Introduction

From the mathematical point of view the problem of the restoration of the images generated by the Hubble Space Telescope (HST) consists of the solution in of the convolution equation

where the kernel represents the Point Spread Function (PSF), the symbol denotes a vector of and is the unknown function, that must be non-negative. As the image is contaminated by noise, the deconvolution of equation (1) by the usual Fourier methods, that is the Fourier inversion of the solution of the equation

where the notation denotes the Fourier transform of , is not effective. Many methods exist, both statistical and deterministic, linear and non-linear, for solving Eq. (1). They are essentially regularization methods, in the sense that the quality of the results, that is the value of the signal-to-noise ratio of the restored images, strongly depends on a non-negative parameter. In the case of iterative methods this parameter is the number of iterations. A different approach, based on the usual Fourier techniques applied to images previously smoothed, could also be adopted. Unfortunately the smoothing techniques are really effective only if the noise is white, but this hypothesis is not acceptable in our case.

In this work we use two different approaches: the first one is based on both the finite dimensional approximation of the unknown function and on the so-called Tikhonov regularization method; the second stems from a coarse smoothing and also exploits the above regularization method.