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Ïîèñêîâûå ñëîâà: arp 220
The Restoration of HST Images and Spectra II
Space Telescope Science Institute, 1994
R. J. Hanisch and R. L. White, eds.
Interactive Deconvolution with Error Analysis
K. Bouyoucef, D. Fraix­Burnet, and S. Roques
Laboratoire d'Astrophysique, CNRS URA 285/GdR 134, Universit# e Toulouse III,
14 Avenue Edouard Belin, 31400 Toulouse, France
Abstract. In the general framework of regularization of inverse problems, we de®ne the
main contours of the reconstruction algorithm IDEA (Interactive Deconvolution with Error
Analysis) that is developed in our group, giving only some methodological principles. The
deconvolution problem is stated in terms of weighted spectral interpolation: the amount
and the nature of the interpolation to be performed is related # in a quantitative manner
# to the choice of a synthetic coverage linked to the target resolution. We illustrate this
deterministic viewpoint on the SN1987A image of HST and compare the result with two
Bayesian approaches: the Richardson­Lucy Method (RLM) and the Maximum Entropy
Method (MEM) in terms of gain in resolution, error propagation, speed of convergence, and
some relevant astrophysical criteria.
1. Introduction
The underlying principle used to restore an image is known generally as a reconstruction or inverse
problem. This usually involves ®nding a method of inverting certain classes y = Ax of integral
equations. In the case of a deconvolution problem, these equations are of the type OE i (z) =
RR
h(z \Gamma z 0 )OE o (z 0 ) dz 0 where h is the kernel associated with the operator A. This is basically
an ill­posed problem (Tikhonov and Arsenin 1977) because it does not ful®ll the three Hadamard
conditions of existence, uniqueness, and stability of the solution. The last condition of stability
causes the main problems because if A \Gamma1 is not continuous, the solutions x are unstable with respect
to slight variations of initial data y. This means in practice that a slight error in the data may lead
to a very large error in the solution.
When the problem is discretized, its transposition into ®nite dimension certainly eliminates
the question of continuity of A \Gamma1 , but the dif®culties of the underlying in®nite problem result again
in numerical instabilities due to the ill­conditioned nature of the matrix of the system. As a result,
the noise is enhanced and the solutions may be singular. Let ffi y be some variation of the data
y. Denoting by ffix the corresponding variation of x, one has kffixk=kxk Ÿ C kffiyk=kyk, where
C = ¯ max =¯ min , ¯ 2
max and ¯ 2
min being the largest and the smallest eigenvalues of the imaging
operator. The condition number C provides then a measure of the dif®culty about the resolution of
equation y = Ax, and hence a measure of the robustness of the problem under consideration.
To circumvent the ill­conditionedness nature of this problem, one is led to postulate that the
properties of the solution are not entirely contained in the equation to be solved. One is then led to
introduce a priori information on the solution which takes a regularizing part in the deconvolution
process. This information is the basis for classes of linear or non­linear methods which are known
as deterministic or Bayesian ones.
2. IDEA: Methodological Principles
The reader will ®nd extensive studies in Lannes et al. (1987a­b) where the method summarized here
is presented in a closed form, and some astrophysical applications in Fraix­Burnet et al. (1989),
Roques et al. (1993) and Bouyoucef et al. (1994). To a ®rst approximation, the experimental data
64

Interactive Deconvolution with Error Analysis 65
OE i are related to the original object OE o , the intensity of the source at some high level of resolution,
by an experimental transfer relation:
OE i (z) = (OE o \Lambda h)(z) + error term
where h is the Point Spread Function. The error term includes random or systematic errors (e.g.,
errors on the determination of the PSF, linearity assumption of the imaging relation, image sampling)
and signal­uncorrelated random noise (telescope, detectors, atmosphere, guiding, etc.).
The accuracy of the approximation considered when writing the convolution equation is con­
trolled in Fourier space by a pointwise image­error bound oe i (u) such that j b
OE i (u) \Gamma h(u) b
OE o (u)j Ÿ
oe i (u). Moreover, OE o can be regarded as a member of a certain family of objects. For each spa­
tial frequency u, it is possible to exhibit a suitable upper bound of the Fourier transform of OE o :
j b
OE o j Ÿ oe 0 (u).
Due to the noise­type and systematic errors characterized by the pointwise Signal­to­Noise
Ratio (SNR) in the frequency space:
SNR(u) = j b
OE i (u)j
oe i (u)
it is preferable to give up trying to determine OE o at its highest level of resolution. One then
de®nes the #object to be reconstructed# OE s as a smoothed version of OE o by a relation of the form
b
OE s (u) = b s(u) b
OE o (u), where b
s(u) is some synthetic transfer function small in the mean square
sense outside some extent H r . This domain is the frequency coverage to be synthesized and
regularizes the effective support H of the transfer function b h. The choice of its diameter is of
fundamental importance because it is closely related to the resolution limit of the reconstruction
process. Intuitively, the greater is H r with respect to H , the greater is the gain in resolution, but at
the same time, the less stable is the deconvolution process. The problem is then to de®ne the best
compromise between the resolution to reach and the stability of the solution. The support of OE s is
contained in some ®nite region V a priori known, which size and shape, determined in an interactive
manner, will prove to play, together with H r , an essential part.
This ®lter b
s ful®lls three conditions:
1. As the support of b h is essentially in H , the support of b s is essentially in H r (say for instance
95% of its energy in H r ). One imposes a condition of the form:
1
kb s(u)k 2
Z
Hr
jb s(u)j 2 du = ¸ 2
2. The support of s must be as small as possible (with respect to the choices of H r and ¸) to
ensure the best possible resolution for OE s .
3. b
s(0) = 1 what secures #ux conservation.
A ®rst approximation b
OE t of b
OE s of the object to be reconstructed is:
b
OE t (u) =
8 !
:
b s(u)
h(u)
if SNR(u) – ff t
0 otherwise
where ff t is some threshold value of the order of unity.
In the frequency domain where the SNR is considered as good (SNR(u) – ff t ), b
OE t is more
or less reliable. One then assigns to the data a weight depending on the values of SNR. Let ff 0
t be
the threshold value beyond which the whole information contained in b
OE t is considered as entirely

66 Bouyoucef, Fraix­Burnet, & Roques
reliable. Let g(u) be the #weight function# characterizing the weight attached to this information.
The function g(u) is de®ned as follows:
g(u) =
8 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? !
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? :
1 outside H r
1 in H r if SNR(u) ? ff 0
t
(information entirely reliable)
SNR(u) \Gamma ff t
ff 0
t \Gamma ff t
in H r if ff t Ÿ SNR(u) ! ff 0
t
(information partially reliable)
0 in H r if SNR(u) ! ff t
(information non reliable)
One supposes that SNR(u) ! ff t outside H r . So de®ned, g characterizes the way of ground­
ing the reconstruction on the information. Our deterministic procedure based on a least squares
minimization de®nes the reconstructed object as the function which minimizes the functional:
q(OE) = kg( b
OE t \Gamma b
OE)k 2 :
The stability of the reconstruction problem is conditioned by the smallest eigenvalue ¯ 2
min of
the imaging operator. It can be analytically estimated by examining some physical parameters: v,
characteristic functions of V , and w = 1 \Gamma g 2 related to the choice of H r and to SNR. Indeed, this
eigenvalue is a function of the #interpolation parameter# :
j = Üš =
`Z
V
v(z)dz
' 1=2 `Z
Hr
(w(u)du
' 1=2
characterizing the amount the interpolation to be performed both is real and in Fourier spaces. One
has the following relation:
¯(j) = 1 \Gamma j 2
1
X
m=0
(\Gamma1) m km j 2m
where the km 's depend on v and w and are of the kind #moment of inertia# relatively to V and H r .
This equation provides useful approximation of the minimum eigenvalue of the imaging operator
occuring in the expression of an upper limit \Theta of the reconstruction error:
\Theta = k\DeltaOEk
kOEk
Ÿ
1
¯ min
ae
kOEk
\Delta
When this analysis is implemented before the reconstruction, one has only an estimation ¯ e of ¯ min
and the estimation OE t of OE. The upper bound of the quadratic reconstruction error is given by:
\Theta e =
1
¯ e
''
kOE t k
\Delta
In these two expressions, ae and '' can be regarded as errors resulting from the noise analysis and
from systematic errors related to the choice of H r .
Note that these majorants correspond to the case where the overall error would be concentrated
in the eigenspace associated with the minimum eigenvalue, what is not the case in practice (the
error is generally distributed amongst all the eigenspaces).
Once the stability conditions are ful®lled, the solution can then be obtained by several well
known minimization iterative methods. We apply the conjugate gradients method (Hestenes et al.,

Interactive Deconvolution with Error Analysis 67
1952) because a particular implementation allows to compute the spectral decomposition of the
imaging operator (then a more precise error can be calculated). The convergence is superlinear,
and the least squares solution OE is reached in a number of iterations of the order of the number of
degrees of freedom of the reconstruction process.
3. Example of Deconvolution: The HST Image of SN1987A
For illustration of IDEA, we present the deconvolution of an HST image of SN1987A because
of its relatively simple structure (a bright core together with a well delimited extended object,
the ring). We compare the result of the deconvolution with IDEA with two Bayesian methods:
the Richardson­Lucy method (RLM) and the Maximum Entropy method (MEM). For these two
methods, we use softwares available in IRAF at STScI.
This 128x128 pixel image is an 822s FOC exposure (f/96 mode, pixel size 0: 00 022) through the
F501N (OIII) ®lter, obtained on 1990 August 24. It is described in Jakobsen et al. (1991) and a
RLM deconvolution of this image is presented in Panagia et al. (1991). We use a PSF obtained on
a star on 1990 August 28 in the same telescope conditions as for SN1987A.
We show in Fig. 1 the original image, and the solutions obtained with IDEA, RLM and
MEM. The RLM deconvolution is obtained after 50 iterations. For the MEM deconvolution, the
result is obtained after 30 iterations with an uniform noise of 2.56, a Poisson noise coef®cient of
0.35, a quadratic noise coef®cient of 0.03, and after ®nal smoothing by 2 pixels. For the IDEA
deconvolution, we choose a gain in resolution of 1.9 yielding an upper limit of 17% to the quadratic
error. As noticed above, the least squares solution is reached in IDEA in a number of iterations
corresponding to the number of degrees of freedom of the reconstruction process (8 in this case).
The supernova star is anisotropic in all images: the RLM and MEM solutions show several
spikes which are residuals of the PSF. But on the IDEA solution this anisotropy is essentially an
elongation at P.A.' 45 ffi that could be physically related to the elliptical projected shape of an
envelope around the supernova (see below).
At ®rst glance, the RLM solution seems to reach the highest resolution. A comparison of the
pro®les of the supernova between the different images is shown in Fig. 2. These pro®les are obtained
by plotting the intensity of each pixel as a function of its distance to the center. A polynomial ®t
of the pro®le of the PSF is presented on Fig. 2a, together with the pro®le of the faint star visible
SE of the supernova. The two bright stars close to the supernova cannot be used as PSF because of
distortions introduced by saturation effects in the camera, but they have nevertheless radial pro®les
indistinguishable from that of the PSF. The ®rst ring caused by the spherical aberration is clearly
seen. The inner part of the PSF has FWHM= 0: 00 06 corresponding to the unaberrated telescope. On
the ®ve other plots, the pro®le of the supernova on the raw and deconvolved images is compared to
the pro®le of the PSF.
Clearly, the supernova on the raw image does not have the pro®le of the PSF, meaning that it is
non­stellar (FWHM= 0: 00 13). This has already been found by Jakobsen et al. (1991), and its width
indeed corresponds to the expected size of the envelope that was ejected in the 1987 explosion.
Hence, the deconvolution should preserve the pro®le of the supernova. This is exactly the case with
IDEA (FWHM= 0: 00 12, Fig. 2b) and MEM (FWHM= 0: 00 11, Fig. 2c). In addition, even if we try to
increase the gain in resolution with IDEA (at the expense of the quadratic error), the pro®le remains
larger than the PSF (FWHM= 0: 00 10). By contrast the supernova on the RLM solution (Fig. 2d) has
exactly the pro®le of the PSF, revealing over­resolution. Because one could argue that the number
of iterations on the RLM solution is too high, we present results with 10 and 30 iterations (Fig. 2e
and f): the supernova is already over­resolved at 10 iterations. So whatever the stopping criterion
that could be implemented in RLM, the solution would be over­resolved.
The total width of the bright structure of the ring W of the supernova (Fig. 1) is 0: 00 09 with
IDEA and 0: 00 08 with RLM (50 iterations). This shows that this structure is slightly resolved. This
con®rms that the #large# pro®le of the supernova seen on the IDEA solution is really due to an
envelope around the star and not to a lack of resolution. We also note that the smoother shape of

68 Bouyoucef, Fraix­Burnet, & Roques
Figure 1. Top left: raw image of SN1987A. Top right: IDEA deconvolution. Lower
left: MEM deconvolution. Lower right: RLM deconvolution. The lut is linear and the
IDEA and RLM images are thresholded at the same levels.

Interactive Deconvolution with Error Analysis 69
Figure 2. a: radial pro®les of the PSF and the star SE of SN1987A. The curve is
a polynomial ®t to the PSF. On the other plots: pro®le of the PSF and pro®le of the
supernova on the raw image compared with pro®les of the supernova after the different
deconvolutions (b: IDEA, c: MEM, d: RLM 50 iterations, e: RLM 30 iterations, f: RLM
10 iterations).

70 Bouyoucef, Fraix­Burnet, & Roques
the supernova after deconvolution with IDEA is more reminiscent of an envelope around the star
than on the other deconvolutions.
On the IDEA solution, the ring is more ®lamentary and less noisy than on the other deconvo­
lutions. It is rather blobby on the RLM solution suggesting some over­resolution as shown in the
analysis above. It is very noisy on the MEM solution probably because MEM fails to gather the
information from the wings of the PSF which are of very low signal­to­noise ratio.
The #uxes of the supernova and the ring measured within the supports used in IDEA are listed
in Table 1. For the raw image, we use the values estimated by Jakobsen et al. (1991) for comparison.
Table 1. Absolute #uxes in 10 \Gamma14 erg s \Gamma1 cm \Gamma2 . The values for the raw image are
taken from Jakobsen et al. (1991).
Image Supernova Ring
Raw 5.9 30
IDEA 5.2 22
RLM 4.3 18
MEM 1.9 10
IDEA is expected to rigorously conserve the photometry and this is veri®ed in this case, taking
into account the uncertainty on the photometry with such a PSF. On another hand, MEM and RLM
clearly loose a signi®cant fraction of the total energy from the wings of the PSF, although the peak
value of the supernova on the RLM deconvolution is much higher than on the other deconvolutions
because of over­resolution.
4. Conclusion
The approach of IDEA allows an interactive choice for the compromise between gain in resolution
and stability of the solution. In addition, the photometry is theoretically conserved. These two
points have been illustrated on the dif®cult example of HST images and the error control allowed
in IDEA clearly provides con®dence in the quantitative analysis of the deconvolved images.
The analysis presented in this communication should lead to a better understanding of the
guiding ideas of the whole regularized restoration methods that can be applied to HST images, and
shows what can and cannot be done in this ®eld whatever the particular technique implemented.
A more extensive presentation of IDEA and a study of HST images of the optical extragalactic
jets in M87 and 3C66B is presented in Bouyoucef et al. (1994). A comparison between our
deterministic approach and a regularized RLM and an algorithm of maximum entropy on the mean
are in progress. Simulations are also used in order to determine the photometry conservation for
low signal­to­noise ratios.
We would like to emphasize that IDEA has been developed independently from the spherical
aberration of the HST, so that it can be used with any image, as for instance with ground­based
astronomical images (see Fraix­Burnet et al. 1989).
Current developments of IDEA include the introduction of multiresolution analysis in the
determination of the signal­to­noise ratio.
Acknowledgments. DFB acknowledges ®nancial support from the Institut National des Sci­
ences de l'Univers (INSU) for this workshop.
References
Bouyoucef, K., Fraix­Burnet, D., Lannes, A., & Roques, S. 1994, to be submitted to A&A

Interactive Deconvolution with Error Analysis 71
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DC