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
