6 Experimental results

We have applied our myopic deconvolution method to real images acquired with PUEO, the Canada-France-Hawaii Telescope adaptive optics system operated at the top of Mauna Kea in Hawaii. PUEO is a low order (19 actuators) curvature based system. Figure 4 shows the image of binary star HD 203024 in H band (1.65 $\mu \rm m$). This image was acquired in June 1996 using the $256\times 256$ Monica near infrared camera provided by University of Montreal. The plate scale is 0.034 arcsec/pixel and the integration time was 3 minutes. The object was bright enough (m_R = 9.5) to be also used as the reference for AO guiding and the WFS could be used at its nominal 1 kHz sampling frequency. During the acquisition, the estimated seeing was 0.76 arcsec (r₀ = 13.5 cm at $\lambda = 0.5~ \mu$m). At the top of Mauna Kea, the wind speed is typically ${\bar v} = 20~ \rm m/s$. A flat field and a background are subtract from the image. The "bad'' pixels (cosmic ray events, aberrant pixels of the camera...) are replaced by a mean of the neighbouring pixels before deconvolution.

  

Figure 4: AO corrected image of binary star HD 203024 in H band (SR $\simeq 33 \%$). $128\times 128$ pixels image corresponding to a 4.35 arcsec field of view

The PSF associated with the image was retrieved from the loop data using the method described in (Veran et al. 1997). The estimated Strehl ratio is $33\%$. For an object of this magnitude, the precision of the estimated PSF is normally limited by the turbulence noise. The PSF estimated from the loop data should be accurate enough for a classical deconvolution. In this particular case however, uncalibrated static aberrations degrade the reconstruction and a myopic deconvolution is necessary.

Using the reconstructed PSF and a classical pixel to pixel deconvolution algorithm such as Lucy-Richardson, the deconvolved object presents two sources which seem to have a faint companion at the very same relative position. This is bound to be an artifact. This artifact can be traced to the presence of a bump on the first Airy ring of the real PSF which does not appear in the reconstructed PSF.

The classical deconvolution (i.e, the deconvolution method using the object reparametrization but assuming that the mean PSF -- estimated with WFS data -- is the real one) allows, of course, the restoration of a binary star (object reparametrization). The problem is that the PSF artifacts are not taken into account. It leads to errors on the estimation of the parameters, especially on the flux.

As shown in Fig. 5, the real PSF can nevertheless be retrieved using our myopic deconvolution algorithm, in spite of the fact that Eq. (9) under-estimates $PSD_{\rm h}$ (the static aberrations are not taken into account in the PSD, because they are not seen by the WFS). After myopic deconvolution, a bump clearly appears at the first Airy ring in the the estimated PSF. We have generally noticed that with our stochastic approach no hyper-parameter is needed. Yet, in this particular case of an under-estimated PSD, one could add an ad-hoc hyper-parameter smaller then 1 on the PSF regularization term in Eq. (5) to loosen the constraint.

An even better accuracy might be obtained, if we were able to quantify the static aberration effects in the Fourier domain in order to add them in the regularization term.

  

Figure 5: Comparison of the WFS reconstructed mean PSF (dotted line) and the myopic estimated PSF (solid line). Normalized square root of x-axis cut are represented

In Table 2, we compare the results of the classical and the myopic deconvolution method. If the estimated star separation is nearly the same in both cases, a significant difference is found in the estimated flux. According to the simulation results, the myopic deconvolution improves the global flux, and therefore the accuracy of the magnitude restoration of each star. It would be useful to have a photometric calibration star to obtain the atmospheric coefficient of absorption and to give the real magnitude of each star; unfortunately no such data are available for this observating night.

  

Table 2: Experimental results: $\Delta {\rm sep}$ is the separation between the two components, pixel unit (first column) and arcsecond units (second column). $\Delta m$ the magnitude difference. Last column gives the total flux of the two components (in detected photons). In comparison, the total flux in the image is 4.86 10⁸ detected photons

In short, the myopic deconvolution allows to take into account unpredictable phenomenons (turbulence noise, uncalibrated static aberrations...) for a better accuracy of the object photometry.