Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.

Original languageEnglish (US)
Pages (from-to)1455-1478
Number of pages24
JournalInverse Problems
Volume20
Issue number5
DOIs
StatePublished - 2004

Fingerprint

Radon Transform
Radon
radon
Attenuation
Inversion
attenuation
Inversion Formula
Trajectories
Transform
inversions
Angle
Dependent
Image reconstruction
Valid
Trajectory
Orbits
Reconstruction Algorithm
Image Reconstruction
Intersect
trajectories

ASJC Scopus subject areas

  • Applied Mathematics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation. / Kunyansky, Leonid.

In: Inverse Problems, Vol. 20, No. 5, 2004, p. 1455-1478.

Research output: Contribution to journalArticle

@article{8c8c363ed14343838cad7ba9663e43a9,
title = "Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation",
abstract = "The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.",
author = "Leonid Kunyansky",
year = "2004",
doi = "10.1088/0266-5611/20/5/008",
language = "English (US)",
volume = "20",
pages = "1455--1478",
journal = "Inverse Problems",
issn = "0266-5611",
publisher = "IOP Publishing Ltd.",
number = "5",

}

TY - JOUR

T1 - Inversion of the 3D exponential parallel-beam transform and the Radon transform with angle-dependent attenuation

AU - Kunyansky, Leonid

PY - 2004

Y1 - 2004

N2 - The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.

AB - The inversion problem for the 3D parallel-beam exponential ray transform is solved through inversion of a set of the 2D exponential Radon transforms with complex-valued angle-dependent attenuation. An inversion formula for the latter 2D transform is derived; it generalizes the known Kuchment-Shneiberg formula valid for real angle-dependent attenuation. We derive an explicit theoretically exact solution of the 3D problem which is valid for arbitrary closed trajectory that does not intersect itself. A simple reconstruction algorithm is described, applicable for certain sets of trajectories satisfying Orlov's condition. In the latter case, our inversion technique is as stable as the Tretiak-Metz inversion formula. Possibilities of further reduction of noise sensitivity are briefly discussed in the paper. The work of our algorithm is illustrated by an example of image reconstruction from two circular orbits.

UR - http://www.scopus.com/inward/record.url?scp=6344253467&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=6344253467&partnerID=8YFLogxK

U2 - 10.1088/0266-5611/20/5/008

DO - 10.1088/0266-5611/20/5/008

M3 - Article

VL - 20

SP - 1455

EP - 1478

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 5

ER -