Basic Questions about Radon Transforms I am currently working on a problem that may be interpreted as recovering an unknown function from its Radon transform. 
Unfortunately I don't have any background in Radon transform, but need to quickly get what I need for solving my problem.
The questions are:  


*

*are there properties (e.g. degree of smoothness) of the Radon transform that are related to the corresponding properties of the original function?  

*is it possible to incorporate restrictions into the reconstruction of a function from its Radon transform? An example for such a restriction would knowledge about the geometry and absorbation properties of bones in computer tomography (that knowledge would amount knowing, how much energy of an x-ray is absorbed in a known interval)?

*does incorporating known restrictions into the recovery of the function improve the quality of the numerical reconstruction of that function?  
I would also appreciate pointers to good "hands on" material on the Radon transform; preferably online articles and free software.
 A: Regarding your first question. The Radon transform on even functions on the sphere $S^d$ is an isomorphism of the Sobolev space $L^2_s(S^d)^+$ (+ stays for even functions) to $L^2_{s+\frac{d-1}{2}}(S^d)^+$ for any real $s$. 
This still remains true if the measure, with respect to which the Radon transform is computed, is not necessarily rotation invariant, but sufficiently close to such a measure in the $C^k$-metric for $k$ large enough. 
This and references to the more classical (rotation invariant) case can be found in the paper 
"On a stability property of the generalized spherical Radon transform" by D. Faifman; Asymptotic geometric analysis, 55–73, Fields Inst. Commun., 68, Springer, New York,
2013.
A: to follow up on my comment with a few more pointers: 
Even if you will not be using the ready-to-use MATLAB toolbox for Radon transform inversion, you will likely want to use MATLAB as a platform for the development and experimentation with your own codes. (It makes little sense to write all the visualisation and data management code yourself, in particular if, as you write, you are in a hurry.)
Various numerical techniques were discussed by Kunyansky in 2012, here you can find his lecture notes and numerical algorithms with examples.
A tutorial on the use of MATLAB to invert Radon transforms can be found here.
If you insist on open-source software, you can try freemat with this inversion toolbox.
A: If you want to incorporate restrictions to your reconstruction scheme, a Bayesian approach might work.
A Bayesian view with a well chosen prior distribution describing what the imaged object should generally look like is a popular and useful way to tackle numerical inverse problems.
If you are not familiar with the subject, you could start by reading about Bayesian inference in Wikipedia and an article by Shouno, Yamasaki and Okada on the Bayesian approach to the Radon transform.
Although I work with Radon-type transforms, I'm not the most knowledgeable person about the numerical side.
There is a list of inverse problems research groups at the homepage of the Finnish centre of excellence in inverse problems, and following these links will most probably lead you to someone who can supply you with a great answer.
(We also have a list of conferences that you may be interested in.)
