scholarly journals Radon Transforms in Hyperbolic Spaces and Their Discrete Counterparts

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Enrico Casadio Tarabusi ◽  
Massimo A. Picardello
Keyword(s):  
Radon Transform ◽  
Hyperbolic Spaces ◽  
Spherical Functions ◽  
Homogeneous Tree ◽  
Radon Transforms ◽  
Convolution Operators ◽  
Convolution Kernels

AbstractIn the hyperbolic disc (or more generally in real hyperbolic spaces) we consider the horospherical Radon transform R and the geodesic Radon transform X. Composition with their respective dual operators yields two convolution operators on the disc (with respect to the hyperbolic measure). We describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree T, separately studied as acting on functions on the vertices or on the edges. This leads to a new theory of spherical functions and Radon inversion on the edges of a tree.

scholarly journals Uniform Estimates for Damped Radon Transform on the Plane

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Youngwoo Choi
Keyword(s):  
Radon Transform ◽  
Lorentz Spaces ◽  
Radon Transforms ◽  
Convolution Operators ◽  
Uniform Estimates ◽  
Sharp Estimates ◽  
Rotational Curvature ◽  
Affine Arclength

Uniform improving estimates of damped plane Radon transforms in Lebesgue and Lorentz spaces are studied under mild assumptions on the rotational curvature. The results generalize previously known estimates. Also, they extend sharp estimates known for convolution operators with affine arclength measures to the semitranslation-invariant case.

scholarly journals Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

2021 ◽  
Vol 27 (5) ◽  
Author(s):  
Francesca Bartolucci ◽  
Filippo De Mari ◽  
Matteo Monti
Keyword(s):  
Radon Transform ◽  
Homogeneous Tree ◽  
Homogeneous Trees ◽  
Regular Representations ◽  
Group Of Isometries

AbstractFollowing previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

Simultaneous source separation using a robust Radon transform

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. V1-V11 ◽  
Author(s):  
Amr Ibrahim ◽  
Mauricio D. Sacchi
Keyword(s):  
Radon Transform ◽  
Real Data ◽  
Radon Transforms ◽  
Accurate Data ◽  
Quadratic Formula ◽  
Penalty Term ◽  
Source Data ◽  
The Cost ◽  
Simultaneous Source ◽  
Incoherent Noise

We adopted the robust Radon transform to eliminate erratic incoherent noise that arises in common receiver gathers when simultaneous source data are acquired. The proposed robust Radon transform was posed as an inverse problem using an [Formula: see text] misfit that is not sensitive to erratic noise. The latter permitted us to design Radon algorithms that are capable of eliminating incoherent noise in common receiver gathers. We also compared nonrobust and robust Radon transforms that are implemented via a quadratic ([Formula: see text]) or a sparse ([Formula: see text]) penalty term in the cost function. The results demonstrated the importance of incorporating a robust misfit functional in the Radon transform to cope with simultaneous source interferences. Synthetic and real data examples proved that the robust Radon transform produces more accurate data estimates than least-squares and sparse Radon transforms.

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.

scholarly journals Sparse Bounds for a Prototypical Singular Radon Transform

2019 ◽  
Vol 62 (02) ◽  
pp. 405-415 ◽  
Author(s):  
Richard Oberlin
Keyword(s):  
Radon Transform ◽  
Radon Transforms ◽  
Singular Radon Transform

AbstractWe use a variant of a technique used by M. T. Lacey to give sparse $L^{p}(\log (L))^{4}$ bounds for a class of model singular and maximal Radon transforms.

scholarly journals SCHUR MULTIPLIERS AND SPHERICAL FUNCTIONS ON HOMOGENEOUS TREES

2010 ◽  
Vol 21 (10) ◽  
pp. 1337-1382 ◽  
Author(s):  
U. HAAGERUP ◽  
T. STEENSTRUP ◽  
R. SZWARC
Keyword(s):  
Fourier Multiplier ◽  
Spherical Functions ◽  
Sufficient Condition ◽  
Homogeneous Tree ◽  
Necessary And Sufficient Condition ◽  
Closed Expression ◽  
Radial Functions ◽  
Schur Multipliers ◽  
Homogeneous Trees ◽  
Necessary And Sufficient

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ||ψ||S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ||⋅||M0A(G) of the radial functions on the free (non-abelian) group 𝔽N on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL2(ℚq) for every prime number q.

scholarly journals A fast butterfly algorithm for generalized Radon transforms

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. U41-U51 ◽  
Author(s):  
Jingwei Hu ◽  
Sergey Fomel ◽  
Laurent Demanet ◽  
Lexing Ying
Keyword(s):  
Integral Operator ◽  
Radon Transform ◽  
Model Space ◽  
Oscillatory Integral ◽  
Fourier Integral ◽  
Low Rank ◽  
Radon Transforms ◽  
Low Rank Approximation ◽  
Data Set ◽  
Generalized Radon Transforms

Generalized Radon transforms, such as the hyperbolic Radon transform, cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We have devised a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low-rank approximation of the kernel function. The overall structure follows the Fourier integral operator butterfly algorithm. For 2D data, the algorithm runs in complexity [Formula: see text], where [Formula: see text] depends on the maximum frequency and offset in the data set and the range of parameters (intercept time and slowness) in the model space. From a series of studies, we found that this algorithm can be significantly more efficient than the conventional time-domain integration.

scholarly journals The Radon Transforms on the Generalized Heisenberg Group

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Tianwu Liu ◽  
Jianxun He
Keyword(s):  
Fourier Transform ◽  
Differential Operator ◽  
Heisenberg Group ◽  
Radon Transform ◽  
Inversion Formula ◽  
The Other ◽  
Radon Transforms

Let Hna be the generalized Heisenberg group. In this paper, we study the inversion of the Radon transforms on Hna. Several kinds of inversion Radon transform formulas are established. One is obtained from the Euclidean Fourier transform; the other is derived from the differential operator with respect to the center variable t. Also by using sub-Laplacian and generalized sub-Laplacian we deduce an inversion formula of the Radon transform on Hna.

GENERALIZED KALOUJNINE GROUPS, UNISERIALITY AND HEIGHT OF AUTOMORPHISMS

2005 ◽  
Vol 15 (03) ◽  
pp. 503-527 ◽  
Author(s):  
TULLIO G. CECCHERINI-SILBERSTEIN ◽  
YURIJ G. LEONOV ◽  
FABIO SCARABOTTI ◽  
FILIPPO TOLLI
Keyword(s):  
Vector Space ◽  
Radon Transform ◽  
Lower Central Series ◽  
Central Series ◽  
Homogeneous Tree

We show that the Lie action of the Kaloujnine group K(p,n) on the vector space (Fp)pn is uniserial. Using some Radon transform techniques we derive a formula for the height of the elements in K(p,n). A generalization of the Kaloujnine groups is introduced by considering automorphisms of a spherically homogeneous tree. We observe that uniseriality fails to hold for these groups and determine their lower central series; finally we discuss in detail Kaloujnine's description of the characteristic subgroups in terms of the (normal) "parallelotopic" subgroups.

Sign in / Sign up

Export Citation Format

Share Document