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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.

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    $\begingroup$ A 1-parameter family may be too small. (In your question you have in fact two parameters, t and z ) The space of affine planes has dimension n. In any case, this type of question has been investigated, and a lot is known, and lots to be discovered. $\endgroup$ Nov 29, 2014 at 10:02
  • $\begingroup$ I refered to the one-parameter family $(F_t)_{t \ge 0}$ which defines the fibers implicitly via $\{x \in \mathbb R^n : F_t(x) = z\}$. So in total, we have the parameter $t \ge 0$ and for each $t \ge 0$ we furthermore can vary between all $z \in \mathbb R$. Your answer on prior work is quite general. Could you please give some references then? $\endgroup$
    – user45183
    Nov 29, 2014 at 10:31
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    $\begingroup$ Have a look at Goncharov's paper Differential equations and Integral Geometry Adv. Math. 131(1997), 279-343 sciencedirect.com/science/article/pii/S0001870897916698 $\endgroup$ Nov 29, 2014 at 12:04
  • $\begingroup$ Check section 7 of the above reference. Also look at the paper by Gelfand-Graev-Shapiro mentioned in Goncharov's paper. $\endgroup$ Nov 29, 2014 at 12:09
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    $\begingroup$ You only know a function of two variables $(t,z)$ and you want to recover a function of $n$ variables. If $n>2$, this is probably impossible or at least very unstable. My answer to your previous question is mostly applicable here as well, but I don't want to reproduce it here. The example I gave there can be modified to give a function family where reconstruction is possible. See here: mathoverflow.net/a/188357/55893 $\endgroup$ Nov 29, 2014 at 22:58

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There is a lot of literature on Radon-type (or X-ray-type) transforms where instead of straight lines one integrates over geodesics in a Riemannian metric or more general curves. See these papers for example:

These papers are about smooth curves. If you want some other kinds of curves (branching or sharp turns), there are papers in these cases as well. Leave a comment if you want more details.

In the formulation of your question you only know a function of two variables and wish to reconstruct a function of $n$ variables. This is probably impossible (and unstable if possible) if $n>2$. The case $n=1$ is probably uninteresting for you. This is related to my answer to your previous question which I don't want to reproduce here.

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