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 exposition of the field of ill-posed inverse problems.
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):