
additior 
either b 
(e.g. ed; 
2-D fea 
this ince 
geometi 
are min 
The visi 
lators o 
alistic 3 
of these 
proach | 
faces an 
els of w 
due to € 
these m 
met. Ge 
ical. In 
disturbi; 
On the 
knowlec 
here use 
stored i; 
interpre 
the obje 
face rec 
interpre 
to objec 
constrai 
reconstr 
jected c 
restorati 
conflict: 
applied. 
The con 
a generi 
models. 
In an op 
are ada] 
determi; 
The go: 
  
! Autor