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 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.

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 Multivariate and abstract approximation theory for Banach space valued functions

2017 ◽  
Vol 50 (1) ◽  
pp. 208-222 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Banach Space ◽  
Degree Of Approximation ◽  
Linear Operators ◽  
Approximation Properties ◽  
Uniform Estimates ◽  
Sharp Estimates ◽  
Differentiable Functions ◽  
Unit Operator ◽  
Fréchet Differentiable ◽  
High Degree

Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

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.

Three-parameter Radon transform based on shifted hyperbolas

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. V39-V48 ◽  
Author(s):  
Ali Gholami ◽  
Toktam Zand
Keyword(s):  
Radon Transform ◽  
Seismic Data ◽  
High Performance ◽  
Fourier Transforms ◽  
Real Data ◽  
Data Sets ◽  
Radon Transforms ◽  
Slice Theorem ◽  
Ground Roll ◽  
Zero Offset

The focusing power of the conventional hyperbolic Radon transform decreases for long-offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets. Furthermore, conventional Radon transforms are ineffective for processing data sets containing events of different shapes. The shifted hyperbola is a flexible three-parameter (zero-offset traveltime, slowness, and focusing-depth) function, which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets. Radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain. We have developed a new method for effective decomposition of seismic data by using such three-parameter Radon transform. A very fast algorithm is constructed for high-resolution calculations of the new Radon transform using the recently proposed generalized Fourier slice theorem (GFST). The GFST establishes an analytic expression between the [Formula: see text] coefficients of the data and the [Formula: see text] coefficients of its Radon transform, with which a very fast switching between the model and data spaces is possible by means of interpolation procedures and fast Fourier transforms. High performance of the new algorithm is demonstrated on synthetic and real data sets for trace interpolation and linear (ground roll) noise attenuation.

scholarly journals Using Few Radon Projections to Recover 3-D NMR Spectrum

2017 ◽  
pp. 149-160
Author(s):  
Fawaz Ibrahim Hjouj
Keyword(s):  
Radon Transform ◽  
Compact Support ◽  
Image Representation ◽  
Radon Transforms ◽  
Back Projection ◽  
Binary Function ◽  
Nmr Spectrum ◽  
Radon Projections ◽  
Mathematical Abstraction ◽  
Minimum Number

Given a regular binary function f on R2 with compact support D, we use translation to form a new binary function g from f so that the image representation of g (x, y) is made up of non-overlapping copies of D. Thus, g is made up of discrete entities that are surrounded by regions of space. We devise a procedure that can determine the translation parameters using a minimum number of Radon projections  g of g . This model is a mathematical abstraction of an application of the Radon transform in Spectroscopy.Keywords: Radon transforms, transformation of an image, Back projection.

Sign in / Sign up

Export Citation Format

Share Document