The famous Crofton formula says that the length of a curve can be calculated by integral of the `line crossing' over the space of all oriented lines. My question is, is there a way to treat this formula as a special case or corollary of the Radon transform theory? If so, how can we express the relationship precisely?
There is a paper here:
that develops a theory of "Gelfand Transforms" which in a sense made precise there is a generalization of both the Radon Transform and the Cauchy-Crofton formula.
You should definitely check these notes generated by three bright undergraduates for an REU project that I supervised a few years ago. I promise you, it will be worth your time.