In observations with any kind of telescope, the relation between
the observational data *d*(*k*) and the intensity distribution *f*(*i*) of
a sky region can be described by the following observation equation
system (modulation equation system):

where the modulation coefficient of the telescope, *p*(*k*,*i*), is the
contribution of a unit intensity source placed at the sky bin *i*
to the *k*th observed data, which represents
the instrument response character. The equation system (4) can be written
in matrix form as

In space high-energy astronomy, the cross-correlation technique is
now widely used to reconstruct the object *f* from the observed data
*d*, i.e. to take the cross-correlation distribution *c* of the data and
the modulation function as a reconstruction of object intensity

The angular resolution of a cross-correlation map is
approximately equal to the intrinsic resolution
of the instrument, i.e. .

The cross-correlation distribution is not a satisfactory reconstruction of the object in RMC imaging. The results from CCM usually have sidelobes. Therefore, faint sources are often covered by sidelobes of strong sources. To solve this problem, the CLEAN technique (Högbom 1974) can be used in the cross-correlation. In this technique, some fraction of flux of bright point sources is subtracted from a cross-correlation image, and more faint sources can be detected. This process continues until a certain closure criterion is satisfied. We may call it CLEAN cross-correlation.

Multiplying the two sides of Eq. (5) by the transpose, ,
of the modulation matrix, we derive the following correlation equation
system

where .

From Eq. (7), one can see that , the relation between *c* and *f*
being described by the correlation Eq. (7). The cross-correlation *c*
is just an image of the object *f* through a modulation (distortion)
of a certain imagery instrument with a point-spread function (PSF)
.
The cross-correlation technique simply uses *c* instead of *f*, completely
ignoring the information included in Eq. (7) about the observation,
which is the main reason of unsatisfactory image quality derived by
cross-correlation deconvolution.

The direct demodulation technique performs a further deconvolution
from *c* by solving Eq. (7) iteratively under proper physical
constraints (Li & Wu 1992, 1993, 1994). The calculation formula of
the direct demodulation algorithm by using Gauss-Seidel iterations is

with the constraint condition

where the lower intensity limit *b*_{i} is the background intensity.
In the case of no priori knowledge of the diffuse background, the intensity
lower limit *b* can be estimated from the data: Use a successive
procedure to subtract the contribution of apparent discrete sources from the
observed data *d* to get background data , calculate
and then iteratively solve the following correlation equation

with a smooth procedure or under a continual
constraint (for details see Li & Wu 1994) to get *b*.
The stability, convergency and global optimum property of the Li-Wu
algorithm of direct demodulation have been proved by the theory of neural
computing (Li 1997).