International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
distortion can clearly exceed the measurement accuracy of VM.
The use of high quality digital equipment allows the
achievement of typical triangulation accuracies of 1:100.000,
which is about 0.05 mm in this case. However, the real
accuracy of the presented network is about 0.25 mm for the
object points (sce Table 2).
Finally, a ‘dangerous’ effect for practical applications can be
observed. Since the adjustment parameters compensate for the
eccentricity error, the bundle adjustment estimates the accuracy
of the object coordinates too optimistically (by factor 10 in this
case), which may lead to misinterpretation, eg in deformation
measurements.
Kager (1981) and Ahn et al. (1999) have developed two
different correction formulae for the ellipse center eccentricity.
Using Equations 16 and 17, the problem can solved via a third
approach. First the implicit ellipse parameters of the imaged
target circle are computed. This allows determination of the
ellipse center coordinates (Equation 21). Using the image space
target center and the ellipse center, the sough-after correction
vector can be calculated. The equality of the three correction
formulas was numerically proven by the authors.
5. DISCUSSION AND CONCLUSIONS
The presented target plane determination algorithm is a fully
automated process which is suitable to any VM system which
employs circular targets. The process has been implemented
and evaluated within the Australis software system. As shown
here, the algorithm can generate accurate target plane
information, which will serve practical applications. The
benefits are the correction of target thickness in the case of
surface inspection and the general increased accuracy in ultra-
precise surveys, since the observation equations of the bundle
adjustment can be described more rigorously.
To date, one has always had to find a compromise between big
(high centroiding accuracy) and small targets (small
eccentricity error). The proposed method resolves this problem
and allows use of big targets without the impact of eccentricity
error.
Further research on real high-precision networks should further
show that the accuracy of the object points can be increased by
considering the eccentricity error. Since the improved accuracy
is not ‘visible’ from internal measures within the bundle
adjustment (as described in Section 4) ultra-precise reference
(checkpoint) coordinates of object points are needed to verify
the improved accuracy.
REFERENCES
Ahn, S.J., Warnecke, H.-J. and Kotowski, R., 1999. Systematic
Geometric Image Measurement Errors of Circular Object
Targets: Mathematical Formulation and Correction,
Photogrammetric Record, 16(93), 485-502.
Ahn, S. J. and Kotowski, R., 1997. Geometric image
measurement errors of circular object targets. Optical 3-D
Measurement Techniques IV (Eds. A. Gruen and H. Kahmen).
Herbert Wichmann Verlag, Heidelberg. 494 pages: 463-471.
Dold, J., 1996. Influence of large targets on the results of
photogrammetric bundle adjustment. /nternational Archives of
Photogrammetry and Remote Sensing, 31(B5): 119—123.
Fraser, C.S., 1997. Automation in Digital Close-Range
Photogrammetry, 1st Trans Tasman Surveyors Conference,
Fryer J (ed), Proceedings of the 1st Trans Tasman Surveyors
Conference, 1: 8.1 - 8.10. Newcastle, NSW: Institute of
Surveyors, Australia.
Fraser, C.S. and Shao, J., 1997. An Image Mensuration Strategy
for Automated Vision Metrology, Optical 3D Measurement
Techniques IV, Heidelberg, Germany: Wichmann Verlag, 187-
197.
Fraser, C. S. and Edmundson, K .L., 2000. Design and imple-
mentation of a computational processing system for off-line
digital close range photogrammetry. ISPRS Journal of Photo-
grammetry and Remote Sensing, 55(1): 94-104.
Kager H., 1981, Bündeltriangulation mit indirekt beobachteten
Kreiszentren. Geowissenschaftliche Mittelungen Heft 19,
Institut für Photogrammetrie der Technischen Universität Wien.
Kraus, K., 1996, Photogrammetrie, Band 2: Verfeinerte
Methoden und Anwednungen, Fred. Duemmlers Verlag.
Mikhail, Edward M. and Ackermann, F. (1996) Observations
and Least Squares, IEP-A Dun-Donnelley.
Photometrix, 2004, Web site: hitp://www.photometrix.com.au
(accessed 15 April 2004).
APPENDIX
In the following the conversion between explicit ellipse
parameters and the implicit form (a general polynomial of the
second degree, see Equation 15) is given. The parametric form
of the ellipse equation can be described by
x) (M.,) ( cos sinó Y Acosa
: | ¢ "| (18)
y M) \-sinÿ cosó A Bcosa
where M,, M, are the centre coordinates, ¢ is the bearing of the
semi-major axis, and A and B are the semi major and semi
minor axes of the ellipse.
The corresponding conversion to the implicit form follows as
— cos’ ¢ sin @
a zi a
B: A
$- 2: cosp-sine | = | (19)
x B® 7
zo d. cav d
A À
f=1-M;-a-M, M, b-M, €
bz
The
