Inversion Formulas for a Cylindrical Radon Transform

2011 ◽  
Vol 4 (3) ◽  
pp. 789-806 ◽  
Author(s):  
Markus Haltmeier
Keyword(s):  
Radon Transform ◽  
Inversion Formulas

METHOD OF MEAN VALUE OPERATORS FOR RADON TRANSFORMS IN THE SPACE OF MATRICES

2008 ◽  
Vol 19 (03) ◽  
pp. 245-283 ◽  
Author(s):  
E. OURNYCHEVA ◽  
B. RUBIN
Keyword(s):  
Radon Transform ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Inversion Method ◽  
Radon Transforms ◽  
Inversion Formulas ◽  
Higher Rank ◽  
Necessary And Sufficient

We extend the Funk–Radon–Helgason inversion method of mean value operators to the Radon transform [Formula: see text] of continuous and Lpfunctions which are integrated over matrix planes in the space of real rectangular matrices. Necessary and sufficient conditions of existence of [Formula: see text] for such f and explicit inversion formulas are obtained. New higher-rank phenomena related to this setting are investigated.

The vertical slice transform on the unit sphere

2019 ◽  
Vol 22 (4) ◽  
pp. 899-917 ◽  
Author(s):  
Boris Rubin
Keyword(s):  
Unit Sphere ◽  
Radon Transform ◽  
Integral Geometry ◽  
Singular Value ◽  
Hypersingular Integrals ◽  
Inversion Formulas ◽  
Spherical Mean ◽  
Singular Value Decompositions ◽  
Vertical Slice ◽  
Class Of Functions

Abstract The vertical slice transform in spherical integral geometry takes a function on the unit sphere Sn to integrals of that function over spherical slices parallel to the last coordinate axis. This transform was investigated for n = 2 in connection with inverse problems of spherical tomography. The present article gives a survey of some methods which were originally developed for the Radon transform, hypersingular integrals, and the spherical mean Radon-like transforms, and can be adapted to obtain new inversion formulas and singular value decompositions for the vertical slice transform in the general case n ≥ 2 for a large class of functions.

scholarly journals Radon transform on affine buildings of rank three

1999 ◽  
Vol 66 (1) ◽  
pp. 66-89 ◽  
Author(s):  
Laura Atanasi
Keyword(s):  
Radon Transform ◽  
Finite Rank ◽  
Affine Buildings ◽  
Locally Finite ◽  
Inversion Formulas

AbstractWe define the Radon transform for functions on the set of chambers of affine, locally finite, rank three buildings. We investigate the problem of the inversion of this transform. Explicit inversion formulas are exhibited for functions which fulfill required summability conditions.

scholarly journals On Radon transforms between lines and hyperplanes

2017 ◽  
Vol 28 (13) ◽  
pp. 1750093 ◽  
Author(s):  
Boris Rubin ◽  
Yingzhan Wang
Keyword(s):  
Radon Transform ◽  
Symmetric Case ◽  
Fractional Integrals ◽  
Radon Transforms ◽  
Full Generality ◽  
Inversion Formulas ◽  
Radial Functions ◽  
Kelvin Transform ◽  
Funk Transform ◽  
Dual Transform

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in [Formula: see text]. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John [Formula: see text]-plane transform in [Formula: see text], the Grassmannian modification of the Kelvin transform, and the Erdélyi–Kober fractional integrals.

scholarly journals Littlewood-Paley g-function and Radon Transform on the Heisenberg Group

2018 ◽  
Vol 11 (1) ◽  
pp. 138
Author(s):  
Zheng Fang ◽  
Jianxun He
Keyword(s):  
Heisenberg Group ◽  
Radon Transform ◽  
Unitary Operator ◽  
Radon Transforms ◽  
Inversion Formulas ◽  
Poisson Integrals

In this paper, we consider Radon transform on the Heisenberg group $\textbf{H}^{n}$, and obtain new inversion formulas via dual Radon transforms and Poisson integrals. We prove that the Radon transform is a unitary operator from Sobelov space $W$ into $L^{2}(\textbf{H}^{n})$. Moreover, we use the Radon transform to define the Littlewood-Paley $g$-function on a hyperplane and obtain the Littlewood-Paley theory.

scholarly journals Inversion Formula for a Radon-Type Transform Arising in Photoacoustic Tomography with Circular Integrating Detectors

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
Sunghwan Moon
Keyword(s):  
Radon Transform ◽  
Inversion Formula ◽  
Photoacoustic Tomography ◽  
Inversion Formulas ◽  
Spherical Radon Transform

This paper is devoted to a Radon-type transform arising in a version of Photoacoustic Tomography that uses integrating circular detectors. The Radon-type transform that arises can be decomposed into the known Radon-type transforms: the spherical Radon transform and the sectional Radon transform. An inversion formula is obtained by combining existing inversion formulas for the above two Radon-type transforms.

Inversion formulas for the $k$ -dimensional Radon transform in real hyperbolic spaces

1991 ◽  
Vol 62 (3) ◽  
pp. 613-631 ◽  
Author(s):  
Carlos A. Berenstein ◽  
Tarabusi Enrico Casadio
Keyword(s):  
Radon Transform ◽  
Hyperbolic Spaces ◽  
Inversion Formulas
Sign in / Sign up

Export Citation Format

Share Document