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 Richardson-Lucy 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
) through the F501N (OIII) filter,
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 coefficient of 0.35, a quadratic noise coefficient of 0.03, and after final 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. that could be
physically related to the elliptical projected shape of an envelope around
the supernova (see below).
At first glance, the RLM solution seems to reach the highest resolution.
A comparison of the profiles of the supernova between the different
images is shown in Fig. 2. These profiles are
obtained by plotting the intensity of each pixel as a function of
its distance to the center. A polynomial fit of
the profile of the PSF is presented on Fig. 2a, together with
the profile 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 profiles indistinguishable from that of the
PSF. The first ring caused by the spherical
aberration is clearly seen. The inner part of the PSF has
FWHM corresponding to the unaberrated telescope. On the five
other plots, the profile of the supernova on the raw and deconvolved images
is compared to the profile of the PSF.
Clearly,
the supernova on the raw image does not have the profile of the PSF,
meaning
that it is non-stellar (FWHM). 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 profile
of the supernova. This is exactly the case with IDEA (FWHM
,
Fig. 2b) and MEM (FWHM
, Fig. 2c).
In addition, even if we try to increase the gain in
resolution with IDEA (at the expense of the quadratic error), the
profile remains larger than the PSF (FWHM
).
By contrast the supernova on the RLM solution (Fig. 2d) has
exactly the profile of the PSF, revealing over-resolution. 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 over-resolved at 10 iterations.
So whatever the stopping criterion that could be implemented in RLM,
the solution would be over-resolved.
The total width of the bright structure of the ring W of
the supernova (Fig. 1) is 0 09 with IDEA and
0
08 with RLM (50 iterations). This shows that this structure is
slightly resolved. This confirms that the ``large'' profile
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 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 filamentary and less noisy than on the other deconvolutions. It is rather blobby on the RLM solution suggesting some over-resolution 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 signal-to-noise ratio.
The fluxes 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.
IDEA is expected to rigorously conserve the photometry and this is verified in this case, taking into account the uncertainty on the photometry with such a PSF. On another hand, MEM and RLM clearly loose a significant 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 over-resolution.