Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.stsci.edu/ftp/instrument_news/FOC/Foc_isr/foc_isr087.ps Äàòà èçìåíåíèÿ: Fri Jan 5 18:18:08 1996 Äàòà èíäåêñèðîâàíèÿ: Sun Dec 23 05:42:16 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: 47 tuc

INSTRUMENT SCIENCE REPORT
FOC­087
TITLE: The New f/96 Geometric Correction Models
AUTHOR: P. Greenfield DATE: 12 September 1995
ABSTRACT
Described are the new geometric distortion models determined for the post­COSTAR f/96 relay
based on multiple, overlapping observations of crowded star fields. New distortion models were
determined for the following f/96 formats: 512zâ1024, 512â512, 256â1024, 256â256, and
128â128 using the method outlined in the previous Instrument Science Report FOC­086. Unlike
the previous observations, the offsets between exposure appear to be as expected. The quality of
the fits are good despite an apparent change of ~0.16% in plate scale over the course of all the
observations. The consistency of fit star positions with the previous model is good. The consis­
tency of the distortion fit with the previous 512â512 distortion fit using the new method is very
good whereas the consistency with the previous 512zâ1024 fit is less so. The evidence is strong
that the photometric variations at the scan line beginning are not a direct result of distortion.
DISTRIBUTION:
FOC Project: B.G. Taylor, R. Thomas
IDT: R. Albrecht, C. Barbieri, A. Boksenberg, P. Crane, J.M. Deharveng,
M.J. Disney, P. Jakobsen, T. Kamperman, I.R. King, C. Mackay,
G. Weigelt.
SSD: C. Cox, W. Hack, R. Jedrzejewski, A. Nota, All Instrument Scientists
SSG: P. Greenfield, P. Hodge. R. White
SESD: M. Miebach, W. Safley
Dir. Office: F. Macchetto
ST/ECF: P. Benvenuti, R.A.E. Fosbury, R.N. Hook, A. Caulet

2
1. Introduction
This report documents the derivation of the new geometric correction calibration files being used
for FOC calibration pipeline processing. The method used is basically that reported in Instrument
Science Report FOC­086 with some minor changes. Only a very brief outline of the method will
be given here with elaboration given only for those aspects different from the original report.
Readers are expected to refer to that report for details on the theory and rationale for the approach
used. An accompanying Instrument Science Report will document the software written to derive
the models.
The basic method consists of spatially overlapping observations of a crowded star field. Presum­
ing the image offsets are known to high accuracy and assuming the relative star positions are con­
stant for the series of observations, it is in principle possible to determine the underlying
distortion model as long as the spatial density of stars is sufficiently high to sample the variations
in distortion well enough. No a priori knowledge is needed for the relative star positions; they are
determined from the fit along with the distortion model parameters. Reseau positions are only
used to register the new distortion model with the old model, otherwise they are irrelevant The
previous two­dimensional polynomial representation for the distortion model has been replaced
by a two­dimensional spline representation to better fit the high spatial frequency components of
the distortion.
The star field selected was the center of 47 Tucanae, which experience has shown can provide a
sufficiently dense field of relatively uniform brightness stars. This last characteristic is particu­
larly important since just a few very bright stars can render significant parts of the field of view
useless because of saturation effects in the FOC.
2. The Observations
The data were obtained from proposal 5750 executed on 11 November 1994. Two previous
attempts to run this program (as proposal 5521) failed; the first because of a faulty guide star, the
second because the necessary warm up time for optimum stability had not been inserted (a mini­
mum of 4 hours after high voltage switch­on). Table 1 lists the exposure information for the pro­
posal. The offset column represents the relative offset in RA and DEC in arcseconds of that
exposure to the default position.
In principle the overlapping exposures only need to be done for the full format and the star posi­
tions derived used to determine the distortions for all the other formats by means of only one
exposure per format. There are two reasons this was not done for all the formats. The first is that
determining the star positions for the full format results in somewhat poorer positions because of
the use of the zoomed pixel mode. We decided to fit the star positions independently for the
512â512 format to get a higher quality fit. A second important reason to use overlapping expo­
sures, even when not necessary to determine the position of the stars, is to increase the ef fective
density of samples of the distortion. Ideally, we would prefer an even higher density of stars than

3
is observed for 47 Tuc (we know of no target which has a higher density of stars this bright). By
using overlapping exposures we effectively increase the density a factor equal to the number of
different offsets. This was done for the smaller formats for this reason (the 128â128 format would
only have had about 20 usable stars in the fields without use of more offsets for example).
These offsets were chosen (by use of POSTARG) to give approximately 250 pixel offsets in x and
y in the image for the full format (and half that for the 512â512 format) so that the four offset
images were placed half way towards each of the four corners of the centered exposure. As it was,
the roll angle was such that they­axis was virtually aligned with north (position angle of0.°715).
Exposure Format
Offset
in RA (")
Offset
in DEC (") Filter
Exp.
Time (s) Target
x2jh0101t 512zâ1024 N/A N/A N/A 3000 DARK
x2jh0102t 512â512 N/A N/A N/A 969 LED
x2jh0103t 512zâ512 0 0 F220W 597 47 Tuc
x2jh0104t 512zâ1024 3.471 --3.529 F220W 597 47 Tuc
x2jh0105t 512zâ1024 --3.529 --3.471 F220W 597 47 Tuc
x2jh0106t 512zâ1024 --3.471 3.529 F220W 597 47 Tuc
x2jh0107t 512zâ1024 3.529 3.471 F220W 597 47 Tuc
x2jh0108t 512zâ1024 0 0 F220W 597 47 Tuc
x2jh0109t 512â512 1.735 --1.764 F220W 418 47 Tuc
x2jh010at 512â512 --1.765 --1.735 F220W 597 47 Tuc
x2jh010bt 512â512 --1.735 1.765 F220W 597 47 Tuc
x2jh010ct 512â512 1.765 1.735 F220W 597 47 Tuc
x2jh010dt 512â512 0,0 0 F220W 597 47 Tuc
x2jh010et 256â256 1.000 --0.008 F220W 597 47 Tuc
x2jh010ft 256â256 --1.000 0.008 F220W 597 47 Tuc
x2jh010gt 256â256 0 0 F220W 597 47 Tuc
x2jh010ht 128â128 0 0 F220W 654 47 Tuc
x2jh010it 128â128 --1.000 0.008 F220W 717 47 Tuc
x2jh010jt 256â1024 0 0 F220W 597 47 Tuc
x2jh010kt 256â1024 --1.000 0.008 F220W 597 47 Tuc
x2jh010mt 512zâ1024 0 0 F220W 600 47 Tuc
x2jh0101t 512zâ1024 N/A N/A N/A 900 LED
Table 1. A summary of the exposures for proposal 5750.

4
3. Data Analysis and Results
The analysis was carried out as described in FOC­086 with the following exceptions. First, there
were two bugs found in the star matching program one of which effectively excluded from the
stars used those used in manually matching pairs of stars between different images. Other than
reducing the number of stars used in the fit, it had no systematic effect on the previous results (the
fraction of lost stars was typically ~10% for the largest formats but was generally larger for
smaller formats). The second bug resulted in a few stars being present twice in the star lists. Both
bugs were fixed and the results presented here were not affected by them. Second, the results
derived in the first report did not use any weighting for the parameter fits. Weighted versions of
the fitting programs were developed and those versions were used for all video formats that did
not use the zoomed pixel mode. The weighting was based on the total counts of each star and the
weighting function used is that derived from Figure 2 in ISR FOC­086. Third, all indications are
that the actual image offsets were very near those that were commanded. Thus it was not neces­
sary to try to determine the offsets based on reseau positions as was done for the previous report.
In other words, the analysis was carried out as planned this time without using any workarounds.
Fourth, a new program was written to exclude stars from full format exposures suspected of being
overflowed, and to exclude all stars closer than 4 pixels to another star in the list of stars.
As in the previous report, the adopted plate scale is 0.01435 arcsec per normal pixel for all the dis­
tortion models (the plate scale is purely arbitrary, the value chosen is close the average plate scale
in the uncorrected images and is the value adopted for the previous post­COSTAR geometric cor­
rection models). The orientation of the corrected image is set by making match the predicted ori­
entation given in the exposure headers (again, this is arbitrary; it was chosen to remain consistent
with past models). The relative position of the geometrically corrected full format was chosen so
that the reseau locations matched up on average with those of the previous distortion corrections.
The relative position of all the other formats was forced to result in alignment with the full format
after accounting for the assumed relative offset between the formats (e.g., 256, 256 pixels for the
512â512 format relative to the full format). These conventions ensure the consistency of align­
ment and plate scale between all formats, something that has not been precisely true for previous
geometric correction models where different formats ended up with slightly different orientations
and offsets. But note that this enforced consistency does not eliminate misalignments or scale
changes between images (of the same or different formats) due to changes in the intrinsic distor­
tion over time).
The spline knot locations are identical to those used in FOC­086 for all the formats except
128â128 which was not done in the previous report. Because of the small number of points in that
format, a two­dimensional polynomial model was used instead for the distortion. (This is a con­
siderable improvement over the previous case for there was no model whatsoever given the inad­
equate number of reseaux present to determine a model.) Table 2 lists the knot locations used for

5
each format. Table 3 lists the number of data points and number of stars involved in the fit for
each format.
3.1. 512 zoomedâ1024
The distortion model for the full format was obtained using an unweighted fit. The rms residuals
for x (in normal pixel units) and y were 0.33 and 0.28 pixels. The plot of the residual vectors is
shown in Figure 1. No spatial correlation of residuals is evident suggesting that the high fre­
Format
x knot
Positions
y knot
Positions
512zâ1024 127 302
240 703
260
331
402
439
473
512â512 159 142
341 370
420
467
256â256 108 324
213 710
256â1024 128 128
Table 2. A list of the x and y knot positions for
all the formats that use spline fits.
Format
Data
Points Stars
512zâ1024 4703 1623
512â512 1544 546
256â256 242 N/A
128â128 39 N/A
256â1024 577 N/A
Table 3. The number of data points
and stars used in the model fits for each
format. The last three formats do not fit
star positions, only the distortion
parameters.

6
quency components have been adequately fit. The resulting model was then applied to correct the
positions obtained from reseaux fits for the full format LED image (x2jh010mt) to undistorted
coordinates. A linear transformation to match the undistorted reseau positions with the reference
positions used previously to determine distortion models was determined by a least squares fit,
and then applied to the reseaux positions to produce a vector plot of residuals. This plot is shown
in Figure 2. There are no clear large scale systematic effects in the residuals. The rms residuals in
x and y are 1.03 and 0.79 pixels. Although smaller than obtained for the previous report, they are
still at least a factor of 2 or 3 larger than expected. I have no explanation for a discrepancy of this
size. The linear transformation that was fit was consistent with a simple rotation and translation
with no evidence for a scale error or skew (the linear transformation fit resulted in a scale change
of 0.2% and a rotation of 0.°09). This strongly suggests that the offsets between the exposures
was very close to that commanded.
One clear test of the reliability of the modeling process can be done by comparing the fit positions
for the stars for the current data with those obtained for the fit reported in FOC­086. Since the rel­
ative orientation between the two observations is known, and the same star can be used to deter­
mine the relative offset, the transformation between the two coordinate systems can be
determined a priori. This was done and then a program to match stars between the two lists was
run. After matching the stars, a linear orthogonal transformation was fit to minimize the residual
differences in position. Ideally, the fit transformation would show no offset, rotation, or scale
change. But detector plate scale changes between different high­voltage switch­ons are not
unusual and in fact the resulting fit shows a effective change of 0.7% between the two sets of posi­
tions. The effective rotation is less than 0.°5. The residual rms inx and y were 0.48 and 0.55 pix­
els which should be considered reasonably good agreement considering the presence of random
errors in both sets of data (this level of residuals implies intrinsic position errors on the order of
0.3­0.4 pixels for the full format fits). If a linear transformation between the two coordinate sys­
tems that allows skew is fit, then thex and y rms residuals become 0.37 and 0.49 pixels indicating
there was a small amount of skew present in the old solution left after matching to the reseau grid.
3.2. 512â512
The fit for the 512â512 format produced significantly smaller residuals. The rms residuals forx
and y were 0.19 and 0.23 pixels. The residual vectors are shown in Figure 3. The star positions
determined here were compared to those determined from the full format fit in a similar way to
those done for the old and new observations. Using only the a priori transformation between the
coordinates (only an offsets since the orientation is presumably identical) thex and y rms residu­
als are 0.34 and 0.35 pixels. Allowing a linear orthogonal transformation reduced these residuals
only slightly to 0.32 and 0.26 respectively. These independent determinations of the star positions
agree quite remarkably and are consistent with the values seen for the fit residuals, i.e., the residu­

7
als of ~0.3 for the 512zâ1024 fit and ~0.2 for the 512â512 fit. That means it is possible to do rela­
tive astrometry on crowded fields to the 3 mas level using this technique.
3.3. 256â256
The fit for the 256â256 format used the positions for the stars obtained from the 512â512 fit to
determine the spline coefficients for this format. The resulting rms residuals forx and y were 0.35
and 0.44 pixels. The plot of the residual vectors is shown in Figure 4. The residuals are higher
here presumably in part because of the contribution of the errors in the previously fit star posi­
tions.
3.4. 128â128
The fit for the 128â128 format used a simple two­dimensional polynomial model for the distor­
tion. The rms residuals forx and y were 0.59 and 0.39 pixels. The plot of the vector residuals is
shown in Figure 5.
3.5. 256â1024
The fit for this format used the star position solutions from the full format and resulted inx and y
rms residuals of 0.35 and 0.38 pixels. The residuals are shown in Figure 6.
4. Consistency of the Distortion Model
Figure 7 shows magnified vectors showing the differences between the 512â512 distortion model
derived for ISR FOC­086 and the latest distortion model. As can be seen, the differences are not
large and appear primarily as a small rotation. The consistency for this format should be consid­
ered very good. The consistency of the full format solutions is not as good and cannot be
explained as a simple rotation (see Figure 8). This discrepancy may arise from the problem with
determining the offsets of the images for the first set of data, but that seems unlikely since errors
in the offsets should only apply a skew to the distortion, it should not introduce smaller scale
changes. It is quite possible that the intrinsic distortion is changing from one high­voltage
switch­on to another. The only way to be sure is to carry out another run of the distortion and plate
scale proposal and determine if the fine scale features in the distortion are changing. The repeat­
ability of the fit star positions strongly imply that the distortion model derived is valid for this
high­voltage switch­on at the very least. For the moment, it must be considered an open issue.
5. Consistency of the Plate Scale
Since the proposal contains two observations taken in the same configuration and pointing taken
several hours apart (x2jh0108t and x2jh010lt), it is possible to check the stability of the distortion
by comparison of the star positions between the two images. The magnified difference vectors are
shown plotted in Figure 9. Fitting a linear orthogonal transformation gives an effective plate scale
change of 0.16% and rotation of 0.°014. The rotation is negligible, but the plate scale change is
more significant and demonstrates that a long warm up time does not make the FOC perfectly sta­

8
Figure 1. The residuals for the full format model. The residuals are shown as magnified vectors showing the
difference between the measured star position and the model­predicted star position. The boxes designate the tails of
the vectors. The vectors are placed at the position in the raw image corresponding to the measured star position. The
overplotted horizontal and vertical lines show the location of thex and y knots that segment the spline fitting region.

9
Figure 2. The difference between the predicted reseau positions and those determined by applying the geometric
correction to the measured reseau positions. A linear transformation (including rotation, scale, and an offset) has been
applied to the measured positions after geometric correction to best match the predicted positions.

10
Figure 3. The residuals for the 512â512 model. The residuals are shown as magnified vectors showing the difference
between the measured star position and the model­predicted star position. The boxes designate the tails of the
vectors. The vectors are placed at the position in the raw image corresponding to the measured star position. The
overplotted horizontal and vertical show the location of thex and y knots that segment the spline fitting region.

11
Figure 4. The residuals for the 256â256 model. The residuals are shown as magnified vectors showing the difference
between the measured star position and the model­predicted star position. The boxes designate the tails of the
vectors. The vectors are placed at the position in the raw image corresponding to the measured star position. The
overplotted horizontal and vertical lines show the location of thex and y knots that segment the spline fitting region.

12
Figure 5. The residuals for the 128â128 model. The residuals are shown as magnified vectors showing the difference
between the measured star position and the model­predicted star position. The boxes designate the tails of the
vectors. The vectors are placed at the position in the raw image corresponding to the measured star position.

13
Figure 6. The residuals for the 256â1024 model. The residuals are shown as magnified vectors showing the
difference between the measured star position and the model­predicted star position. The boxes designate the tails of
the vectors. The vectors are placed at the position in the raw image corresponding to the measured star position. The
overplotted horizontal and vertical lines show the location of thex and y knots that segment the spline fitting region.

14
Figure 7. The difference between the new distortion model vectors and the distortion model vectors (ISR FOC­086)
for the 512â512 format shown magnified on a grid with a spacing of 16 pixels. Because of the large distortion effects
near the edges, the distortion differences are not shown near the edges.

15
Figure 8. The difference between the new distortion model vectors and the distortion model vectors (ISR FOC­086)
for the 512zâ1024 format shown magnified on a grid with at spacing of 16 pixels (dezoomed). Because of the large
distortion effects near the edges, the distortion differences are not shown near the edges.

16
ble. The amount of the drift in this case should not seriously affect the consistency of the solution
since if we presume the drift was basically linear in plate scale versus time, the drift over any one
of the series for a given format would be approximately a quarter of that observed between these
two observations (which spanned over six hours). That would imply a net drift over a series of
about 0.04% or about 0.5 pixels difference in the spacing of stars located at opposite edges of the
format which is sufficiently small for the purposes of the fit.
6. Geometric Correction Files
The new distortion models were used to generate geometric correction files. Table 4 shows the
new files and the files they replaced. The new files should be considered the ones to use for all
post­COSTAR data. They were installed into the calibration pipeline on 19 March 1995. Because
there is a different optical distortion resulting from the use of COSTAR, the flat fields used for
pipeline calibration needed to be modified to match the current optical distortion correction. A
``differential'' geometric correction file was constructed to apply to the old flat field files to pro­
duce flat field files that match the current optical distortion correction. The new flat field correc­
tion files generated are listed in Table 5 with the files they replaced. The new flat field correction
files were generated and installed into the calibration pipeline the same time the geometric correc­
tion files were.
7. Residual Photometric Effects
One of the motivations driving the new approach to determining the distortion model was the pos­
sibility of correcting for the fine scale variations in the flat field seen near the beginning of the
scan line (the alternating bright and dark vertical bars seen near the right hand edge, see Figure 10
for example). The canonical explanation for these scan start­up variations was that they were the
result of variations in the scan rate at the start of a scan. More specifically, variations in scan rate
result in distortion, primarily in thex­direction. The varying size of pixels along the scan line
direction results in varying apparent sensitivity as seen in flat fields. If this is indeed the case, a
proper geometric correction would remove these flat field features.
The obvious question is: have these features been removed by the new geometric correction?
Looking at a 512â512 LED flat field that has been geometrically corrected using the new model
shows clearly that these features have not gone away as a result of the new geometric correction.
This raises the question of whether the model is inadequate to model such variations or whether
they do not arise from distortion. Figure 11 shows the average of all rows betweeny=200 and
y=500 for both the raw image (upper curve) and the geometrically corrected image (lower curve).
The corrected curve has been displaced by 0.4 to clearly separate it from the upper curve. Both
have been normalized by the average value. The vertical lines mark the location of thex knots
(which refer to their location relative to the raw image coordinates, i.e., the top curve). The begin­
ning of scan variations show up clearly on the right edge of the plot starting with one large oscil­
lation (~20% peak­to­peak) followed by three smaller ones (~10% peak­to­peak).

17
Figure 9. The difference in positions for stars shown as magnified vectors for two repeated exposures of the same
field in 47 Tuc separated by more than 6 hours.

18
Before concluding that these photometric features are not cause by distortion it is necessary to
estimate whether they are measurable as distortion using the new method. Assuming that the dis­
tortion is sinusoidally periodic we can determine the peak distortion given the size of the photo­
metric variation and its period in pixels. It is straightforward to show that the peak distortionD
(deviation from a the average plate scale) for a one­dimensional sinusoidal distortion has the fol­
lowing relation to the periodP and amplitude of photometric variationA (expressed as a fraction
of the average level) D=AP/2p. For the smaller oscillations A is 0.05 and P is 16 resulting in a
peak distortion of 0.12 pixels which would only be marginally noticeable in the residuals. The
peak residuals for the first oscillation should be twice as large, i.e., 0.25 pixels, still relatively
small but should be observable. Nevertheless it is not evident in the residuals.
This analysis by itself would not be terribly convincing by itself that the photometric oscillations
are not a result of distortion. The major piece of evidence that argues against distortion as the
explanation is a larger scale feature, the general 10% to 15% increase in brightness seen between
pixels 410 and 470 on top of which the oscillations are present. This enhancement is clearly asso­
ciated with the oscillations and is sizeable over many pixels. If it were due to distortion it should
have been modeled, for unlike the small scale oscillations, it is of a size scale comparable to the
knot spacing. A 10% average increase in brightness over 50 pixels would result in a 5­pixel dis­
New Cal.
Filename Format
Cal. File
Replaced
f3715310x.r5h 512zâ1024 e8q0912mx.r5h
f371529ex.r5h 512â512 e8q09106x.r5h
f3715276x.r5h 256â256 e8q09090x.r5h
f371522px.r5h 128â128 N/A
f371522ex.r5h 256â1024 N/A
Table 4. The new geometric correction reference files and
the files they replace. These files took effect 19 March 1995 in
the calibration pipeline. They should be considered the files to
use for all post­COSTAR FOC observations.
New Cal.
Filename Wavelength
Cal. File
Replaced
f3716027x.r2h 1360 å c2410530x.r2h
f3716029x.r2h 4800 bbe14193x.r2h
f371602cx.r2h 5600 bbe14196x.r2h
f371602dx.r2h 6600 bbe14198x.r2h
Table 5. The new flat field correction reference files and the
files they replace. These files took effect 19 March 1995 in
the calibration pipeline. They should be considered the files
to use for all post­COSTAR FOC observations.

19
tortion, a huge amount. Yet the corrected curve shows that there is relatively little change in the
relative brightness. This shows that the model did not remove the effect, and the residuals show
no evidence that the model missed the presumed geometric effect. This is powerful evidence that
the photometric effects at the beginning of the scan line are not due entirely to distortion effects;
in fact, it is apparent that very little of it, if any, is due to distortion effects. Whatever the cause, it
must be indirectly related to the scan line sweep rate, that is, a small variation in scan rate has
some large indirect effect on sensitivity.
Figure 10. A 512â512 LED flat field that has been geometrically corrected using the latest geometric correction
model. Note the photometric variations near the beginning of the scan line (the right hand edge) that appear as vertical
features have not been removed as a result of the geometric correction.

20
8. References
1. Greenfield, P., Instrument Science Report FOC­086,Deriving the Geometric Correction
from Crowded Fields, 15 February 1995.
2. Greenfield, P., Instrument Science Report FOC­088,A Description of the Software used to
Derive the Geometric Correction, 12 September 1995.
9. Appendix
Tables 4­8 list the distortion model coefficients for the different formats. Refer to Instrument Sci­
ence Report FOC­088 for an explanation of how to use the spline coefficients to apply the distor­
tion correction. Since the spline fits generate a distortion correction which has an arbitrary offset,
applying an offset is necessary to register the correction with other formats (and by implication,
set the overall registration, which I have taken to match the previous full format distortion correc­
tion). The coefficients for the 128â128 format are used in a 2­dimensional polynomial which is
defined as
f = c 0 + c 1 x + c 2 y + c 3 x 2 + c 4 xy + c 5 y 2 .
Figure 11. The average of rows 200 through 500 of the 512â512 LED flat field. The top curve shows the average for
the raw image while the lower curve shows the average for the geometrically corrected image. Both curves have been
normalized by the average value in the averaged region. The lower curve has been offset by 0.4 to clearly separate it
from the upper curve. The vertical lines show the location of thex knots which are valid only for raw image coordinates.

21
0 1 2 3 4 5 6 7 8 9 10
x: 0 ­335.3777 ­294.4028 ­220.0122 ­137.7627 ­74.6514 ­21.9114 35.8766 83.3888 121.8391 140.8901 158.4359
1 ­335.5470 ­294.6411 ­221.1190 ­139.2623 ­76.5960 ­24.7699 32.5531 81.3252 117.7703 141.4733 150.3053
2 ­336.6700 ­295.9409 ­223.9097 ­143.4431 ­80.8980 ­28.6400 28.7800 75.9615 114.4685 134.4923 148.5208
3 ­340.4296 ­300.1554 ­226.9872 ­145.2825 ­82.7836 ­30.6043 26.3263 75.2299 112.3768 135.3084 145.8864
4 ­348.1082 ­307.0546 ­232.7321 ­150.1066 ­86.0276 ­32.5104 27.2513 75.8467 115.6780 136.6391 151.0307
5 ­349.6896 ­308.8469 ­236.4470 ­152.0013 ­87.2978 ­32.9190 27.1183 78.7278 118.5663 143.1199 154.0493
y: 0 ­606.8826 ­607.1485 ­609.8506 ­614.5410 ­620.2610 ­624.4877 ­633.3167 ­632.3627 ­642.3163 ­638.1720 ­638.9640
1 ­501.7147 ­501.5342 ­505.1399 ­510.4167 ­516.2632 ­519.3739 ­526.0953 ­527.4296 ­531.3765 ­530.2935 ­528.0785
2 ­258.8111 ­260.4648 ­264.8695 ­270.1564 ­275.8738 ­278.5222 ­284.4624 ­281.5574 ­286.2589 ­282.4141 ­276.9131
3 98.6047 94.3756 86.4956 81.2116 76.5211 76.2930 72.5325 77.8506 74.8378 81.6228 83.3578
4 363.4758 359.7860 347.1910 341.6883 336.1764 337.0558 335.0323 341.5379 341.2943 345.0512 353.8827
5 495.7200 482.1450 468.7441 460.6934 455.0268 455.1176 451.8534 457.8040 454.6922 462.1972 462.0476
Table 6. The distortion model coefficients for the 512zâ1024 format. These spline coefficients are used to
translate the raw, distorted, zoomed coordinates into undistorted zoomed coordinates. To generate the
translation between dezoomed coordinates it necessary to divide the x coordinate by 2, use the above defined
transformation and then multiply the resulting x coordinate by 2. Finally an offset of 713.75, 605.05 must be
added to the x,y coordinates.
0 1 2 3 4 5 6 7
x: 0 ­277.8163 ­230.6939 ­127.9230 ­1.9516 94.0992 149.5876 178.9515 192.4354
1 ­278.6607 ­230.6110 ­128.6591 ­3.4270 93.6152 147.0999 177.2429 191.2976
2 ­279.6118 ­234.7947 ­130.7464 ­5.2146 90.4180 146.4820 174.9105 189.2814
3 ­284.6721 ­235.8037 ­134.2955 ­7.6732 89.6568 143.2984 172.0329 189.5963
4 ­287.4205 ­240.9640 ­136.5205 ­9.7565 86.9572 143.2423 173.0443 184.5473
5 ­289.2265 ­242.5367 ­137.6575 ­10.9817 87.2798 142.6642 170.0594 191.7626
y: 0 ­117.8632 ­118.9170 ­123.9690 ­128.1461 ­132.2248 ­131.4605 ­137.3464 ­129.4594
1 ­68.9852 ­71.1555 ­74.8196 ­80.5303 ­84.1685 ­83.6478 ­87.4229 ­77.9673
2 53.4630 52.5330 46.9123 43.7284 40.4486 39.9230 38.5806 40.8678
3 227.1560 224.2442 221.3669 215.1190 213.5247 214.6154 210.4180 221.3669
4 353.6387 352.2208 345.6836 343.4112 341.0799 341.9911 340.1939 344.2387
5 404.0305 401.3763 397.2847 392.1207 390.7592 393.4785 388.2157 398.7874
Table 7. The distortion model coefficients for the 512â512 format. After applying the
transformation defined by these coordinates is necessary to add the offset 300.85, 131.72 to the
resulting coordinates.

22
0 1 2 3 4
x: 0 ­163.513 ­124.600 ­46.8961 28.6725 70.0034
1 ­165.032 ­124.428 ­50.5662 31.8930 64.2634
2 ­165.271 ­127.407 ­47.1502 26.5015 70.6125
3 ­169.135 ­126.024 ­53.5315 29.0873 64.5162
4 ­166.434 ­130.732 ­48.4659 24.2038 66.8202
y: 0 4.00661 3.90794 ­0.204523 ­0.798520 ­1.32415
1 48.1036 46.7047 44.8028 40.4506 42.5996
2 133.506 129.464 129.261 126.892 129.382
3 216.608 218.638 214.633 209.708 213.094
4 263.556 259.084 259.110 256.014 258.178
Table 8. The distortion model coefficients for the 256â256 format. After applying
the transformation defined by these coordinates is necessary to add the offset
172.85, 3.72to the resulting coordinates.
0 1 2 3 4 5
x: 0 ­311.384 ­277.802 ­214.822 ­134.213 ­91.1200 ­74.5506
1 ­315.303 ­281.894 ­217.848 ­140.138 ­96.2336 ­82.4975
2 ­322.152 ­288.489 ­225.454 ­147.441 ­103.860 ­88.5535
3 ­326.260 ­292.254 ­229.833 ­150.227 ­107.051 ­94.6133
4 ­336.418 ­301.237 ­236.485 ­156.716 ­111.490 ­93.7263
5 ­340.370 ­303.482 ­240.657 ­157.099 ­113.944 ­96.6036
y: 0 ­601.382 ­602.797 ­605.890 ­609.125 ­609.415 ­607.435
1 ­492.857 ­494.001 ­495.813 ­499.217 ­499.635 ­498.949
2 ­253.216 ­255.653 ­257.316 ­260.771 ­260.525 ­260.020
3 89.7133 88.7008 86.9394 84.6335 84.7821 88.9437
4 336.853 336.913 332.982 332.785 331.670 334.570
5 454.543 450.273 449.126 445.020 446.465 447.250
Table 9. The distortion coefficients for the 256â1024 format.After applying the transformation
defined by these coordinates is necessary to add the offset 329.10, 605.05 to the resulting
coordinates.

23
x y
c 0 2.935 4.9309
c 1 0.861428 ­4.276848e­2
c 2 ­1.210785e­2 0.996674
c 3 3.650710e­4 1.652092e­4
c 4 ­1.437217e­5 ­3.533065e­5
c 5 3.802776e­5 ­1.406670e­5
Table 10. The distortion coefficients for the 128â128 format.