The generalized Radon transform maps a function $f \in L^1(\mathbb R^n)$, usually interpreted as a density of an object, to its integral value over an $(n-1)$-dimensional affine subspace.

To be more precise, if $H$ is an affine hyperplane parameterized via $H = \{ x \in \mathbb R^n \; : \; \langle \omega, x \rangle = s \}$, then \begin{align*} (Rf)(H) = (Rf)(\omega, s) = \int_{H} f \; \text{d}S. \end{align*}

It is a well-known result (cf Helgason Sigurdur: Integral geometry and Radon transforms) that knowing the Radon transform for all affine hyperplanes is sufficient to determine $f$.

My question is whether knowing the Radon transform for really all affine hyperplanes is really necessary to determine $f$ uniquely.

For example for $n=2$ the hyperplanes are clearly lines and the transform is bettern known as X-Ray transform. For this transform we know that an infinite number of directions of "rays" (plus all possible displacements from the origin) is already sufficient, i.e. for $n=2$ the condition is not necessary.

How does this behave in the more general cases, i.e. $n>2$ ?

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    $\begingroup$ Did you check Natterer tinyurl.com/mgg2ntx as well? Be aware of the fact that you can define the Radon transform on L^1-functions just in an a.e. sense, so I'm sure the result means it is sufficient to know "almost all" affine hyperplanes. $\endgroup$ – tomglabst Jan 18 '14 at 9:04

I finally found the results I was searching for in "Andrew Markoe: Analytic tomography. Cambridge ; New York : Cambridge University Press, 2005."


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