3. Analysis of deconvolved HST images

In this section, we compare the results obtained with the three deconvolution techniques (IDEA, RLM and MEM) using images obtained by the Hubble Space Telescope before the COSTAR correction.

First we present results on the supernova SN 1987A image because of its relatively simple structure (a bright core together with a well-delimited extended object, the ring). These results have been partially discussed in Bouyoucef et al. (1994). Second, the jet of M 87 is an example of a weak extended object aside a very bright unresolved source. And third the jet of 3C 66B is a difficult case because of its very low signal-to-noise ratio.

All images are from the FOC with the f/96 mode (pixel size ) and were retrieved from the HST archives. For RLM and MEM we used the softwares available in IRAF.

3.1. SN 1987A

This pixel image is a 822 s exposure through the F501N (OIII) filter obtained on August 24, 1990. It is described by Jakobsen et al. (1991) and a RLM deconvolution of this image is presented in Panagia et al. (1991). We use a PSF obtained on August 28, 1990 in the same telescope conditions as for SN 1987A.

We show in Fig. 2 (click here) the original image, and the solutions obtained with IDEA, RLM, and MEM. The RLM deconvolution is obtained after 50 iterations. For the MEM deconvolution, convergence is obtained with a uniform noise of 2.56, a Poisson noise coefficient of 0.35, a quadratic noise coefficient of 0.03 (30 iterations); the result is smoothed by a Gaussian of FWHM equal to 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. The least-squares solution in IDEA is reached after 8 iterations. A multiresolution analysis has been used for the choice of the support (see Sect. 2).

Figure 2: a) raw image of SN 1987A. b) IDEA deconvolution. c) MEM deconvolution. d) RLM deconvolution. The grayscale levels tables are below each figure. IDEA and RLM images are thresholded at the same levels. The images are

The supernova star is anisotropic in all images: the RLM and MEM solutions show several spikes which are residuals of the PSF. On the IDEA solution this anisotropy is essentially an elongation at PA \ that could be physically related to the elliptical projected shape of an envelope around the supernova (see below).

A comparison of the profiles of the supernova between the different images is shown in Fig. 3 (click here). 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 in Fig. 3 (click here)a, together with the profile of the faint star visible SE of the supernova. 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.

Figure 3: a) radial profiles of the PSF and the star SE of SN 1987A. The curve is a polynomial fit to the PSF. On the other plots: profile of the PSF and profile of the supernova on the raw image compared with profiles of the supernova after the different deconvolutions (b) IDEA, c) MEM, d) RLM 50 iterations, e) RLM 30 iterations, f) RLM 10 iterations)

Clearly, the center of the supernova on the raw image has not 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 somehow preserve the profile of the center of the supernova. This is exactly the case with IDEA (FWHM , Fig. 3 (click here)b) and MEM (FWHM , Fig. 3 (click here)c); whereas the supernova on the RLM solution (Fig. 3 (click here)d) has exactly the profile of the PSF, revealing over-resolution. One could argue that the number of iterations on the RLM solution is too high. We present results with 10 and 30 iterations (Fig. 3e and f): the supernova is already over-resolved at 10 iterations. We also note that the smooth shape of the core after IDEA deconvolution, compared to other methods, is more reminiscent of an envelope around the star.

The total width of the bright structure of the ring of the supernova (Fig. 2 (click here)) is 0 09 with IDEA and 0 08 with RLM (50 iterations). This shows that this structure is slightly resolved and confirms that the ``large'' profile of the supernova seen on the IDEA solution is due to an envelope around the star.

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 at very low signal-to-noise ratio.

3.2. M 87

Because of the limited field of the FOC, two images are necessary to cover the entire jet. We call them ``center image'' (where the nucleus is present) and ``jet image'' (where the outer part of the jet is visible). They represent two different deconvolution problems because of the presence of a very bright unresolved source in the center image and a faint filamentary extended structure in the jet image. These images are presented in Boksenberg et al. (1992); they were obtained on April 5-6, 1991 and are F220W exposures of 1197 s each. Here, the PSF is simulated with TINYTIM (simulation package of the HST's PSF, see Krist 1992).

In Figs. 4 (click here) and 5 (click here), we show the raw center image and the raw jet image, the RLM deconvolutions (40 iterations), the MEM deconvolutions (50 and 36 iterations respectively, with uniform noise of 0.69 and 1.0, poisson noise of 0.1 and 0.05) and the IDEA deconvolutions (gain in resolution of 1.8 and 1.7 with upper limit of 17.5% and 13% for the quadratic error; the solution is reached in 2 and 5 iterations respectively). The MEM solutions are clearly noisier and consequently miss faint details in the jet. The RLM solutions appear the most resolved, but as seen in SN 1987A analysis, the result may be biased by over-resolution. Moreover, the IDEA solutions do reveal the same level of detail while also showing fainter structures.

The presence of numerous artifacts (especially rings) around the center of the galaxy is due to the simulated PSF.

3.3. 3C 66B

The HST observations of this jet through the F320W filter were first presented by Macchetto et al. (1991). In this work, we use images obtained through the F220W filter. Four 1197 s exposures were obtained on March 18, 1991. As the jet was barely visible on the individual frames, we averaged the four exposures to increase the signal-to-noise ratio. The PSF is simulated with TINYTIM as if it were for an individual frame and thus does not represent the exact PSF of the four exposures average.

In Fig. 5 (click here), we show the raw image, the RLM deconvolution (40 iterations), the MEM deconvolution (50 iterations, uniform noise of 4.6 and poisson noise of 0.2) and the IDEA deconvolution (gain in resolution of 1.8 with an upper limit of 18% for the quadratic error; the solution is reached after 2 iterations).

The MEM solution appears noisy and the jet is extremely weak. The RLM and the IDEA deconvolutions give very similar solutions; except that the B knot (brightest knot at about from the nucleus, Fraix-Burnet et al. 1989) is filamentary on the IDEA deconvolution whereas it is blobby on the RLM one. This can be probably explained by over-resolution (see Sect. 3.1). In addition there are less artifacts around the center of the galaxy on the IDEA deconvolution than on the RLM deconvolution. The FWHM measured on the faint star North of the galaxy center is about for the raw, MEM, RLM and IDEA images. It is also the same for the center of 3C 66B except for the RLM solution which has FWHM , indicating over-resolution on bright parts of the image.