ISSI Visiting Scientists Programme

**RHESSI IMAGING**

The process of image formation can be described by the integral equation [2]:

where g(**x**) is the recorded noisy image, f(**y**) is the original scene
and K(**x**,**y**) is the so called Point Spread Function (PSF) of the
imaging system. The effect of the PSF is called *blurring* and it
represents the image of a point source located at the point **y** (impulse
response function).

If the PSF is space invariant (i.e. K(**x**,**y**)=K(**x**-**y**)),
the image restoration problem becomes the deconvolution problem

Since RHESSI imaging system [3] has nine rotating collimators, characterized by
nine different PSFs (and therefore by nine different resolutions and
signal-to-noise ratios) the RHESSI imaging problem reduces to a multiple
deconvolution problem, i.e., the problem of restoring an object using nine
different images of it. Our idea is to address such problem taking into account
the following methodological issues:

1. instead of summing up the images and using an averaged PSF, a better
resolution is achievable by appropriately combining the nine images and PSFs
according to the approach followed in [4];

2. a better accuracy is achievable by using a priori information on the
solution (such as its positivity or its compact support [5]) and on the
statistical properties of the noise affecting the data [1].

**REFERENCES**

[1] Anconelli B., Bertero M., Boccacci P. and Carbillet M. 2005, Restoration of
interferometric images IV: an algorithm for super-resolution of stellar systems,
*A&A* (in press)

[2] Bertero M. and
Boccacci P. 1998, *Introduction to inverse problems in imaging*, IOP,
Bristol

[3] Hurford G. J. et al.
2002, The RHESSI Imaging Concept, *Solar Physics* **210**, 61-86

[4] Piana M. and Bertero
M. 1995, Regularized deconvolution of multiple images of the same object, *J.
Opt. Soc. Am. A* **13**, 1516-1523

[5] Piana M. and Bertero
M. 1997, Projected Landweber method and preconditioning, *Inverse Problems*
**13,** 441-463

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