Tomography with a finite set of projections: Singular value decomposition and resolution

The problem of tomography with a finite set of projections has been the object of many investigations. In particular the formulae for the generalized solution and the singular values have been obtained in the case of equispaced projections. These results imply that the problem of determining the generalized solution is well-posed even if it may be ill-conditioned. In this paper we derive several properties of the singular values and singular functions both in the general case and in the case of equispaced projections. We use these results to identify the singular functions related to the aliasing effects in the generalized solution and to estimate the resolution achievable when these effects have been eliminated by means of a suitable filtering. It turns out that the resolution essentially depends on the number of projections and not on the noise, if the number of projections is smaller than a certain upper limit (depending on the noise), which can be quite large. In the case of equispaced projections, the resolution coincides with that estimated by the asymptotic theory even when the number of projections is rather small.