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We have the Radon transformation, in the most common form:

$g(\phi,r)=\int\int f(x,y)\delta(x\sin\phi-y\cos\phi-r)dxdy$

Question 1:

So what is the closed form of the derivative of Radon transform w.r.t $\phi$? Also the derivation details.

Also, I want to consider the derivative in the discrete form, i.e.,

$g(\phi,r)=\sum_x\sum_y f(x,y)\delta(x\sin\phi-y\cos\phi-r)$

in order to do numerical implementation. Not sure if I will have problems in implementation - but any advice welcome!

Question 2:

How to derive a closed form solution for an expression of $\phi$, from the Radon transformation, (maybe) with the use of the derivative of the Radon transformation.

I have searched for some literatures but seems couldn't find a satisfied answer.

Thanks in advance!

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