All Questions
Tagged with inverse-problems integral-transforms
13 questions
2
votes
1
answer
265
views
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 ...
- 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 ...
- 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 ...
- 21.8k
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
...
- 1,465
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 ...
4
votes
1
answer
271
views
Injectivity of the Funk transform for nonsmooth functions
Let $S^{n-1}$ be the unit sphere in $\mathbb R^n$ and $\Gamma_n$ the collection of great circles on it.
Assume $n\geq3$.
The Funk transform of a function $f:S^{n-1}\to\mathbb R$ is a map $Ff:\Gamma_n\...
- 8,086
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. $(...
user45183
23
votes
1
answer
1k
views
Geodesics in finite groups
It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for ...
- 8,086
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{...
user45183
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 ...
user45183
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)...
- 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\...
- 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.
...
- 1,329