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# 2. Imaging a binary star by using PDFs

A binary system, for which none of the stars is individually resolved by the telescope, is the most simple object that can be considered for image reconstruction. Its perfect image is made of two points of intensities I1 and I2 , separated by a vector of position corresponding to the angular separation.

Let us denote as the instantaneous monochromatic speckle pattern produced at the focus of the telescope by a point-source (i.e. a single star unresolved by the telescope, or a reference star). is therefore the point-spread function (PSF) if one considers a unit mean intensity. Assuming isoplanatism, the observed binary star speckle pattern can be written as:

where: , and I0 is the intensity of the binary system corresponding to its overall magnitude.

The relevant information for the imaging of the binary is contained in the three parameters I1 , I2 and , or equivalently in the three parameters I0 , and . Unless very accurate photometry is performed, we cannot access the absolute value of I0 , so the imaging parameters to retrieve are and . Whereas (or equivalently ) and the value (greater or not than 1) of give a point in the orbit of the binary, an accurate value of leads to relative photometry of the system. The object of our analysis will be therefore to obtain with no ambiguity and . Let us now show how an analysis of the PDFs can achieve this goal.

Let us first denote the intensity value taken by and that of , where describes the intensity distribution in the speckle pattern at a position , and is a space-lag. As we assume stationarity in space, the second-order statistics of are completely defined (Lee 1960) by the twofold PDF .

The quantity measures the probability that has an intensity value lying in the elementary interval while , of the same speckle pattern, has an intensity value lying in the interval .

As discussed by Aime et al. (1990), there is a strong difference between twofold PDFs of speckle patterns produced by a point-source and a binary star. For a given value of , the observed PDFs appear as joint occurrence histograms of the discretized values and , and can be represented as gray-level images. As we shall see in what follows, the twofold PDF of a point-source has an overall symmetrical structure in and whatever the value of . Whereas for close to the star separation vector of the binary, the corresponding twofold PDF of the double star speckle pattern has an arrow-head shape with a trend towards a direction . There is a unique relationship between the shape of the twofold PDF and .

Carbillet et al. (1996a) presented a calibration procedure that uses a parametric approach leading to an estimation of the two parameters and from one-dimensional near-infrared speckle data. We present here a new approach that is found to give better results for two-dimensional visible speckle data. The separation d (modulus of ) and the position angle PA (with a quadrant indetermination) need within the present framework to be determined by the by now classical power spectrum analysis and visibility function calculus of Labeyrie's technique. We will now focus on the most accurate way possible of determining (and the absolute quadrant) by using an analysis of the PDF's slices computed for (or equivalently ).

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