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Comparison between closure phase and phase referenced interferometric image reconstructions
Nuno Gomes
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´ ´ , Paulo J. V. Garcia , Eric Thiebaut , Stephanie Renard , Mercedes Filho

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European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Straúe 2, Garching bei Munchen, D-85748 Munchen, Germany ¨ ¨ Laboratorio de Sistemas, Instrumentac~o e Modela¸~o em Ci^ncias e Tecnologias do Ambiente e do Espa¸o (SIM), Faculdade de Engenharia da Universidade do Porto (FEUP),Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal ´ ¸a ca e c Faculdade de Ci^ncias da Universidade do Porto (FCUP), Rua do Campo Alegre, s/n, 4169-007 Porto, Portugal e Laboratoire d'Astrophysique, Observatoire de Grenoble (LAOG), 414, Rue de la Piscine, Saint-Martin d'H´res, France e Centre de Recherche Astrophysique de Lyon (CRAL), Observatoire de Lyon, 9, Avenue Charles Andr´, 69561 Saint-Genis Laval C´dex, France e e Centro de Astrof´sica da Universidade do Porto (CAUP), Rua das Estrelas, 4150-762 Porto, Portugal i

ABSTRACT
We compare the quality of interferometric image reconstructions for two different sets of data: square of the visibility plus closure phase (e.g. AMBER like case) and square of the visibility plus visibility phase (e.g. PRIMA+AMBER or GRAVITY like cases). We used the Multi-aperture image Reconstruction Algorithm (MiRA) for reconstructions of test cases under different Signal-to-Noise Ratios (SNRs) and noisy data (squared visibilities and phases). Our study takes into account noise models based on the statistics of visibility, phase and closure phase. The final images were then compared to the original ones by means of positions and fluxes. For astrometry, the precision is typically of tens of miocroarcseconds and, for the photometry, typically of a few percent. Although both cases are suitable for image restorations of real interferometric observations, the results indicate a better performance of phase referencing (V 2 + visibility phase) in a low SNR scenario.
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Introduction
In optical interferometry, data is obtained in a sparse coverage of the Fourier plane, not in the form of an image. By means of visibility and closure phase information and supported by physical models, modern optical interferometers yield the possibility to obtain reconstructed images of real ob jects. The Phase-Referenced Imaging and Micro-arcsecond Astrometry (PRIMA) dual-feed facility and the GRAVITY experiment (7734-33) will offer a phase referenced imaging mode, where data consisting on spectrally dispersed visibilities and phases can be used to generate images. Therefore, two scenarios for interferometric image reconstruction arise from current facilities: power-spectrum (square of the visibility amplitude) + closure phase data and power-spectrum + absolute phase data. We devised a simple method to perform a formal comparison between images belonging to both cases.
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S e t up
A synthetic image of a cluster of eight stars was built with the freeware programming language Yorick. The cluster was used to create several Optical Interferometry FITS exchange format (OIFITS) files, each corresponding to a different set of SNR and noisy data. All files were used as input to MiRA. A set of optimal parameters (initial guess, number of steps and regularisation) was found and kept for all the restorations, allowing one to compare the resulting images under the same conditions. Using Starfinder and SExtractor, the astrometry and the photometry of the images were measured. We computed the distances and the relative fluxes between each element of the cluster and the brightest star. These data were used to evaluate the quality of the reconstructions and to compare the images with the original one.
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Figure 2: Contour plots of the reconstructed images. Top row: AMBER case; Bottom row: PRIMA+AMBER case. Left: N 107; Middle:
N 105; Right: N 103. The contour levels are at 1, 10, 40 and 90%.

Results

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Tables 2 and 3: Astrometry of the reference and third set (N 103) of reconstructed images, using (left) closure phase and (right)
visibility phase information. The "Reference" column refer to the synthetic image. The distances are in respect to the brightest star.

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Figure 1: Contour plot of the synthetic cluster of stars used as a model for the reconstructions (left ) and uv -coverage with the VLTI
interferometer (right ). The simulated stars are discs with a Gaussian intensity profile. The cluster is approximately 22.4 mas wide and is embedded in a FOV of 80 mas â 80 mas (500 â 500 pixels); the pixel size is approximately equal to 0.16 mas. The contour levels are at 1, 10, 40 and 90%. The uv -coverage corresponds to a 6 ATs configuration (A0-B1-D2-G1-J2-M0).

We consider three scenarios for the SNR, which is controlled by the total number of photons N reaching the array of telescopes: N 107, N 105 and N 103 photons. The errors for the power-spectrum and closure phase, in photon and detector regimes, were based on the work developed by Tatulli and Chelli (2005). For the absolute phase, the errors were calculated according to the model of Colavita et al. (1996). For the power-spectrum, the detector noise regime is considered (N 1), while for the closure phase, both the photon (N 1) and detector noise regimes are taken in to account. Some approximations were implemented: Strehl equal to 1 and Strehl error equal to 0 (fully adaptive optics corrected), transmission in the optical fibre equal to 1 and the fraction of light selected for photometry at the output of the beam splitter was neglected. All errors were randomly added to the data by means of an uniform distribution. For each group of three realisations corresponding to a specific number of photons, we computed the mean of the medians of the SNR of the power-spectrum (V 2), phase () and closure phase (3) data points.

Tables 4 and 5: Photometry of the reference and third set (N 103) of reconstructed images, using (top) closure phase and (bottom)
visibility phase information. The "Reference" column refer to the synthetic image. The flux ratios are in respect to the brightest star.

Conclusions
One of the biggest problems of image reconstruction is the calibration of the visibilities. In our work, we considered stochastic errors but it is possible that calibration errors, which change between observation nights, might dominate the uncertainties. In that perspective, this simulation is not realistic. Under the imposed conditions, MiRA was able to fairly reconstruct the first five stars. Relative positions are correct, shapes are well reproduced and most of the flux is restored. The flux ratio between these stars is equivalent to m = 3. In the phase referencing case, at least six stars were restored, which corresponds to m 4. Only for the faintest stars, with fluxes less than 4% of the brightest star, reconstructions are of inferior quality: in the lower SNR scenarios, their positions and fluxes are not well determined and, sometimes, they are not even restored at all. The results seem to indicate that when using FFTs in MiRA, phase referencing case gives better results than closure phase case in a low SNR scenario.

Table 1: Mean of the medians of the SNRs of the power-spectrum (V 2), phase () and closure phase (3).

Image Restorations
For each SNR scenario, three restored. MiRA was configured smoothness regularisation and pixels length (100 mas) and the the simulation. OIFITS were generated and a corresponding image under a positivity constraint, using a edge-preserving a normalised image. The images are squares of 500 pixel size equal to 0.20 mas. We used = 2.2 µm for

References
Colavita et al., 1999, The Astrophysical Journal, 510, 505­521 Tatulli, Eric and Chelli, Alain, 2005, Optical Society of America Journal, 22, 1589­1599 Thi´baut, Eric, 2008, SPIE, 7013, 70131I­70131I­12 e

Contact: Nuno Gomes | ngomes@eso.org | ESO | Karl-Schwarzschild-Str. 2 | D-85748 Munich