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2. Methods of inversion

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 kth 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 tex2html_wrap_inline1036 is approximately equal to the intrinsic resolution tex2html_wrap_inline1038 of the instrument, i.e. tex2html_wrap_inline1040.

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, tex2html_wrap_inline1042, of the modulation matrix, we derive the following correlation equation system
where tex2html_wrap_inline1044.

From Eq. (7), one can see that tex2html_wrap_inline1046, 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) tex2html_wrap_inline1056. 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 bi 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 tex2html_wrap_inline1070, calculate tex2html_wrap_inline1072 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).

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