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An identity for triplets of double Hilbert transforms, with applications to the attenuated Radon transform
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2012 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 28, no 12, 125007- p.Article in journal (Refereed) Published
Abstract [en]

We consider an elementary identity for double singular integrals in the plane and show that one can apply this to deduce inversion and product formulae for the Hilbert transform and inversion formulae for the affine and weighted Radon transforms. We will be able to allow many of the previously known weights for which there is an inversion formula for the weighted Radon transform and also pose some new conditions on which weights that can be used.

Place, publisher, year, edition, pages
2012. Vol. 28, no 12, 125007- p.
Keyword [en]
Inversion-Formula
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-109617DOI: 10.1088/0266-5611/28/12/125007ISI: 000312103100008Scopus ID: 2-s2.0-84870466873OAI: oai:DiVA.org:kth-109617DiVA: diva2:583471
Funder
Knut and Alice Wallenberg Foundation, KAW 2005.0098
Note

QC 20130108

Available from: 2013-01-08 Created: 2013-01-08 Last updated: 2017-12-06Bibliographically approved
In thesis
1. On Invertibility of the Radon Transform and Compressive Sensing
Open this publication in new window or tab >>On Invertibility of the Radon Transform and Compressive Sensing
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis contains three articles. The first two concern inversion andlocal injectivity of the weighted Radon transform in the plane. The thirdpaper concerns two of the key results from compressive sensing.In Paper A we prove an identity involving three singular double integrals.This is then used to prove an inversion formula for the weighted Radon transform,allowing all weight functions that have been considered previously.Paper B is devoted to stability estimates of the standard and weightedlocal Radon transform. The estimates will hold for functions that satisfy an apriori bound. When weights are involved they must solve a certain differentialequation and fulfill some regularity assumptions.In Paper C we present some new constant bounds. Firstly we presenta version of the theorem of uniform recovery of random sampling matrices,where explicit constants have not been presented before. Secondly we improvethe condition when the so-called restricted isometry property implies the nullspace property.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. vii, 30 p.
Series
TRITA-MAT-A, 2014:02
Keyword
Radon transform, invertibility, compressive sensing, stability estimates
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-141837 (URN)978-91-7501-998-7 (ISBN)
Public defence
2014-03-28, D3, Lindstedtsvägen 5, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Knut and Alice Wallenberg Foundation, KAW 2005.0098
Note

QC 20140228

Available from: 2014-02-28 Created: 2014-02-25 Last updated: 2014-02-28Bibliographically approved

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Citation style
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