The Radon transform in two-dimensions is well studied. It maps a sufficiently nice function $f: \mathbb R^2 \to \mathbb R$ to its line integral along a certain line $L$, i.e.

\begin{align*} Rf(L) := \int_{L} f(x) \; \text{d}S. \end{align*}

The line $L$ can easily be parameterized by its angle $\alpha$ (say to the $x_1$-axis) and its displacement $s$ to the origin. We use same the notation and also write $Rf(\alpha,s)$.

It is a well-known result that if we know the Radon transform of a function $f$ for an open set of angles and all displacements, then $f$ can be uniquely determined. (By the Fourier-Slice theorem, the Fourier transform is given for a cone (indeed an open set in $\mathbb R^2$) and by the Paley-Wiener theorem knowing the Fourier transform on an open set is enough (Holomorphy argument)).

Now when it comes to the generalized Radon transform (on hyperplanes) however, I have not been able to find such "limited angle" result. Of course in the higher-dimensional case it does not make sense to talk about angles, but I think there should be some kind of analogy such as: "if one has enough freedom to spin the hyperplanes in $\mathbb R^n$, then uniqueness is given.

Literature and comments are greatly appreciated.Thank you.


1 Answer 1

  • MR1723736 Helgason, Sigurdur The Radon transform. Second edition. Progress in Mathematics, 5. Birkhäuser Boston, Inc., Boston, MA, 1999. xiv+188 pp.

  • MR2743116 Helgason, Sigurdur Integral geometry and Radon transforms. Springer, New York, 2011. xiv+301 pp. ISBN: 978-1-4419-6054-2

  • $\begingroup$ thank you for your references. The second reference introduces the $d$-plane transform but only discusses the limited data issue for $d=1$, i.e. the X-Ray transform. It is shown that infinite directions of lines in $\mathbb R^n$ are sufficient. That result is nice as it generalizes the result from $n=2$ to any dimensions, however I still can't find results for $d \ne 1$. $\endgroup$
    – user45183
    Jan 15, 2014 at 16:58

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