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Let $f : \mathbb R^3 \to \mathbb R$ be an integrable function. Let $\eta$ be a one-dimensional subspace of $\mathbb R^3$. We denote $p+\eta$ the affine subspace (a line) which is obtained by translation of $\eta$. The Radon transform for lines is then defined by \begin{align*} (Rf)(\eta, p) = \int_{p + \eta} f \, \text{d}S. \end{align*} Denote the function $R_\eta f$ defined by $(R_\eta f)(p) = (Rf)(\eta,p)$ the radiograph along the direction $\eta$. It is a well-known result that if we know $(R_\eta f)$ for infinitely many distinct one-dimensional subspaces, then we can uniquely reconstruct $f$.

Now define the Radon transform along planes \begin{align*} (P f)(\eta, q) = \int_{q + \xi} f \, \text{d}S, \end{align*} where $\xi$ is now a two-dimensional subspace (and thus $q+\xi$ are planes). Suppose one has an infinite number of distinct $\xi$ which intersect in the one-dimensional subspace $L$, see the figure below. enter image description here Clearly one cannot reconstruct $f$, but intuitively I would expect that one can then get partial information about $f$, such as the values \begin{align*} \int_{p+L} f \, \text{d}S = (Rf)(L,p) \end{align*} from the functions $P_\xi f$. However I haven't found a way to make this rigorous.

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  • $\begingroup$ It is not true that if you know the line transform for infinitely many lines in three-space then you can recover the function. The best you can do is to know the value of the transform over certain three-dimensional manifolds in the space of lines. $\endgroup$ Jan 29, 2014 at 20:45
  • $\begingroup$ It is well-known for the X-ray transform in n-dimensional space, that if one knows the radiographs (!) for infinitely many directions, then one can uniquely reconstruct the function under scrutiny. $\endgroup$
    – user45183
    Jan 29, 2014 at 22:22
  • $\begingroup$ what's your reference for this? $\endgroup$ Jan 30, 2014 at 11:25
  • $\begingroup$ Sigurd Helgason: Integral Geometry and Radon Transforms, Proposition 7.8 and Andrew Markoe: Analytic Tomography, Theorem 3.144. $\endgroup$
    – user45183
    Jan 30, 2014 at 14:27
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    $\begingroup$ @alvarezpaiva: It seems that you are confused with the notion of radiographs. I did not say that knowing $\int_L f dS$ for an infinite number of distinct lines L is sufficient. Rather, I said that knowing $\int_{p+L} f dS$ for all displacements p and an infinite number of distinct lines L. For brevity one calls $R_L f$ defined by $(R_L f)(p) = (Rf)(L,p) = \int_{p+L} f dS$ a radiograph. Clearly for your specific example, one can reconstruct the function on all slices and by putting the slices together one gets the whole function. $\endgroup$
    – user45183
    Jan 30, 2014 at 14:39

1 Answer 1

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Define \begin{align*} \widetilde{f}(p) := \int_{p+L} f \, \text{d}S = (Rf)(L,p). \end{align*} By doing so we are basically back in the left picture. We know the integral of $\widetilde{f}$ along all the lines $\eta_i$, simply because \begin{align*} R_\eta \widetilde{f} = R_\xi f, \end{align*} which we know by assumption.

Thus by the uniqueness theorem for the X-Ray transform we may reconstruct $\widetilde{f}$, which is what I asked for.

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