| BZs >gnuplot.ps


Acta Polytechnica Vol. 51 No. 2/2011

Advanced Processing of Images Obtained from Wide-field
Astronomical Optical Systems

M. Řeřábek, P. Páta

Abstract

The principal aim of this paper is to present a general view of the special optical systems used for acquiring astronomical
image data, commonly referred to as WFC or UWFC (Ultra Wide Field Camera), and of their transfer characteristics.
UWFC image data analysis is very difficult in general, not only because the systems have so-called space variant (SV)
properties. Images obtained from UWFC systems are usually incorrectly presented due to a wide range of optical
aberrations and distortions. The influence of the optical aberrations increases towards the margins of the field of
view. These aberrations distort the point spread function of the optical system and rapidly cut the accuracy of the
measurements. This paper deals with simulation and modelling of the UWFC optical systems used in astronomy and
their transfer characteristics.

Keywords: PSF, UWFC, Zernike polynomial, aberration, space variant, deconvolution algorithm.

1 Introduction
The properties of UWFC astronomical systems along
with specific visual data in astronomical images con-
tribute to complicated evaluation of acquired im-
age data. These systems contain many different
kinds of optical aberrations, which have a negative
impact on the image quality and imaging system
transfer characteristics. Therefore, for precise as-
tronomical measurements (astrometry, photometry)
over the entire field of view (FOV), it is very im-
portant to comprehend how the optical aberrations
affect the transfer characteristics of the optical sys-
tem. Another question that arises is how the as-
tronomical measurements depend on optical aber-
rations. A definition of the accurate point spread
function (PSF) at any point in the FOV of the op-
tical system could help us to restore the original
image. Optical aberrations models for linear space
invariant and variant (LSI/LSV) systems are there-
fore outlined in this paper. These models based on
Zernike polynomials serve us as suitable tools for
understanding how to estimate and fit the wave-
front aberration of a real optical system, and give
us an idea of intrinsic PSF. When the model of
the PSF of a real acquisition system is known, we
can use it for restoring the original image or for
a precise evaluation of the astronomical measure-
ments. Two experiments are presented in this pa-
per. The first describes PSF model retrieval when
we have original image data. The second focuses on
using the PSF model in known deconvolution algo-
rithms with which we can restore the original im-
age.

2 Astronomical image data
processing and acquisition

Most astronomical image data is provided by auto-
matic robotic astronomical systems. The main idea
of these automatic astronomical optical systems is
based on long-term image data collection. The data
is acquired with specific characteristics. These in-
clude especially focal length, spectral bands and sen-
sor parameters. According to the focal length that
is used, we can distinguish image data acquired in
the primary focus of a big telescope (deep sky), a
wide-field lens or a “fish-eye” lens for all sky data
imaging. Astronomical image data contains specific
visual data. An astronomical image usually consists
of the dark background of the starry sky, together
with bright points and clumps, which represent stars
and galaxies. Image data can be divided into four
groups [7]: Flat Field, Dark Frame, Light Image and
Deep Sky Light Image. Before processing and eval-
uating astronomical images, it is necessary to make
corrections of these images. Dark frame compensa-
tion and flat field compensation are two of the most
frequently used pre-processing methods.

Real data from double-station video observation
of meteors [10, 11] is used for our simulations and
modelling. UWFC image data analysis is very dif-
ficult in general. There are many different kinds of
optical aberrations and distortions in these systems.
Moreover, the objects in ultra wide-field images are
very small (a few pixels per object dimension). The
influence of the optical aberrations increases most to-

90



Acta Polytechnica Vol. 51 No. 2/2011

Fig. 1: Block diagram of image processing in a video observation system

wards the margins of the FOV (high angular distance
from the optical axis of the system). These aber-
rations distort the PSF of the optical system and
rapidly cut the accuracy of the measurements. The
optical aberrations are dependent on spatial data,
which affects the transfer characteristics of optical
systems and makes them spatially variant.

3 Stellar object profile

There are two common functions for fitting the pro-
files of stars — Gaussian and Moffat [7, 11]. The aim
is to match a star’s profile as well as possible with
the Gaussian or Moffat profile, and then to store the
parameters of the fit. If the star is ideal, it will be
represented by a small “dot” — PSF (Point Spread
Function). Unfortunately, because of many differ-
ent distortions (see below), the dot is ”blurred” all
around, and the star’s profile is close to the Gaus-
sian function. The more it is blurred the worse is the
PSF of the whole imaging system. If the system is
linear and space-invariant, the PSF of all stars will
be the same. The centre of a star usually has a profile
closer to the Gaussian function, while the more dis-
tant parts of a star are closer to the Moffat profile.
Hence the ideal fitting function combines Gaussian
and Moffat profiles. It is obvious that the centre of
the star is fitted very well, whereas the more distant
parts are fitted only poorly. Figure 1 presents a sim-
plified block diagram of various factors affecting the
final image and stellar profile. We assume the first
two blocks for our purposes; however, we consider
that these blocks are placed in a black box. In our
experiments, we do not take into account a model
of atmospheric turbulence. Other blocks implicate
disruptive effects, which bring unwanted artefacts in
PSF. One of the most difficult effects is brought on
by recording and compression. In order to improve
the subjective image quality, the video tape recording
and even JPEG compression use a sharpening mech-
anism which results in rapid transients at the edge of
PSF. These parts of the electro-optics transfer system
are not considered in our simulations and modelling.

4 Optical aberration
modelling

Optical aberration can be described using a so-called

wave aberration function. The wave aberration [1, 5]
function is defined as the distance (in optical path
length) from the reference sphere to the wavefront
in the exit pupil measured along the ray. Zernike
polynomials are used for describing high order wave-
front aberrations with high precision. Zernike poly-
nomials are normally expressed in polar coordinates
(ρ, θ) in the exit pupil, where 0 ≤ ρ ≤ 1, 0 ≤
θ ≤ 2π. A Zernike polynomial consists of two fac-
tors, the normalization factor N mn and the radial
polynomial R|m|n (ρ). Zernike polynomials are defined
as [1, 4]:

Zmn (ρ, θ) =

⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩

N mn R
|m|
n (ρ) cos(mθ)

for m ≥ 0, 0 ≤ ρ ≤ 1, 0 ≤ θ ≤ 2π

−N mn R
|m|
n (ρ) sin(mθ)

for , m < 0, 0 ≤ ρ ≤ 1, 0 ≤ θ ≤ 2π,

(1)

for a given n, the number m can take values of
−n, −n + 2, −n + 4, . . . , n. The normalization fac-
tor N mn and the radial polynomial R

|m|
n (ρ) can be

expressed as

N mn =

√
2(n + 1)
1 + δm0

,

R|m|n (ρ) = (2)
n−|m|
2∑

s=0

(−1)S (n − s)!
s![0,5(n + |m|) − s]![0,5(n − |m|) − s]!

ρn−2s.

where δm0 = 1 for m = 0, δm0 = 0 for m �= 0. The
wavefront aberration function may be expressed by
Zernike polynomials [1, 5, 7] as

W (ρ, θ) =
k∑
n

n∑
m=−n

W mn Z
m
n (ρ, θ) =

k∑
n

{
−1∑

m=−n
W mn

(
−N mn R

|m|
n (ρ) sin(mθ)

)
+

n∑
m=0

W mn

(
N mn R

|m|
n (ρ) cos(mθ)

)}
, (3)

where k is the polynomial order of the expansion, and
W mn is the coefficient of the Z

m
n mode in the expan-

sion, i.e. it is equal to the RMS wavefront error for
that mode.

91



Acta Polytechnica Vol. 51 No. 2/2011

a) b) c)

Fig. 2: Visualization of coma and astigmatism. a) The testing image. b) SV image distorted by coma. c) SV image
distorted by astigmatism

5 Effect of wavefront
aberration on the PSF

Optical systems with all aberrations compensated are
called diffraction limited. The influences of aberra-
tions on image quality can be expressed as the wave-
front error at the exit pupil, and their effects on
the transfer characteristics can be expressed as the
change in the PSF size and shape. Changes in the
size and shape of PSF blur the image. When an
imaging system is diffraction limited, the PSF con-
sists of the Fraunhofer diffraction pattern of the exit
pupil [2]. The relation between the object and the
image of a space invariant optical system can be ex-
pressed by the convolution in the spatial domain (the
object irradiance distribution is convolved with the
impulse response) [2]. The PSF of the LSI optical
imaging system can be expressed as

P SF (u, v, δ, ϕ) =∣∣∣∣F T
{

p(x, y) exp

(
−i

2π
λ

W (ρ, θ, δ, ϕ)

)}∣∣∣∣
2

, (4)

where p(x, y) defines the shape, size and transmission
of the exit pupil, exp[(−i2π/λ)W (ρ, θ, δ, φ)] accounts
for the phase deviation of the wavefront from a ref-
erence sphere, and (δ, ϕ) are the polar coordinates at
the object plane — these coordinates equal (0, 0) all
the time if we consider an LSI optical system.

If we assume the spatial variant system with
anisotropic PSF, then the aberration wavefront is dif-
ferent for each source point, see Figure 2. The real
image is always obtained from the spatially variant
optical system. An example of an image acquired
from a real SV optical system is shown in Figure 3a,
where can we also see the influences of aberrations in
the middle of the image and at the edge. The PSF
of this optical system is different in the middle of the
image and at the edge, due to optical aberrations.
The differences of the PSFs in the two positions are
compared in Figure 3b and Figure 3c.

a)

b)

c)

Fig. 3: a) Input imagedata fromthesystemat theOndře-
jov base. b) The object profile near the optical axes. c)
Object profile on the edge of FOV

92



Acta Polytechnica Vol. 51 No. 2/2011

Fig. 4: Ellipticity (an ideal PSF deformation) of the profiles of stars as a function of the angle distance from the middle
of an all-sky image

a) b)

Fig. 5: a) Dependence of the astrometric error (in pixels) on wavefront aberration error (in fraction of λ), b) RMS error
(in levels) versus wavefront aberration error (in fraction of λ)

6 Effect of Wavefront
Aberration on Astrometric
Measurements

Unlike aperture photometry, precise astrometry de-
mands high quality images. Even slight distortion
may cause inaccurate determination of the position
or movement of a stellar object. The error depends
on the magnitude of the measured star — the greater
the magnitude, the greater the error can be. Even a
quite small compression rate may cause a wrong de-
tection, especially when considering faint stars.

If the system is space-variant, the PSF of the stars
in an image differs. The profiles of stars are not cir-
cular but rather elliptical, especially in the case of
inferior lenses and greater distance from the middle
of the image. The ellipticity is qualified by σx and σy
(while considering the Gaussian function), where σx
and σy are the distance from the centre of the Gaus-
sian at which the Gaussian is equal to exp(−0,5) of
its central value. σx and σy are the major and minor
half-axis of the ellipse, perpendicular to each other.
The ellipticity in the radial direction can grow signifi-
cantly with the angle distance from the middle of the
image, especially for wide-angle images, see Figure 4.

All of the measurements in this section were done
in IRAF (Image Reduction and Analysis Facility)
software [13]. This software system enables the user
to measure, reduce and analyse astronomical data
and write her/his own scripts. The program is “open
source”. The reader can find further information at
http://iraf.noao.edu. SkyCat can be used for visual-
ization of images and for access to catalogues, for
example http://archive.eso.org/skycat.

The effect of coma wavefront aberration on the
precision of astrometric measurements (e.g. posi-
tion of objects) is demonstrated in Figure 5a. We
can see that the coma aberration has no influence on
the precision of astrometric measurements for values
of wavefront aberration error less than λ/50. The
measurement error grows with the wavefront aberra-
tion error, and for wavefront aberrations error values
greater than λ/10 it is no longer acceptable. The
second graph represents the effect of coma wavefront
aberration on the RMSE of a distorted image in sig-
nal levels, see Figure 5b. This effect is insignificant
for wavefront aberration error values less than λ/100.
RMSE further increases with wavefront aberration
error. For wavefront aberration error greater than
λ/10, the increase in RMSE values is over 30 %, and
is also no longer acceptable. We assume only the

93



Acta Polytechnica Vol. 51 No. 2/2011

a) b)

Fig. 6: a) Dependence of the brightness stars on distance from centre of FOV, b) Dependence of the magnitude error of
stars on the distance from centre of FOV

coma aberration included in the optical shift invari-
ant system.

7 Experiments and Results

7.1 Estimation of aberration

If we want to use the deconvolution algorithm for pre-
cise restoration of the SV UWFC system, we need to
know what the PSF looks like. One approach is to
find the PSF of the optical system from empirical
image data. Here we describe the procedure for ob-
taining the model of PSF from real image data.

Our experiments involve analysing real image
data and comparing it with the modelled image data.
Real data from the double-station video observation
project is used for our experiments. The stars vary
not only their own position, but also their brightness
and shape. The space variant properties are repre-
sented in Figures 6a) and 6b), which show how the
magnitude and the magnitude error vary in depen-
dence on distance from the centre of FOV. In princi-
ple, the brightness of stars has a decreasing character.

Within the double-station video observation
project we obtain a sequence in non-compressed AVI
format from the observation system. A star which
changes its position from the edge to the middle of
the image, or vice versa, is taken as a suitable for
our purposes. At first, several frames from each se-
quence form the uncorrected testing image, which is
determined as the median of the used frames. In the
next step, we pre-process the image using flat field
and dark frame correction. We can also use noise
and background removal. The next step is to iso-

late the star at discrete times when it changes its
own position from the middle of FOV to the edge
of FOV. We therefore cut the stars as sub-images
with 15 by 15 pixel proportions. Then the compar-
ison of the real stellar profile and the model profile
is implemented [12]. A generalized block diagram
of the algorithm that estimates the optical aberra-
tions of the real system is shown in Figure 7. The
input parameters are the wavefront aberration (i.e.,
the Zernike coefficients) and PSF computed accord-
ing equation (4). Then the real object profile is com-
pared with the suggested model — both of them have
the same FWHM [7]. The suitability of the result-
ing PSF for the model is evaluated on the basis of
the RMS error between the modelled PSF and the
real object profile. The result for the chosen star is
presented in Figure 8. As can be seen, this model as-
sumes only two varying aberrations (coma and astig-
matism) with a constant defocus value.

Fig. 7: Generalized block diagram of an estimation algo-
rithm

a) b) c) d)

Fig. 8: Results of estimation. a) Model of the star profile, b) Original of the chosen star, c) Differential image for the
estimated and real star profile, d) Dependence of the RMS error profile on coma and astigmatism wavefront aberration
error
94



Acta Polytechnica Vol. 51 No. 2/2011

a) b) c)

Fig. 9: a) Model of the PSI optical system, b) Source testing image, c) Image passed through the SV optical system with
coma aberration only

a) b) c)

Fig. 10: Image restored using the Wiener deconvolution algorithm, b) Deconvolution using the Lucy — Richardson
algorithm, c) Deconvolution using the maximum likelihood algorithm

7.2 Model of spatially variant
deconvolution

The principal difficulty in spatially variant (SV) sys-
tems is that the Fourier approach can no longer be
used for restoring [6] the original image. If we want
to use the Fourier method for the deconvolution pro-
cess, we need to split the original image. The transfer
characteristics of each part are described by unique
PSF. This system is referred to as a partially space
invariant system [8] and we use it in our experiments.
Such a system has parametric PSF — for each value
of a parameter (in our case it is the coordinate of the
object at the object plane) PSF takes a different size
and shape according to the aberrations that are con-
tained. Thus we obtain a number of PSFs, one for
each region referred to as an isoplanatic patch [8],
within which the system is approximately invariant.
To describe the imaging system fully, the impulse re-
sponse appropriate for each isoplanatic patch should
be specified.

The wavefront aberration function for the LSV
optical system can be described as

W (ρ, θ, δ, ϕ) =
k∑
n

n∑
m=−n

W mn (δ, ϕ)Z
m
n (ρ, ϕ−θ), (5)

where W mn (δ, ϕ) is the RMS wavefront error for aber-

ration mode m, n and for object polar coordinate
(δ, ϕ). Using the partially space invariant system al-
lows us to describe the transfer characteristics in in-
dividual patches by Fourier approaches, according to
equation (4). Thus we obtain a number of space in-
variant PSFs, one for each isoplanatic patch. Now we
can also use the Fourier approach for image deconvo-
lution. The optical system is divided into 40 isopla-
natic patches in our experiments, see Figure 9. Three
deconvolution algorithms [3, 4] — the Wiener, Lucy-
Richardson and maximum likelihood algorithm —
are used for restoring the original image, see Fig-
ure 10.

8 Conclusions
This paper has discussed ways of improving astro-
nomical measurements. The first approach is to use
the novel star profile model for restoring the original
image, which is undistorted and is suitable for precise
astrometry using ordinary fitting models (Gaussian,
Moffat). The second approach for improving astro-
nomical measurements is to use a different evalua-
tion and detection algorithm. This involves creating
a new PSF fitting model based, e.g., on the Zernike
polynomial and using it for detection and astrometric
measurements. The situation is complicated by fact
that all wide field astronomical observational systems

95



Acta Polytechnica Vol. 51 No. 2/2011

are SV. Therefore, the shape of a stellar object varies
across the entire FOV, and Gaussian or Moffat fitting
procedures are useless for precise astrometric mea-
surements, especially towards the margins of FOV.

An experiment for estimating the optical aber-
rations of real optical systems was implemented. A
comparison of the model PSF and the real object
profile was performed using the RMS error. Obvi-
ously the model should have more input parameters,
especially more varying optical aberrations. SV de-
convolution was addressed, and a model of the par-
tially space variant optical system was implemented.
Only the “brute force” method was used for restoring
the test images. The results of various deconvolution
algorithms were demonstrated.

Acknowledgement

This work has been supported by grants no.
GA205/09/1302 “Study of sporadic meteors and
weak meteor showers using automatic video inten-
sifiers cameras”, and GA P102/10/1320 “Research
and modelling of advanced methods of image quality
evaluation” of the Grant Agency of the Czech Re-
public.

References

[1] Born, M., Wolf, E.: Principles of Optics. 2nd ed.
London, Pergamon Press, 1993.

[2] Goodman, J. W.: Introduction to Fourier Op-
tics. Boston, McGraw-Hill, 1996.

[3] Campisi, P., Egiazarian, K.: Blind Image De-
convolution: Theory and Applications. CRC,
2007.

[4] Janson, P. A.: Deconvolution of Images and
Spectra. 2nd edition. Academic Press, London,
1997.

[5] Welford, W. T.: Aberration of the Symmetrical
Optical System. Academic Press, London, 1974.

[6] Shannon, R. R., Wyant, J. C.: Applied Op-
tics and Optical Engineering. Academic Press,
Lodon, 1992.

[7] Starck, J., Murtagh, F.: Astronomical Image
and Data Analysis. Springer Verlag, Berlin,
2002.

[8] Trussel, H. J., Hunt, B. R.: IEEE Trans. Acous-
tic Speech Sig. Proc., 608, 157, 1978.

[9] Maeda, P. Y.: Zernike Polynomials and their
Use in Describing the Wavefront Aberrations of
the Human Eye. 2003.

[10] Hudec, R., Spurny, M., Krizek, P., Páta, P.,
Řeřábek, M.: Low-Cost Optical All-Sky Mon-
itor for Detection of Bright OTs of GRBs In:
Gamma-Ray Burst: Sixth Huntsville Sympo-
sium. Austin: American Institute of Physics,
2009, p. 215–217. ISBN 978-0-7354-0670-4.

[11] Řeřábek, M., Páta, P., Koten, P.: Process-
ing of the Astronomical Image Data obtained
from UWFC Optical Systems. In Proceedings
of SPIE – 7076 – Image Reconstruction from
Incomplete Data V. Washington: SPIE, 2008.
ISBN 978-0-8194-7296-0.

[12] Řeřábek, M., Páta, P.: Advanced processing
of astronomical images obtained from optical
systems with ultra wide field of view In: Ap-
plications of Digital Image Processing XXXII.
Bellingham (Washington State): SPIE, 2009,
p. 1–4. ISBN 978-0-8194-7733-0.

[13] IRAF – Image Reduction and Analysis Facility,
http://iraf.noao.edu/, 2007.

M. Řeřábek
P. Páta
E-mail: rerabem@.fel.cvut.cz
Czech Technical University in Prague
Faculty of Electrical Engineering
Department of Radio Engineering
Technická 2, 166 27 Prague 6, Czech Republic

96