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Stability estimates with a priori bound for the inverse local Radon transform
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Stockholm University.
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We consider the inverse problem for the 2-dimensional weighted local Radon transform $R_{m}[f]$, where $f$ is supported in $y\geq x^2$ and $R_{m}[f](\xi,\eta)=\int f(x,\xi x+\eta)m(x,\xi,\eta)dx$ is defined near $(\xi,\eta)=(0,0)$. For weight functions satisfying a certain differential equation we give weak estimates of$f$ in terms of $R_{m}[f]$ for functions $f$ that satisfies an a priori bound.

##### Keyword [en]
Radon transform, local injectivity, stability estimates
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
OAI: oai:DiVA.org:kth-141834DiVA: diva2:698798
##### Note

QS 2014

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