In a previous paper, (Bertero et al., Opt. Acta. vol.3., p.923 (1984)), it was shown that the resolution of the type-II confocal scanning microscope may be improved by recording the full image and by inverting the data. In this paper, the authors propose a numerical method which can be easily implemented and, in principle, applied to the practical problem. They show that by using a rather small number of data points on the image plane, it is possible to obtain the improvement in resolution (by a factor of two) predicted in their previous analysis.

In three previous papers the authors have considered the problem of Laplace transform inversion when the unknown function is of bounded, strictly positive support. The improvement in resolution due to a priori knowledge of the support was quantified and methods for the choice of an optimum sampling of data were given. The authors discuss some undesirable edge effects encountered in practice with these methods and indicate a way to refine such calculations by considering the problem of Laplace transform inversion in weighted L2 spaces. A smoothly varying weight takes into account a partial knowledge of the localisation of the solution which can be estimated a priori from the knowledge of its first and second moments which are easily derived from the data before inversion. In such a way the reconstructed solution is forced to be small where it is likely to be small and the troublesome edge effects found in the previous methods are suppressed. The results show somewhat surprising improvements in typical inversions. The extensions of the method required for the analysis of sampled and truncated experimental data are also discussed and applied.

The Landweber method is a simple and flexible iterative regularization algorithm, whose projected variant provides nonnegative image reconstructions. Since the method is usually very slow, we apply circulant preconditioners, exploiting the shift invariance of many deblurring problems, in order to accelerate the convergence. This way reasonable reconstructions can be obtained within a few iterations; the method becomes competitive and more robust than other approaches that, although faster, sometimes lead to lower accuracy. Some theoretical analysis of convergence is given, together with numerical validations.

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