I stumbled upon several sources already dealing with the Radon transform. However, when it comes to the inverse Radon transform, it is only said that it can be obtained via the central slice theorem etc in a very hand-wavy manner. As a consequence, it is not clear to me how to obtain the inverse Radon Transform.
Can anyone recommend some literature where the author(s) are elaborating on the details/proofs?
Can you please state which sources you have read and what was so "hand-wavy" about the proofs of the Radon transform inversion?
If you're mathematically inclined and are serious about the Radon transform, tomography, and integral geometry, I recommend the following textbooks.
My doctoral dissertation was on a related transform called the cone-beam transform. I made good use of these books while doing my research and found them all to be quite useful.
The list above has been ordered from easy to difficult. If you're just starting out, I recommend beginning with #1. It is written by engineers and has a few notations and derivations that can make a mathematician frown, but I strongly recommend it to beginners since it does a good job of laying out the foundation of the 2D Radon transform and its inversion by the filtered backprojection method, in both parallel-beam and fan-beam geometries. If you're interested in the numerical implementation of the inverse transform, then this is the book for you. However, it does not discuss the general n-dimensional Radon transform and barely touches any of the important properties of the 2D transform.
#2 is written by a physicist. I believe Stanley Deans is one of the pioneers in the field. I didn't read this book extensively in grad school, but recently I've started studying this book. I think it is one of the best books out there on Radon transform, slightly more advanced than #1. It is accessible and self-contained. I think anyone with an undergraduate degree in math or physics should be able to read it. It even provides in its appendix a translation of Radon's original paper which introduced the transform and its inversion formula for n=2 and n=3. You will find in the book several proofs of the Radon transform inversion. They are easy to follow and I didn't find any of them to be "hand wavy". However, (with all due respect) the level of rigor in this book is what is expected from a physicist doing mathematics. Regardless, from reading this book, not only will you become knowledgeble about the Radon transform on Euclidean spaces, but you will also learn about other topics like the theory of distributions, spherical harmonics, and many more. This book will definitely expand your knowledge!
#3 and #4 are classics in tomography. Frank Natterer is a mathematician and another pioneer in computed tomography. I didn't really find these books too useful in the beginning when was just beginning my journey with the Radon transform. Natterer's writing style is too dry and he tends to overwhelm you with many things at once. I was initially intimidated by his compact notation of symbols which can be really difficult for someone reading this material for the first time (I think #4 is better to start with than #3). On the positive side, these books are mathematically rigorous compared to #2. Most of the proofs that you need to know are provided. These books also cover similar integral transforms like the cone-beam transform and their inversion formulas. I recommend these books only if you have some familiarity with the Radon transform and are doing serious research in mathematical tomography.
#5 is probably the most difficult to read as it's very abstract and most proofs, at least in the beginning chapters, are left to the reader. This book was translated from its original version in Russian. I.M. Gelfand is a mathematical giant who made significant contributions to the field of integral geometry, in addition to many other areas (if you didn't know this already!). The book gives a proof of the Radon transform inversion which is due to the co-authors Gelfand and Graev. I have so far read only the first chapter of this text. The rest of the book goes into further generalizations of the Radon transform, like in the complex domain and in spaces of constant curvature. It also covers topics in representation theory, which I'm not familiar with at all. I hope to learn more from this book in the near future. This is a very advanced pure mathematics book, and you have to be fluent in functional analysis, differential geometry, and even some algebra to fully understand the content.
I hope you will find at least one of these books helpful! :)
