Recently we reported [1–3] on the theory and the technique of reconstructing three- dimensional images of flaws and inclusions from an extremely limited number of cone- beam X-ray projections. The number of projections is chosen between two and seven and they are achieved in an observation angle smaller than 180 degrees. We introduced an approach using the Bayesian reconstruction (BR) with Gibbs prior in the form of mechanical models like noncausal Markov fields. As it was pointed out the convergence of the iteration reconstruction procedure depends on the chosen prior functional within a compact set of solutions. We investigated the capabilities of three types of a priori functional, which are represented by Gibbs energies. Corresponding to the supported structures, they were named (i) cluster support, (ii) plane support and (iii) phase support. While examining the phase support we made an effort to estimate the influence of Gaussian white noise on the quality of restoration. The noise was generated artificially and superimposed to the two dimensional x-ray images. It was shown that the algorithm was stable despite the disturbance of the noise. On the other hand it was observed that an increasing noise level leads to a noticeable deterioration of the quality of the restored image. The restoration of images from extremely incomplete and noisy data is a strict practical demand in many cases. This explains the effort to investigate the influence of noise to the reconstruction results.