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Radon transform range theorem and radial functions

(UPDATED for rapid decay considerations + new question) In dimension 2, the Radon transform range theorem states that a rapidly decaying (Schwartz) function $g(t,\theta)$ can be represented as a ...
phaedo's user avatar
  • 123
3 votes
0 answers
119 views

Radon transform on complex Grassmannians

Let $Gr_{i,n}$ denote the Grassmannian of complex linear $i$-dimensional subspaces in the Hermitian space $\mathbb{C}^n$. Let $1\leq i<n/2$. Consider the Radon transform between space of functions ...
asv's user avatar
  • 21.8k
1 vote
0 answers
71 views

Kernel of Radon transform in $\mathbb{R}^3$

Consider the Radon transform from the space of functions on the manifold of affine lines in $\mathbb{R}^3$ to functions on the manifold of affine 2-planes in $\mathbb{R}^3$: $$(Rf)(H):=\int_{l\subset ...
asv's user avatar
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4 votes
1 answer
137 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

I asked this question on math stackexchange, without any reply yet. Link:https://math.stackexchange.com/questions/1401580/under-what-hypothesis-is-the-x-ray-transform-john-transform-operator-bounded ...
Learning math's user avatar
6 votes
1 answer
196 views

Radon transform between affine grassmannians

Let $\overline{GR}(n,k)$ be the manifold of all affine k-dimensional subspaces in $R^n$, and let $R:C^{\infty}_c(\overline{GR}(n,k))\to C^{\infty}_c(\overline{GR}(n,l))$, $0\le k<l\le n-1$, be the ...
Dmitry Ryabogin's user avatar
2 votes
0 answers
108 views

Combining microlocal Helgason's support and Holmgren's theorem to prove injectivity of limited-angle Radon transform

This questions is slightly related to Kashiwara's watermelon theorem and Microlocal version of Helgason's (support) and Holmgren's theorems, in which I asked for some references. Now I ...
Qwertuy's user avatar
  • 251
3 votes
1 answer
131 views

General Radon-type inverse problem

Let $f : \mathbb R^n \to \mathbb R$ be a density which is sufficiently smooth and can also be restricted to have compact support for now. Let $t \ge 0$ and $F_t : \mathbb R^n \to \mathbb R$, i.e. $(...
user avatar
4 votes
1 answer
236 views

Invertibility of an inverse problem

Let $p$ be a scalar field $p: \mathbb R^n \to \mathbb R$. I encountered the problem of reconstructing an unknown density $p$ from its integral values $$I(t,z) = \int_{V_t} p(x) dS$$ along a one-...
user avatar
5 votes
2 answers
204 views

Reconstructing set of points from one-dimensional images

Consider a set of $N$ points in $n$-dimensional space, i.e. \begin{align*} \{x_1, \dots, x_N\} \subset \mathbb R^n. \end{align*} Let us be given a finite family of non-injective matrices \begin{...
user avatar
3 votes
1 answer
217 views

Partial recovery from Radon transform

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 ...
user avatar
2 votes
1 answer
266 views

"Limited angle" in n-dimensional Radon transform?

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)...
zeno44's user avatar
  • 41
4 votes
1 answer
258 views

Interpretation of the integral "with respect to a plane wave" in terms of Radon transform

This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform: $\newcommand{\R}{\mathbb{R}}$ $$ Rf(\varphi,s)=\int_{-\infty}^\infty f(s\...
icurays1's user avatar
  • 203
7 votes
1 answer
478 views

Inversion of Radon transform by incomplete data: specific case

Let $R[f](p,t)$ denote the Radon transform of smooth function $f(x) \colon \mathbb{R}^n \to \mathbb{R}$ with compact support in $\mathbb{R}^n_+$: $$ R[f](p,t) = \int\limits_{x \cdot p = t} f(x) dx. ...
Appliqué's user avatar
  • 1,329