3 Image simulation and reconstruction test

In this section, we present the application of the reconstruction method of Sect.2 to numerical simulations of LBT imaging, taking into account the telescope performance and adding realistic estimates for the detector. In a first step, the construction of simulated interferometric observations was carried out using an ideally AO corrected PSF within the whole field of view. In addition we considered perfect optics and co-phasing of the two pupils. The PSF's were therefore modeled as cosine-modulated Airy functions. In the near future we plan to obtain a more realistic PSF modeling by taking into account the attainable level of AO correction. The simulated interferometric observations, performed in R-band, were obtained as the convolution of the target with the PSF corresponding to each parallactic angle, adding sky-background emission. Independent Poisson noise realizations were then computed, and realistic detector RON added for each parallactic angle. In all simulations presented below, we assumed 1000s integration time per parallactic angle, $30 \%$ efficiency (mirrors + optics + detector), a sky brightness of 20.80 ${\rm mag/arcsec^2}$ and a RON equivalent magnitude of 35.8 ${\rm mag/pix}$.

Since the large extension ( $\sim 30 \hbox{$^{\prime\prime}$ }$ in R-band) of the foreseen AO corrected LBT field, it will be theoretically possible to obtain a sufficiently bright reference star in the field for PSF calibration. Indeed the average density of stars with $m_v \leq 21$ is about 0.9 per sq arcmin at $90\hbox{$^\circ$ }$ galactic latitude, and almost ten times greater at lower galactic latitudes. Therefore the PSF's used in these restorations were assumed without noise-contamination. A few applications carried out on both point-like (binary star) and extended objects are presented hereafter.

3.1 Reconstructed images of binary star objects

We have applied the reconstruction method to binary stars of different relative magnitude, and outlined effects of the SNR on both astrometric and photometric precision of the reconstruction. In the two examples presented here, we have chosen extreme magnitudes and magnitude differences. Note that we have considered 4 equidistant parallactic angles ( $0\hbox{$^\circ$ }$, $45\hbox {$^\circ $ }$, $90\hbox{$^\circ$ }$ and $135\hbox{$^\circ$ }$), and each star was located on a pixel of the $64 \times 64$ pixels array, which leads, with the sampling of 4 pixels per fringe adopted, to a $0\,\hbox{$.\!\!^{\prime\prime}$ }10$ field of view. The separation of the binaries were fixed to a value of 14.1 pixels, corresponding to 22.6 mas i.e. about 3 times the diffraction angular resolution limit, and the orientation angle to $45\hbox {$^\circ $ }$. The relative photometry was computed by measuring the peak pixel values of the reconstructed sources since the sources were initially located at integer pixel locations.

Figure 3: Residuals of the magnitude difference for the mR=29-30 binary (top) and the mR=27.5-30 binary (bottom) reconstructions as a function of the iteration number, and for 5 independent noise realizations

In the first case, $\Delta m_R=1$ and m_R=29 for the main component. This leads to an average peak SNR of only 5.5 for the simulated interferograms. If we do not consider fringe pattern overlapping between main star and companion, the peak SNR for this latter is only 2.2 (see Fig.1). Concerning the reconstructed object, the binary location is fully retrieved, while we will comment separately on the photometric accuracy.

In the second case, $\Delta m_R=2.5$ and m_R=27.5 for the main component. This leads to an average peak SNR of 11.3 for the simulated interferograms, and 1.1 for the companion. As in the first case, the reconstruction leads to a fully retrieved "astrometric'' position of the original object. The relative magnitude, after 1000 iterations, of the reconstructed object is 3.1, a good result when the difficulty of detecting this faint companion in the noise level is considered (see Fig.2).

In both cases, we measured a magnitude difference larger than the true magnitude difference. We noticed however that this discrepancy was significantly reduced in tests with higher SNR. In Fig.3, we show the variation of the photometric accuracy with iteration number. The residuals of the magnitude difference appear to remain stable after several hundred of iterations in both cases. Note also that, even though the algorithm convergence takes place essentially in <10² iterations (Fig.4), the photometric accuracy seems to continue to improve, but this fact is only a consequence of performing photometry on the peak pixel. For the purpose of the present work, we were satisfied to verify this stability, and did not concern ourselves with a criterion to stop the iterations. In practical applications, the algorithm will be stopped according to the usual considerations on noise.

Figure 4: Variation of the error metric with iterations number for the mR=29-30 binary (solid line) and the mR=27.5-30 binary (dashed). Error metric is defined as the sum, for all parallactic angles, of the Euclidian distance between the simulated interferogram and the convolution of the result with the PSF

Figure 5: Numerical simulation of interferometric imaging of an extended object with LBT. Top left: one of the observed interferograms of the $0\,\hbox{$.\!\!^{\prime\prime}$ }14$ field of view of a mR = 19 galaxy (parallactic angle = $30 \hbox {$^\circ $ }$), obtained after an integration time of 1000s. Bottom row shows the simulated target, unfiltered (left) and band-pass limited to a 22.65m perfect circular aperture (right). Top right: result of the algorithm for 6 interferograms of 1000s integration time each after 150 iterations

3.2 Reconstructed image of an extended object

We also tested the ability of the algorithm to reconstruct images of extended objects from LBT interferograms. The target used here is an image of the spiral galaxy NGC 1288 rebinned in a $128 \times 128$ pixels array that, with the sampling adopted, corresponds to a field of view of $0\,\hbox{$.\!\!^{\prime\prime}$ }20$ in R-band. The image was apodized in order to avoid the effect of edge discontinuities in the restoration. In this example, the reconstruction was based on simulated observations at 6 equidistant parallactic angles ( $0\hbox{$^\circ$ }$, $30 \hbox {$^\circ $ }$, $60\hbox{$^\circ$ }$, $90\hbox{$^\circ$ }$, $120\hbox{$^\circ$ }$ and $150\hbox{$^\circ$ }$). With the 1000s integration time assumed per parallactic angle, the total integration time is about 1.7hours. The magnitude of the galaxy was set to m_R = 19, which leads to approximately 2 10⁷ photons per long exposure image and a peak SNR of 80. One can notice the sharper aspect of the reconstructed galaxy shown in Fig.5 with respect to the theoretical diffraction-limited image. This is due to the behaviour of the algorithm in cases of high SNR. Also, a closer inspection shows that the nucleus does not appear as smooth as in the original picture. This is a consequence of the phenomenon of "noise amplification'', which basically arises for such a maximum-likelihood deconvolution algorithm from the difference of converging rate between extended objects and point-like features (White [1994]). For our aim, we did not concern ourselves with this problem, but different approaches concerning the solution of this drawback of the LR algorithm can be found in the literature (Lucy [1994]; White [1994]; Waniak [1997]).