Let $f : \mathbb R^3 \to \mathbb R$ be an integrable function. Let $\eta$ be a one-dimensional subspace of $\mathbb R^3$. We denote $p+\eta$ the affine subspace (a line) which is obtained by translation of $\eta$. The Radon transform for lines is then defined by \begin{align*} (Rf)(\eta, p) = \int_{p + \eta} f \, \text{d}S. \end{align*} Denote the function $R_\eta f$ defined by $(R_\eta f)(p) = (Rf)(\eta,p)$ the radiograph along the direction $\eta$. It is a well-known result that if we know $(R_\eta f)$ for infinitely many distinct one-dimensional subspaces, then we can uniquely reconstruct $f$.
Now define the Radon transform along planes
\begin{align*}
(P f)(\eta, q) = \int_{q + \xi} f \, \text{d}S,
\end{align*}
where $\xi$ is now a two-dimensional subspace (and thus $q+\xi$ are planes). Suppose one has an infinite number of distinct $\xi$ which intersect in the one-dimensional subspace $L$, see the figure below.
Clearly one cannot reconstruct $f$, but intuitively I would expect that one can then get partial information about $f$, such as the values
\begin{align*}
\int_{p+L} f \, \text{d}S = (Rf)(L,p)
\end{align*}
from the functions $P_\xi f$. However I haven't found a way to make this rigorous.
