Relation between Radon transform and Fourier transform

On page 2 of chapter 1, the Radon transform of a function $$f: \mathbb{R}^n \to \mathbb{C}$$ on a hyperplane $$\xi$$ is defined as $$\hat{f}(\xi) = \int_{\xi} f(x) dm(x)$$, where $$dm$$ is the Euclidean measure on the hyperplane $$\xi$$.

Later on on page 4, a relationship between Fourier transforms and Radon transform is noted, that $$\tilde{f}(s \omega) = \int_{-\infty}^{\infty} \int_{\langle x, \omega \rangle = r} f(x) e^{-i s \langle x, \omega \rangle} dm(x) dr$$ where $$\tilde{f}$$ is the Fourier transform of $$f$$, $$\omega$$ is a unit vector and $$s \in \mathbb{R}$$.

My questions:

1. What exactly is meant by "Euclidean measure on the hyperplane on $$\xi$$"? The notes do not define it and Google doesn't seem to turn up anything.

2. I do not understand how the formula for $$\tilde{f}(s \omega)$$ is obtained. Here, the Fourier transform is written as $$\tilde{f}(\omega) = \int f(x) e^{-i \langle x, \omega \rangle} dx$$, but I do not see how it can be transformed into the equation involving the Radon Transform.

• You're writing the integral as an integral over a plane that is a distance of $r$ from the origin with normal $\omega$ and then an integral in $r$. The integral over the plane is with respect to the ordinary Euclidean measure on the plane. – DisintegratingByParts Mar 13 at 0:50
• Thanks for the reply. I made a small typo in the formula that's now been fixed. With respect to your answer, I'm not sure I understand what "ordinary Euclidean measure on the plane" means. I do see what the interpretation of right-hand side of the equation, but I still don't really see the relation to the LHS. I think it would be most helpful to me to see what precisely what measure $dm$ refers to. – mathmajor55567 Mar 13 at 1:29
• Euclidean measure on a plane $x_1=d$ for $(x_1,x_2,\cdots,x_n)\in\mathbb{R}^n$ would correspond to integration with respect to $dx_2 dx_3 dx_4\cdots dx_n$ on an $\mathbb{R}^{n-1}$ plane. For a more general plane, you can always apply a rotation to put it back into this configuration. The rotation preserves measure. A translation in a variable also preserves measure. – DisintegratingByParts Mar 13 at 1:49