The dual horospherical Radon transform as a limit of spherical Radon transforms

2003 ◽  
pp. 135-143 ◽  
Author(s):  
J. Hilgert ◽  
A. Pasquale ◽  
E. B. Vinberg
Keyword(s):  
Radon Transform ◽  
Radon Transforms

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.

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.

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 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.

Sampling two‐dimensional seismic data and their Radon transforms

Geophysics ◽  
1989 ◽  
Vol 54 (10) ◽  
pp. 1318-1325 ◽  
Author(s):  
Virgil Bardan
Keyword(s):  
Radon Transform ◽  
Seismic Data ◽  
Amplitude Spectrum ◽  
Triangular Grid ◽  
Radon Transforms ◽  
Rectangular Grid ◽  
Fourier Plane ◽  
Data Set ◽  
Band Region ◽  
Sample Points

2‐D seismic data are usually sampled and processed in a rectangular grid, for which sampling requirements are generally derived from the usual 1‐D viewpoint. For a 2‐D seismic data set, the band region (the region of the Fourier plane in which the amplitude spectrum exceeds some very small number) can be approximated by a domain bounded by two triangles. Considering the particular shape of this band region, I use 2‐D sampling theory to obtain results applicable to seismic data processing. The 2‐D viewpoint leads naturally to weaker sampling requirements than does the 1‐D viewpoint; i.e., fewer sample points are needed to represent data with the same degree of accuracy. The sampling of 2‐D seismic data and of their Radon transform in a parallelogram and then in a triangular grid is introduced. The triangular sampling grid is optimal in these cases, since it requires the minimum number of sample points—equal to half the number required by a parallelogram or rectangular grid. The sampling of 2‐D seismic data in a triangular grid is illustrated by examples of synthetic and field seismic sections. The properties of parallelogram grid sampling impose an additional sampling requirement on the 2‐D seismic data in order to evaluate their Radon transform numerically; i.e., the maximum value of the spatial sampling interval must be half of that required by the sampling theorem.

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.

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