Let (X, g) be a Riemannian manifold and Y be a variety of closed submanifolds Y ⊂ X. Consider the integral transform for the variety Y: $$Mf\left( Y \right) = \int_Y {fd} V\left( Y \right)\,,\,Y \in Y,$$ (6.1) where dV (Y) is the volume element on Y induced by the metric g. The reconstruction problem is to find the function ffrom data of M f∣Y. More complicated versions of (6.1) arise in applications. A weight function ω = ω (x, Y) (known or unknown) can appear in the integral, the “image” f can be a section of a tensor bundle, like a differential symmetric or skew symmetric form.