A key technical step in the proof of the Radon Transform support theorem is proving that the result holds for radial functions. The proof presented in class was elementary but technical and, in my opinion, bewildering. Not the sort of thing I’d come up with.
Yes, I know, I need to work on my skills.
Instead of doing that, I came up with another proof that I find more satisfying. It is fairly simple, and uses nothing more than the Paley-Wiener theorem and a bit of Calculus.
Theorem if , , , and is radial, then has compact support and .
The rest of this post will be the proof. Here is a quick sketch:
- is analytic of exponential type in , by the Paley-Wiener Theorem.
- , so the exponential type bounds we got in for in (1) transfer to bounds on the pure partial derivatives of
- If is radial, then at the origin
for all multi-indices
- This shows that the mixed partials of are dominated by the pure partials at the origin, and the exponential type estimates in (2) hold for all partials
- So is analytic of exponential type. Thus has compact support. Continue reading
Now we want to study the range of the Radon transform. This is important for our inverse problem: when we want to reconstruct a function from its Radon transform, all we really have to work with is a finite number of error-prone samples of the Radon transform. We need to analyze this statistically to find the function “most likely” to have produced the observed data. We can only do this if we know what sort of functions can be Radon transforms of other functions.
In other words, we can only do this if we know the Range of . Continue reading
With the tools of Sobolev spaces at our disposal, we are ready to go back to one of our basic questions: is our solution to the Radon transform inverse problem stable? In other words if\
, do we know that .
If we don’t know this, we can’t trust our reconstruction — even tiny rounding errors could be disastrous.
A Formal Look
Let’s consider the case when is odd (so that is even). If , we know that , so
This suggests that . To prove it, we need to define that space. Continue reading
We will define Sobolev spaces using the Fourier transform and .
Definition For let be
Claim Continue reading
The Radon Inversion formula developed in the last section is nice, but there is a problem. It only applies to smooth functions, and the functions we are interested in are not smooth. Heart tissue does not smoothly change into lung tissue; body parts have sharp boundaries. If we want to use this for applications like medical imaging we must extend the results to a larger domain. Continue reading
To keep the notes coherent, I’ll repeat some notes from the last lecture and start getting into new material when considering Radon inversion in even dimensions. Continue reading