Processing binary stars by speckle interferometry (Labeyrie 1970)
leads to a 180
ambiguity in the measured position angle (PA).
This is known as "quadrant ambiguity''. Several techniques of speckle
imaging can solve the problem, among which the
techniques of Knox-Thompson (Knox & Thompson 1974), shift-and-add
(Bates 1982) and speckle masking (Weigelt 1991). A review of these
techniques has been made by Roddier (Roddier 1988). As they aim to
reconstruct the image of any extended object from
its specklegrams, these techniques usually require a lot of computer
resources and processing time. They are not really well adapted to
the double star problem: observers want to measure the separation
and the PA of many stars a night and need a fast
(near real-time) processing. Several techniques have been suggested for
this purpose; for example the Directed Vector Autocorrelation
(Bagnuolo et al. 1992) which provides both the separation and absolute PA,
the "fork'' algorithm (Bagnuolo 1988)
based on the analysis of four equidistant points in the double star's
specklegrams or the probability imaging technique (Carbillet 1996b)
based on the computation of twofold probability density functions of
the specklegrams. These later techniques require
a prior knowledge of the star separation which is usually measured
from the power spectrum.
We propose a technique based upon the computation of a quantity very close to the autocorrelation function (AC): the cross-correlation (CC) between the specklegrams and their square. This function can be written as a slice of the triple correlation obtained for a speckle masking vector equal to zero. It is a two-dimensional function. For a double star, this quantity at first glance looks like the AC: a central peak surrounded by two smaller ones. These secondary peaks, identical in the AC, are asymmetric for the CC, allowing a quick diagnostic of the relative position of the two stars. The CC is almost as easy to compute as the AC, does not require the prior estimation of the power spectrum, and is then suitable for real-time processing. It also permits, under some hypothesis which will be developed in the text, the determination of the magnitude difference between the stars.
This paper is organized as follows. Section 2.1 defines the statistical
function we use, and derives relevant expressions for the double star.
Section 2.2 describes the technique proposed to process real star data.
We shall see in particular that the
object-image convolution relation valid for the AC does not apply here
and
we propose a solution to overcome this difficulty. Section 3 is
devoted to low-light level and photon bias. Application of the CC
technique is investigated for clipped photon-counting
specklegrams (where the number of detected photons
is "0'' or
"1'').
Figure 1: Schematic representation of a double star O(x) (left),
its AC 
(middle) and the CC 
between O(x) and
its square (right). The arrows represent Dirac delta distributions.
Note the asymmetry of the CC, where the ratio between
the intensities of the two peaks in (-d) and (+d) is exactly the
intensity ratio of the stars