A Spline Inversion Formula for the Radon Transform

1989 ◽  
Vol 26 (2) ◽  
pp. 456-467 ◽  
Author(s):  
Michael Lautsch
Keyword(s):  
Radon Transform ◽  
Inversion Formula

The d-plane radon transform on the torus $$ \mathbb{T}^n $$

2011 ◽  
Vol 14 (2) ◽  
Author(s):  
Ahmed Abouelaz
Keyword(s):  
Radon Transform ◽  
Inversion Formula ◽  
Flat Torus ◽  
Smooth Functions ◽  
Suitable Function

AbstractWe define and study the d-plane Radon transform, namely R, on the n-dimensional (flat) torus. The transformation R is obtained by integrating a suitable function f over all d-dimensional geodesics (d-planes in the torus). We specially establish an explicit inversion formula of R and we give a characterization of the image, under the d-plane Radon transform, of the space of smooth functions on the torus.

An Inversion Formula of the Radon Transform on the Heisenberg Group

2004 ◽  
Vol 47 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Jianxun He
Keyword(s):  
Heisenberg Group ◽  
Radon Transform ◽  
Inversion Formula ◽  
Weak Sense

AbstractIn this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

scholarly journals AN INVERSION FORMULA FOR THE GAUSSIAN RADON TRANSFORM FOR BANACH SPACES

2013 ◽  
Vol 16 (04) ◽  
pp. 1350034 ◽  
Author(s):  
IRINA HOLMES
Keyword(s):  
Banach Spaces ◽  
Radon Transform ◽  
Inversion Formula ◽  
Bargmann Transform ◽  
Abstract Wiener Spaces ◽  
Wiener Spaces

We prove a disintegration theorem for the Gaussian Radon transform Gf on Banach spaces and use the Segal–Bargmann transform on abstract Wiener spaces to find a procedure to obtain f from its Gaussian Radon transform Gf.

A new slant on seismic imaging: Migration and integral geometry

Geophysics ◽  
1987 ◽  
Vol 52 (7) ◽  
pp. 943-964 ◽  
Author(s):  
D. Miller ◽  
M. Oristaglio ◽  
G. Beylkin
Keyword(s):  
Wave Equation ◽  
Radon Transform ◽  
Inversion Formula ◽  
Acoustic Scattering ◽  
Seismic Profile ◽  
Seismic Inversion ◽  
Borehole Data ◽  
Velocity Models ◽  
Generalized Radon Transform ◽  
Zero Offset

A new approach to seismic migration formalizes the classical diffraction (or common‐tangent) stack by relating it to linearized seismic inversion and the generalized Radon transform. This approach recasts migration as the problem of reconstructing the earth’s acoustic scattering potential from its integrals over isochron surfaces. The theory rests on a solution of the wave equation with the geometrical‐optics Green function and an approximate inversion formula for the generalized Radon transform. The method can handle both complex velocity models and (nearly) arbitrary configurations of sources and receivers. In this general case, the method can be implemented as a weighted diffraction stack, with the weights determined by tracing rays from image points to the experiment’s sources and receivers. When tested on a finite‐difference simulation of a deviated‐well vertical seismic profile (a hybrid experiment which is difficult to treat with conventional wave‐equation methods), the algorithm accurately reconstructed faulted‐earth models. Analytical reconstruction formulas are derived from the general formula for zero‐offset and fixed‐offset surface experiments in which the background velocity is constant. The zero‐offset inversion formula resembles standard Kirchhoff migration. Our analysis provides a direct connection between the experimental setup (source and receiver positions, source wavelet, background velocity) and the spatial resolution of the reconstruction. Synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data. The best resolution is obtained from a zero‐offset experiment that surrounds the region to be imaged.

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.

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