Ground-based optical navigation seeks to determine the angular position of a star, Solar System body, or laser-emitting spacecraft relative to objects with well-known coordinates. Measurement accuracies of 25 nrad would make optical techniques competitive with current radio metric technology. This article examines a proposed design for a wide-field astrograph and concludes that the deviation of an image centroid from the ideal projection can be modeled to the desired accuracy provided that the field of view does not exceed 5 deg on a side

Owen, W. M., Jr.

Shaklan, S. B.

English

NASA Technical Reports Server

51
TDA Progress Report 42-109 May 15,1992
N 9 2 -2 93 70
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Geometric Distortion Analysis of a Wide-Field Astrograph
W. M. Owen, Jr.
Navigation Systems Section
S, B. Shak!an
Optical Sciences and Applications Section
Ground-based optical navigation seeks to determine the angular position of a
star, Solar System body, or laser-emitting spacecraft relative to objects with well-
known coordinates. Measurement accuracies of 25 nrad would make optical tech-
niques competitive with current radio metric technology. This article examines a
proposed design for a wide-field astrograph and concludes that the deviation of an
image centroid from the ideal projection can be modeled to the desired accuracy
provided that the field of view does not exceed 5 deg on a side.
I. Introduction
Astrometry, the science of measuring angular positions
of celestial objects, is currently in renaissance thanks to
new instrumentation such as the Multichannel Astromet-
ric Photometer [1], the Hipparcos spacecraft [2], and the
Mark III stellar interferometer [3]. These devices, whether
orbiting or ground-based, have increased the precision of
an angular measurement by two orders of magnitude com-
pared to conventional photographic or transit-circle tech-
niques.
Optical angular measurements accurate to 25 nrad
(5 mas) could revolutionize JPL's spacecraft navigation as
well. Ground-based observations of asteroid 951 Gaspra,
while good to only 0.1 arcsec, nevertheless contributed
greatly I to the successful flyby of that asteroid by Galileo
1 D. K. Yeomar_ and M. S. Keesey, "Updated Orbit and Ephemeris
for Asteroid 951 Gaspra," 3PL Interoffice Memorandum 314.6-1332
(internal document), Jet Propulsion Laboratory, Pasadena, Cali-
fornia, August 30, 1991. More detailed accounts of the Gaspra
encounter navigation will be presented at the AIAA/AAS Astro-
dynamics Conference in Hilton Head, South Carolina, in August
1992.
in 1991. Measurements of the satellites of Jupiter and of
Saturn could provide accurate ephemerides of these ob-
jects, thereby improving the power of onboard navigation
images taken by the Galileo or Cassini spacecraft. And
direct optical measurements of the laser light emitted by
a future spacecraft can determine its trajectory position
relative to its target.
One of the tasks necessary in defining a future opti-
cal navigation system is to assess the capability of vari-
ous instruments to deliver relative positions at the 5-mas
(25-nrad) level s over a relatively wide (5- or 6-deg) field
of view. The instrument itself must perform at that level;
the problem of measuring images in the focal plane is a
separate issue.
This article reports on the optical characteristics of one
candidate, a proposed wide-field astrograph designed by
the United States Naval Observatory. This instrument
2 G. Null, "Wide-Field Telescope Selection Results," JPL Inter-
office Memorandum 314.8-815 (internal document), Jet Propulslon
Laboratory, Pasadena, California, December 4, 1991.
87
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has a clear aperture of 36 cm, a focal length of 3.6 m,
a plate scale of 57.3 arcsec/mm, and a 6-deg field of view;
the focal length in particular is similar to those of the
51-cm astrographs at Lick Observatory and Yale Univer-
sity Observatory.
Analysis of this instrument involved simulating stellar
images at various places in the focal plane, finding the cen-
troids of each image, and developing a model to account
for systematic differences between the observed positions
of these centroids and their ideal positions as predicted by
a pinhole projection. The observed centroids can deviate
from the ideal projection by almost 0.4 arcsec, necessitat-
ing the use of higher order terms in the transformation
model. However, most of the deviation can be accommo-
dated by one third-degree radial term, and the remainder
is on the order of 2-7'mas.
The authors conclude that the instrument under consid-
eration will indeed be suitable for 5-mas astrometry over
a 5-deg x 5-deg field of view.
II. Image Generation
The Controlled Optics Modeling Package (COMP) was
used to simulate images in the focal plane, at steps of
0.5 deg (roughly 31 ram) from -3 deg to +3 deg in both z
and y. The symmetry of the problem allowed generation of
only those 28 images in the first quadrant for which x > y,
marked by tile filled dots in Fig. 1; the others follow from
these. A total of 132 = 169 image points was therefore
effectively obtained.
The COMP program contains both ray trace and dif-
fraction analysis capabilities. The advantages of this pro-
grain are that it can automatically calculate highly aber-
rated or off-axis diffraction patterns while providing direct
access to optical parameters for modification and sensitiv-
ity testing. COMP is not a design tool, but an analysis
subroutine that can automatically generate sets of images
or sensitivity tables.
For this astrograph study, 902 rays were traced through
the system to produce each image. The rays were traced
to the focal plane, then back-propagated to the cxit pupil
onto a spherical reference surface whose center of curva-
ture was defined by the intersection of the chief ray and
the focal plane. The spherical phase term was removed,
and a Fourier transform was performed in order to cal-
culate a monochromatic image. The inaage was stored as
a 100 x 100 array of brightness values, with each num-
ber representing the intensity within a 1-pro square in
88
the focal plane. Subsequent wavelengths were treated in
the same fashion, except that the reference sphere was al-
ways centered on the same point as the first one. The five
monochromatic images (at A = 705.6, 650.0, 610.0, 587.6,
and 546.0 nm) for each source position were then coadded
to produce the final image files.
Images formed by the astrograph are diffraction limited
on-axis, and show a slight coma at 3 deg off-axis. The
first Airy ring is not broken, indicating weak aberration.
Chromatic aberration was negligible.
One important question is to determine how much dis-
tortion is introduced by the finite number of rays, the fi-
nite number of Fourier grid points, the stepwise change in
aperture representation due to vignetting, and pixellation
effects in the image plane. One metric of the amplitude of
these effects is to compare the ray centroids to diffraction
centroids. One finds that diffraction centroids are within
0.8 mas of ray centroids throughout the field, which is well
within the accuracy required for this study. A second met-
ric is simply to increase the number of rays and grid points.
Again, the distortion did not significantly change.
A second question involves the accuracy of the poly-
chromatic approximation in this modeling. Because only
five wavelengths were used to sinmlate a broad band, the
composite Airy pattern is not a true representation of the
broadband image. However, an empirical study showed
that the major difference occurred only near the first and
second ring minima. Because the images remain highly
symmetric throughout the field, the small error does not
significantly shift the centroids. Further, the ring minima
regions are not particularly important in image centroid-
ing.
III. Image Centroiding
hnage centroids for each image file were determined by
using a data number (DN) filter. Each image is modeled as
the sum of a point-spread function (PSF) and a constant
background; the PSF itself is smoothed from a square ar-
ray of brightness values. The coordinates of the center of
the PSF, the amplitude of the PSF, and the background
level are the unknowns in an iterated linear least-square
process. The on-axis image was used as the PSF for all
images. The brightness values (DN's) in each pixel of an
image form the observation set. The variance in the DN
in each pixel was composed of two parts: a constant to
represent read noise, and a part proportional to the DN
level itself to represent photon statistics. The resulting
centroidscarrya formalerrorof -4-0.025 pm in the focal of (i + j) or (2k + 1) as appropriate. The formal errors of
plane (1.4 mas on the sky). The centroids were first com- ttle coefficients are at most 1 in the final digit shown.
puted in pixel and line coordinates within each image file,
then transformed into millimeter coordinates in the focal
plane.
IV. Plate Constants
Given the image centroids determined above and the
ideal coordinates of each image, one seeks a transforma-
tion from measured to ideal coordinates. Following [4]
or [5], denote the measured coordinates by (x, y) and the
ideal coordinates by (_, r/). Model the transformation as a
polynomial in each coordinate:
_= E EaijxiyJ
i j
and similarly for r/. The two coordinates are handled in-
dependently, and symmetry allows one to examine only _.
The aij are the solution parameters (the plate constants,
for in classical astrometry these are found for each pho-
tographic plate from reference star images on that plate).
The best model is generally considered to be the one with
the fewest parameters such that the fit is acceptable.
From symmetry, any displacement of an image centroid
from its ideal location must be in the radial direction, as
ill this analysis all the astrograph lenses were perfectly
aligned. Aberration theory in geometrical optics also dic-
tates that the displacement must be proportional to an
odd power of the separation r of the image from the op-
tical axis. Thus, the leading nonlinear term in x is xr _,
which is represented in the plate model by the two coef-
ficients a30 and a12. Trial solutions confirmed that the
only nonzero terms in the plate model were those with i
odd and j even. Accordingly, one can construct a more
restrictive model,
in which tile only displacements allowed were proportional
to odd powers of r.
Table 1 gives coefficients for various solutions: those in
terms of aij on the top, and those in terms of bk at the
bottom. Each of the maximum degree numbers represents
the highest degree of the solution; this is the largest value
Table 2 summarizes the post-fit residuals for each case.
The residuals generally reach a maximum in the corners of
the field, but there is a secondary maximum not far from
the origin. The table gives these values as well as the rms
residual for all 169 images.
It is evident that low-order fits do not model the actual
image centroid locations well in the corners of the full 6-deg
x 6-deg field. The analysis was therefore repeated using
only a 5-deg x 5-deg field by omitting the first and last
row and column of images. The results appear in Tables 3
and 4. Now a third-degree fit meets the 5-mas requirement
even in the corners of the field.
V. Conclusions and Discussion
The ideal projection is obviously not sufficient for a full
6-deg x 6-deg field, as distortions in this instrument reach
0.385 arcsec (in each coordinate) in the corners of the field.
A third-degree fit removes most of the distortion, leaving
residuals on the order of 2-7 mas or 10-35 nrad. In order
to achieve the design goal of 25 nrad, it will be necessary
to include the fifth-degree term or else to restrict the field
of view to 5-deg square.
It is clear from the size of the residuals (Table 2) that
it should be sufficient to model the astrograph by
= box + blr2x + b2r4x
the full generality of the aij should not be necessary. This
is true, however, only if the optics are perfectly aligned.
Any one of tile five lenses can move in any direction by
up to 8 pm without introducing more than 5 mas of ad-
ditional aberration? Random motions of this size are not
likely (except for thermal expansion), as the lenses would
be rigidly mounted in a cell. IIowever, it is not at all
clear that the assembly process can be controlled to that
tolerance. Although one can expect stability, one should
also expect the instrument to be misaligned, and these
inevitable misalignments will modify the radial distortion
pattern of the perfectly aligned system. Accordingly, the
3 S, Shaklan and J. Yu, "Images Formed With 36-cm Astrograph:
Results to Supersede OSAS DFM 91-166," JPL Interoffice Mem-
orandum OSAS DFM 91-173 (internal document), Jet Propulsion
Laboratory, Pasadena, California, November 8, 1991.
89
full model in terms of a,j should be employed if the instru-
ment is built.
In practice, most astrometrists are loath to include
terms higher than third degree in their plate solutions,
simply because most star fields do not contain enough
reference stars to support the higher terms. Also, as
mentioned above, in most cases the plate constants are
determined independently for each plate, because non-
instrumental effects such as atmospheric refraction and
dispersion will randonfly corrupt the instrumental distor-
tions. These real-world considerations do not invalidate
this study. Rather, the instrumental distortion model
should be found by testing the optics ill the laboratory,
and the model parameters and their covariance used as
a priori information when reducing actual observations.
Outside the scope of this article is the requirement tc
measure image location to the 5-mas level. For this in-
strument, this requirement translates into a measurement
precision (and accuracy) of about 0.1 /am. Maintaining
this accuracy over a region nearly 40 cm on a side may be
challenging.
In conclusion, the candidate astrograph design suffers
from radial distortion but that can be modeled with one
parameter provided that the field of view is 5 deg on a side.
Including terms to fifth degree will produce a model good
to 1 mas and will allow the use of a 6-deg field, but the
coefficients may be difficult to determine in practice. The
instrument design is therefore viable provided that detec-
tors can be found to exploit its excellent optical qualities.
Acknowledgment
The work of Jeffrey Yu to help create the image files for this analysis is gratefully
acknowledged.
References
[1] G. D. Gatewood, "The Multichannel Astrometric Photometer and Atmospheric
Linfitations in the Measurement of Relative Positions," Astron. J., vol. 94, no. 1,
pp. 213-224, July 1987.
[2] M. A. C. Perryman et al., The Hipparcos Satellite, European Space Agency,
Publication SP-1111, vols. I-III, Paris, France, June 1989.
[3] M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston,
D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, and G. H. Kaplan,
"The Mark III Stellar Interferometer," Asfron. Aslrophys., vol. 193, pp. 357-371,
1988.
[4] A. KSnig, "Astronomy with Astrographs," in Astronomical Techniques: Vol. 2,
Stars and Stellar Systems, edited by A. Hiltner, Chicago: University of Chicago
Press, pp. 461-486, 1962.
[5] H. Eichhorn, Astronomy of Star Positions, New York: Frederick Ungar Publish-
ing Co., pp. 68-76, 1974.
9o
Table 1. Solution parameters for 6-deg field.
Maximum degree
Parameter
1 3 5 7
alo 0.9999655948 1.000008736 1.000009976 1.000010197
a3o --0.0011189094 -0.0011969770 -0.0012239529
a12 -0.0011063839 --0.0011871044 -0.0012123322
aso 0.0011528 0.002152
a32 0.0018920 0.003283
a14 0.0009628 01001705
aTo --0.0117
a25 --0.0196
a34 --0,0147
a16 --0.0070
bo 0.9999655948 1.000008630 1.000009925 1.000010154
bl -0.0011116554 -0.0011895162 -0.001214510
b2 0.00097727 0.0017079
ba -0.00615
Table 2. Residual statistics for 6-deg field.
Using aij
RMS residual, pm
RMS residual, mas
V¢orst central residual, pm
Worst central residual, mas
_,Vorst corner residual, #m
_¥orst corner residual, mas
Maximum degree
3 5 7
2.1277 0.0349 0.0043 0.0021
121.91 2.00 0.25 0.12
3.233 0.073 0.014 0.007
185.2 4.2 0.8 0.4
6.720 0.118 0.009 0.003
385.0 6.8 0.5 0.2
Using bk
Maximum degree
1 3 5 7
RMS residual, _m
RMS residual, mas
Worst central residual, _m
"Worst central residual, mas
Worst corner residual, #m
_¥orst corner residual, mas
2.1277 0.0369 0.0050 0.0027
121,91 2.11 0.28 0.15
3.233 0.078 0.015 0,009
185.2 4.5 0.9 0.5
6.720 0.125 0.010 0.004
385.0 7.2 0.6 0.2
91
Table 3. Solution parameters for 5-deg field.
Maximum degree
Using a 0
1 3 5 7
axo 0.9999779128 1.000009343 1.000010126 1.000010264
a3o -0.0011415289 -0.001211976 -0.00122835
a12 -0.0011294587 -0.001200062 -0.00122944
aso 0.0014983 0.001925
u32 0.0023167 0.004635
a ] 4 0.0011932 0.002338
a7o 0.00
a25 -0.04
a34 --0.04
a16 --0.01
bo 0.9999779128 1.000009268 1.000010085 1.000010271
bl -0.0011344615 -0.0012033140 -0.001232187
b2 0.00121582 0.002412
b3 -0.0143
Table 4. Residual statistics for 5-deg field.
Using a U
Maximum degree
1 3 5 7
F/MS residual, I*m
RMS residuM, mas
X,Vot*t central residual, pm
Worst central residual, mas
Worst corner residual, pm
Worst comer residual, mas
1.0925 0.0153 0.0025 0.0014
62.59 0.88 O. 15 0.08
2.034 0.040 0.009 0.006
116.5 2.3 0.5 0.3
3.841 0.052 0.005 0.003
220.1 3.0 0.3 0.2
Using bk
Maximum degree
1 3 5 7
RMS residual, pm
1RMS residual, mas
Worst central residuM, /_m
V_'m_t central residual, mas
Worst corner residual, #m
x,Vox_t corner residual, mas
1.0925 0.0163 0.0030 0.0018
62.59 0.94 0.17 0.10
2.034 0.055 0.010 0.005
116.5 3.2 0.6 0.3
3.841 0.043 0.005 0.003
220.1 2.5 0.3 0.2
92
y0 0 0 0 0 0 0 0 0 0 0 0 @
0 0 0 0 0 0 0 0 0 0 0 @ @
0 0 0 0 0 0 0 0 0 0 @ @ @
0 0 0 0 0 0 0 0 0 @ @ @ @
0 0 0 0 0 0 0 0 II II @ @ @
0 0 0 0 0 0 0 @ @ @ @ @ @
0 0 0 0 0 0 @ @ @ @ @ @ @ ----_ x
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
Fig. 1. Layout of Images In the astrograph focal plane. Points
are placed at 0.5-dog intervals from --3 to -_-3 dog in each co-
ordinate. Filled circles correspond to Images aclually produced;
open circles were derived from symmetry considerations.
93


Geometric distortion analysis of a wide-field astrograph

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