I second the recommendations in the comments which suggest looking at the tomography literature and at the characterization of the problem as an ill-posed inverse problem. I, in particular, recommend these two references.
The first one contains a detailed exposition of the recovery of $f$ in the special case $n=2$ (chapter 5):
That book is quite neat in the sense that it presents a very wide and rigorous exposition of the field of ill-posed inverse problems. Thus, it constitutes an excellent entry point to the field.
There are also a lot of papers in statistics on that particular problem. The second reference that I recommend is a paper which might be of particular interest to you (since it is an econometric paper that studies inversion of Radon transforms)transforms for the purpose of some non-parametric estimation problem):
- Hoderlein, Klemelä and Mammen (2007). Reconsidering 2010). Analyzing The Random Coefficient Model Nonparametrically
Edit: I updated a link above for the random coefficient modelactual published paper location of the second reference. A 2007 working paper version is available at the following link.