In this paper one-dimensional notation will be used for simplicity,
the extension to two dimensions being trivial.
The intensity of a double star O(x) can be modeled as the sum of
two unit impulses distant d and weighted by the intensity ratio
, i.e.:
We denote as 
the cross-correlation (CC) between O(x)
and its square. is defined as
This function is a slice of the triple correlation of O(x) defined
as (Weigelt 1991)
we have 
.
For a double star, becomes
This quantity may be compared with the AC 
of the double star
O(x)
Both and are composed of a central peak surrounded
by two smaller ones distant d (see Fig. 1 (click here)). For the
AC, these two peaks are symmetrical whatever the value of .
This is why Labeyrie's speckle interferometry
cannot give the relative positions of the two stars when observing
a binary system. The CC presents two asymmetrical peaks
of ratio . The relative position of the peaks is those of the
stars in O(x). Using this quantity in double
star's speckle interferometry, rather than AC, should give the position
angle (PA) of the binary without any ambiguity.
In the Fourier domain, the cross spectrum (CS) 
between
O(x) and its square is the Fourier transform of . It is a
complex quantity whose real and imaginary parts are:
Both are sinusoidal functions of period 
. The amplitude
of the real and of the imaginary part gives the value of
without any ambiguity. But information concerning the relative
position of the stars is fully contained in the imaginary
part of . Let s be the slope of 
at the origin:
We note that s<0 when 
and 
when 
.
See Fig. 2 (click here) for illustration.
Figure 2: Real and imaginary parts of the CS between a double star O(x)
and its square. Both figures are for an intensity ratio 
between the two stars. Up: brighter star on the left, down: brighter
star on the right. The real part of the CS
is not sensitive to this orientation contrary to the imaginary part:
its slope at the origin is positive in the first case and negative
in the second one
from speckle data
We denote as I(x) the instantaneous double star's specklegrams
and S(x) the corresponding point-spread function (PSF). Assuming
isoplanatism, we can write
We denote as 
the CC of I(x) and the CC of
S(x).
where 
denotes ensemble average. From
Eqs. (2 (click here)) and (3 (click here)), we have
and 
.
Unfortunately we cannot
find between and 
the simple
convolution relation that exists between the corresponding
full triple correlations. Inserting the value of I(x) of
Eq. (8 (click here)) into Eq. (9 (click here)), a simple calculation gives:
This can be written as a convolution product plus a bias term
where the bias term is
It is difficult to estimate and subtract this bias from speckle
data because of the presence of the unknown factors 
and 
. Nevertheless we shall see that 
vanishes
if we consider zero-mean specklegrams in the case of a star
separation large with respect to the speckle size s.
We call 
and 
the zero-mean
specklegrams of the PSF and the double star:
We respectively denote as 
, 
,
and 
the mean
(with obviously 
), the AC, the CC and the triple
correlation of . We denote as 
the CC of . From Eqs. (12-14) we have
Let us consider the term 
. We have
where 
is the mathematical expectation of 
.
We have assumed that 
, so and 
are uncorrelated. We can distinguish 3 cases:
: and 
are correlated;
is uncorrelated both with and , so:
: and are correlated;
is uncorrelated with the two others, so:
, and are
uncorrelated, so:
Figure 3a-g): Cross-correlation/spectrum computed on simulated speckle
patterns. The calculus has been made on two sets of 200 images, one
for the double star and one for the reference star. The double star
is 10 pixels separation oriented along the x-axis.
The intensity ratio is 0.5. The simulation has been made for a Fried
parameter r0=20 cm, a telescope diameter of 2.60 m and a wavelength
nm. a) is a typical double star's specklegram,
b) is the two-dimensional object's CC
and c) is a cut along the 
axis. Notice the
asymmetry
of the two secondary peaks. Lower pictures are the real d) and
imaginary f) parts of , while the curves e) and
g) are the corresponding cuts along the ux axis. Note
the sign of the slope at the origin of
Figure 4a-f): This figure shows an application to the bright star
(see text for details). Computation was made for
1089 short exposure (20 ms) frames of the double star and 2993
on the reference star HR7536. The mean value of each specklegram has
been estimated as an average of the intensity over the image,
then subtracted. a) and b) are the two-dimensional CC
and its cut along the axis (the coordinate system has
been rotated so that interesting features are along the horizontal
axis). The asymmetry of the secondary peaks gives the relative
position of the stars. e) and f) are the real and imaginary
parts of the CS , c) and d) are cuts along the
ux axis. The imaginary part of is a sine
function with positive slope at origin: the brightest star is on the
left
The term 
is obtained by changing
d into -d in the above expressions. We see that in most
cases the bias vanishes under the hypothesis . It is
important to remark that this previous calculus is valid only
under the space-stationarity hypothesis, i.e. if mS is the same on the
whole image. This is valid only if we take the central part of the
speckle pattern.
Let us assume that . We can then write the approximation
and in the Fourier domain:
Estimating from speckle data is very similar to
classical speckle interferometry processing. The cross-spectra
are estimated as ensemble averages (
denoting the Fourier
Transform):
Note that 
is a real function (assuming the statistical
properties of the ideal point-spread speckle pattern are isotropic in
space).
Numerical simulations of speckle data are presented in
Fig. 3.
This technique has been applied successfully to the newly
discovered double star MOAI 1
(Carbillet et al.
1996a). Figure 4 shows another application to
the star 
Sge. Observations were made on September, 1994
with the 2 m Telescope Bernard Lyot (TBL) of the Pic du Midi observatory,
using the speckle camera of the Aperture Synthesis group of Observatoire
Midi-Pyrénées (André et al. 1994) and an ICCD detector.