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I am having a hard time trying to understand apparently two different definitions of the dual Radon Transform. I am reading simultaneously the book "Mathematics of computerized tomography", by Frank Natterer, and "Radon transform on homogeneous spaces", by Sigurdur Helgason (by now, only the first chapter of each book).

I will try to be thorough in the exposition of my doubt(s). To this end, let me explain a little bit the treatment of this subject in each author, to save you from the trouble of having to look at the books mentioned.

(Note: in the sequel, $\omega_{n}$ is the surface area of the unit sphere in $\mathbb{R}^{n},$ $\mathcal{P}$ is the set of all hyperplanes, $R(f)(\pi)=\int_{\pi}f(y)dm(y)$ is the Radon transform, $R(f)(\theta,s)=\int_{\{ \langle x,\theta \rangle =0 \}}f(x+s\theta)dm(x),$ is also the Radon transform, seen as a symmetric function over the cylinder )

Let me start with Natterer. He defines the dual Radon transform as follows: first, a continuous function $g(\pi)$ in the set of hyperplanes can be identified with a symmetric continuous function in $S^{n-1} \times \mathbb{R}$ of the form $g(\theta, s),$ where $\theta$ is the normal vector to the hyperplane and $s$ is the distance from the plane to the origin. Then Natterer defines the dual Radon transform as: $ R^{\#}(g)(x):=\displaystyle \int _{S^{n-1}} g(\theta ,\langle x,\theta \rangle ) dS(\theta).$ He strongly motivates this definition, so I take it is not a misprint. Is with this definition that he derives an important formula (in particular, to derive the inversion theorem): $$R^{\#}R(f)(x)=\omega_{n-1}\int_{\mathbb{R}^{n}} \vert x-y \vert ^{-1}f(y) dy.$$

Now, let me very briefly explain what Helgason does: The dual transform is now defined as $R^{\#}(g)(x)=\displaystyle \int_{\{ \pi \in \mathcal{P} \hspace{1mm}:\hspace{1mm} x \in \pi \} }g(\pi) d\mu,$ where $\mu$ is the unique normalized measure invariant under rotations. Helgason readily says that this must be $\displaystyle c\int_{S^{n-1}} g(\theta,\langle x,\theta \rangle) dS(\theta)$ for certain constant $c,$ which seems very logical but I don't seem to be able to prove rigorously. Of course, for Natterer $c=1,$ but the point is that this is not so for Helgason. For Helgason, is trivial that his definition implies that $R^{\#}(g)(x)=\displaystyle \int_{O(n)} g(x+k \pi_{0}) dk,$ where $dk$ is the Haar measure on the orthogonal group and $\pi_{0}$ is any fixed hyperplane containing the origin (so, another doubt, how do we prove this?).

Now, an argument based upon the uniqueness of normalized rotationally invariant measures shows that $c=\frac{1}{\omega_{n}}.$ Due to this, Helgason derives the formula: $$R^{\#}R(f)(x)=\frac{\omega_{n-1}}{\omega_{n}}\int_{\mathbb{R}^{n}} \vert x-y \vert ^{-1}f(y) dy.$$

My feeling is that Helgason's point of view is more correct, but I cannot completely grasp his arguments, and he seems a little thick in his exposition at some points.

I am worried, because I do not know what's going on with Natterer's book. So, to sum up, I want to clarify:

1º: The arguments used by Helgason, which I have marked as doubts.

2º: The convention used by Natterer, which seems less logical than Helgason's. Also, I'd like to find out which convention or criterion is more widely employed.

As a last point, if you have any suggestions about other books which I could use in my study of the Radon Transform, I will appreciate them!

Thank you.

EDIT: From a quick overview of several lecture notes available online, I take that Natterer's convention is pretty common. This seems weird to me, for that convention is incompatible with the natural definition of Helgason $R^{\#}(g)(x)=\displaystyle \int_{\{ \pi \in \mathcal{P} \hspace{1mm}:\hspace{1mm} x \in \pi \} }g(\pi) d\mu.$

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