On unique recoverability of convex and visible compacta from their projections

The question is: Under what conditions does the congruence of the projections of two compacta on every two-dimensional plane imply the congruence of the compacta themselves? The results are extended to the cases of $(n-2)$-visible and $(n-2)$-convex compacta, and even to the case when it is known the projections of the compacta are merely similar.