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Inverse Problems in Tomography and Fast Methods for Singular Convolutions

Andersson, Fredrik LU (2005)
Abstract (Swedish)
Popular Abstract in Swedish

I denna avhandling behandlas två delvis överlappande teman; inversa problem av tomografisk karaktär samt snabba och noggranna metoder för applicering av faltningsoperatorer.



Inom det första temat så studeras det inversa problemet inom Dopplertomografi och för detta ändamål så introduceras Dopplermomenttransformen. Genom att undersöka egenskaperna hos denna transform så visas dels entydighetsresultat och dels presenteras en metod för numerisk rekonstruktion. Vi fortsätter i samma andemening genom att fokusera på klassisk datortomografi till vilken en snabb inversionsmetod för Radontransformen utvecklas. Vår metod är av filtrerad-tillbakaprojektionstyp, den praktiskt mest använda... (More)
Popular Abstract in Swedish

I denna avhandling behandlas två delvis överlappande teman; inversa problem av tomografisk karaktär samt snabba och noggranna metoder för applicering av faltningsoperatorer.



Inom det första temat så studeras det inversa problemet inom Dopplertomografi och för detta ändamål så introduceras Dopplermomenttransformen. Genom att undersöka egenskaperna hos denna transform så visas dels entydighetsresultat och dels presenteras en metod för numerisk rekonstruktion. Vi fortsätter i samma andemening genom att fokusera på klassisk datortomografi till vilken en snabb inversionsmetod för Radontransformen utvecklas. Vår metod är av filtrerad-tillbakaprojektionstyp, den praktiskt mest använda inom området, men har en ordnings lägre tidskomplexitet än motsvarande standardimplementationer. Vidare behandlas diffraktionstomografiområdet genom att utveckla en snabb och noggrann metod för att lösa framåtproblemet för vågutbredning i inhomogent medium.



För detta ändamål används de metoder för snabb applicering av faltningsoperatorer som utgör avhandlingens andra tema. Till att börja med behandlas snabba Gausstransformer med komplexa parametrar och senare vidareutvecklas vissa av idéerna där för att konstruera ett kontinuerligt ramverk för snabb applicering av faltningsoperatorer. (Less)
Abstract
There are two, partially interlaced, themes treated in this thesis; inverse problems of tomographic type and fast and accurate methods for the application of convolution operators.



Regarding the first theme, the inverse problem of Doppler tomography is considered and the Doppler moment transform is introduced for that purpose. By investigating the properties of the transform, we prove results regarding uniqueness and develop a numerical method for reconstruction. Continuing on the tomography track, we turn focus to X-ray tomography and construct a fast numerical method for the inversion of the Radon transform. Our method is of filtered back-projection type, the most commonly used in practice, but has one order lower time... (More)
There are two, partially interlaced, themes treated in this thesis; inverse problems of tomographic type and fast and accurate methods for the application of convolution operators.



Regarding the first theme, the inverse problem of Doppler tomography is considered and the Doppler moment transform is introduced for that purpose. By investigating the properties of the transform, we prove results regarding uniqueness and develop a numerical method for reconstruction. Continuing on the tomography track, we turn focus to X-ray tomography and construct a fast numerical method for the inversion of the Radon transform. Our method is of filtered back-projection type, the most commonly used in practice, but has one order lower time complexity than the standard methods. Moving on to the field of diffraction tomography, we develop a fast method for solving the forward scattering problem in inhomogeneous media.



To this end, we employ the methods for fast and accurate application of singular convolution operators that constitute the second main theme of the thesis. First, we consider the problem of fast Gauss transform with complex parameters, and then develop further some of the results to construct a continuous framework for fast application of convolution operators. (Less)
Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Professor de Hoop, Martijn, Colorado School of Mines, Golden, CO 80401, USA
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Mathematics, Matematik, Unequally spaced FFT, Integral transforms, Singular integrals, Lippmann?Schwinger equation, Helmholtz equation, Inverse problems, Fast Gauss transform, Fast summation, Convolution, Radon transform, Tomography
pages
183 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Room MH:C,Centre for Mathematical Sciences, Sölvegatan 18, Lund Institute of Technology
defense date
2005-05-30 13:15
external identifiers
  • other:ISRN: LUTFMA-1019-2005
ISSN
1404-0034
ISBN
91-628-6519-6
language
English
LU publication?
yes
id
e54dd707-914b-4baf-b562-f53adf387f6b (old id 544905)
date added to LUP
2007-09-27 14:39:37
date last changed
2016-09-19 08:44:59
@phdthesis{e54dd707-914b-4baf-b562-f53adf387f6b,
  abstract     = {There are two, partially interlaced, themes treated in this thesis; inverse problems of tomographic type and fast and accurate methods for the application of convolution operators.<br/><br>
<br/><br>
Regarding the first theme, the inverse problem of Doppler tomography is considered and the Doppler moment transform is introduced for that purpose. By investigating the properties of the transform, we prove results regarding uniqueness and develop a numerical method for reconstruction. Continuing on the tomography track, we turn focus to X-ray tomography and construct a fast numerical method for the inversion of the Radon transform. Our method is of filtered back-projection type, the most commonly used in practice, but has one order lower time complexity than the standard methods. Moving on to the field of diffraction tomography, we develop a fast method for solving the forward scattering problem in inhomogeneous media.<br/><br>
<br/><br>
To this end, we employ the methods for fast and accurate application of singular convolution operators that constitute the second main theme of the thesis. First, we consider the problem of fast Gauss transform with complex parameters, and then develop further some of the results to construct a continuous framework for fast application of convolution operators.},
  author       = {Andersson, Fredrik},
  isbn         = {91-628-6519-6},
  issn         = {1404-0034},
  keyword      = {Mathematics,Matematik,Unequally spaced FFT,Integral transforms,Singular integrals,Lippmann?Schwinger equation,Helmholtz equation,Inverse problems,Fast Gauss transform,Fast summation,Convolution,Radon transform,Tomography},
  language     = {eng},
  pages        = {183},
  publisher    = {Centre for Mathematical Sciences, Lund University},
  school       = {Lund University},
  title        = {Inverse Problems in Tomography and Fast Methods for Singular Convolutions},
  year         = {2005},
}