Let $f : \mathbb R^n \to \mathbb R$ be a density which is sufficiently smooth and can also be restricted to have compact support for now.

Let $t \ge 0$ and $F_t : \mathbb R^n \to \mathbb R$, i.e. $(F_t)_{t \ge 0}$ is a one-parameter family of mappings.

Now as given information consider the integrals $$I(t,z) := \int_{(F_t)^{-1}(\{z\})} f \; dS$$ for all $t \ge 0$ and all $z \in \mathbb R$. In other words, we know all integrals along fibers of points $z \in \mathbb R$ under mappings $F_t$. Under which conditions on the family $(F_t)_{t \ge 0}$ can we guarantee uniqueness of the reconstruction problem? Is this general problem already considered in integral geometry or the study of Radon transforms?

Clearly, if $F_t$ is restricted to be linear, then the integral transform is the classial Radon transform. In this question we would like to consider the more general case of arbitrary (curved) fibers.

Differential equations and Integral GeometryAdv. Math. 131(1997), 279-343 sciencedirect.com/science/article/pii/S0001870897916698 $\endgroup$