cross centre to a resolution of 0.5 pixel, thereafter 
least squares template matching is used to 
determine the centre to higher precision (~ 0.05 
pixel, or 0.7 pm). 
As the camera is being moved relative to the stage 
system during this procedure, the pixel coordinates 
must be corrected in accordance with the profile 
measurements above before a transformation can 
be derived. Since the physical reality is a 
perspective relationship between two planes, a two- 
dimensional projective transformation should be 
used. However, if the two planes are parallel, then 
this reduces to a affine transformation; this is in fact 
the case on the S9AP in Zürich. 
The transformation is derived by least squares 
adjustment. A 25 position grid is usually used in 
preference to one with 81 positions; the difference 
in the parameters and the RMS residual is 
negligible. This value for the Sony camera on the 
right-hand stage lies at slightly more than 1.0 pin 
(0.07 pixel) when the central stage cross is used. 
The RMS at other stage crosses, particularly in the 
comers, is lower at around 0.9 pm; this is probably 
due to the relatively stable nature of the change in 
the profile values in these regions of the stage. 
The pattem of residuals show very little systematic 
tendencies, and hence, for the reason already 
explained, distortion in the lens system can be 
discounted. 
3.3.3 Application of the Calibration 
Using the above derived calibration, and assuming 
that an affine transformation is sufficient, the 
relationship between pixel coordinates and full 
stage coordinates is represented by: 
x c = Ax„ + t + x„ + Ax„ 
s p p c c 
where A 
x p 
X c 
Ax c 
x s 
is the transformation matrix 
is the transformation shift vector 
are the pixel coordinates 
are the camera stage coordinates 
are the profile corrections 
are the full stage coordinates 
Note that the profile corrections are expressed in 
the stage coordinate frame. 
3.4 Indicator of Global Accuracy 
The accuracy of the derived transformation is only 
meaningful in the area where the calibration was 
performed. To determine a global measure, the 
accuracy of the stage calibration must also be 
considered. This is achieved using a procedure in 
which all 25 engraved stage crosses are visited and 
their coordinates determined by measurement in 
the digital image, using the same method as in the 
calibration determination. The resulting pixel 
coordinates are then transformed back into the full 
stage frame using all calibration information. The 
relationship of these coordinates to the factory 
calibrated values will indicate the global accuracy. 
This assessment is done by means of an affine 
transformation determined by least squares 
adjustment, as this can also reveal factors relating 
to stability. The residual error of the 
transformation is a little higher than that of the 
manual instrument calibration, at around 1.3 pm as 
compared to 1.0 pm for the right stage. 
3.5 Stability of the Calibration 
The stability of the calibration relates to its 
constancy over time. Aspects of stability must 
consider effects primarily from two sources: the 
S9AP measurement system and the CCD camera 
and frame-grabber combination. In both cases, the 
main cause of instability will be temperature 
fluctuations. 
The stability of the measurement system is 
assessed by a check procedure on the instrument 
calibration. The result is an affine Pansformation 
representing the change from the current 
parameters. During periods of rapid temperature 
change the drift can be significant - 5.0 pm has 
been observed within a two-hour period. For this 
reason, digitisation is usually done during periods 
of temperature stability, when drift within a half 
day has been observed to be no more than 1.2 pm, 
and up to 5.0 pm over a week. Significant changes 
in the scales or shears have never been observed, 
and hence any drift is compensated for by an up-to- 
date calibration of the CCD camera, or by an inner 
orientation to the analogue imagery. 
Drift in the CCD camera and frame-grabber 
combination is a known phenomenon and hence is 
expected. Again temperature change - either 
ambient or directly of the electronic components - 
is a significant influencing factor (Gulch, 1984; 
Dahler, 1987). The latter of these can be 
minimised by starting up the system some time 
before use, and thereafter running it continuously 
for as long as the project lasts. Drift due to the 
ambient temperature may still be observed, 
however. During the same period of temperature 
change mentioned above, a drift of the measuring 
mark coordinates of 0.4 pixel (6 pm on the stage) 
was observed in the x direction. Otherwise, the 
characteristic has been observed to be similar to 
that of the instrument calibration: short-term drift is 
around the 1 pm level, whilst over a period of some 
days larger values of up to 10 pm can be observed. 
Since a stage cross is used to determine a 
calibration, any change in the translation 
parameters will be due to a combination of both of 
the above instabilities, and will also show up in the 
result of the global accuracy test (section 3.4). 
During digitisation, this combined effect of drift