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On Invertibility of the Radon Transform and Compressive Sensing
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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. , p. vii, 30
##### Series
TRITA-MAT-A ; 2014:02
##### Keyword [en]
Radon transform, invertibility, compressive sensing, stability estimates
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
ISBN: 978-91-7501-998-7 (print)OAI: oai:DiVA.org:kth-141837DiVA, id: diva2:698802
##### 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
##### List of papers
1. An identity for triplets of double Hilbert transforms, with applications to the attenuated Radon transform
Open this publication in new window or tab >>An identity for triplets of double Hilbert transforms, with applications to the attenuated Radon transform
2012 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 28, no 12, p. 125007-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.

##### Keyword
Inversion-Formula
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-109617 (URN)10.1088/0266-5611/28/12/125007 (DOI)000312103100008 ()2-s2.0-84870466873 (Scopus ID)
##### 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
2. Stability estimates with a priori bound for the inverse local Radon transform
Open this publication in new window or tab >>Stability estimates with a priori bound for the inverse local Radon transform
##### 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
Radon transform, local injectivity, stability estimates
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-141834 (URN)
##### Note

QS 2014

Available from: 2014-02-25 Created: 2014-02-25 Last updated: 2014-02-28Bibliographically approved
3. On the Theorem of Uniform Recovery of Random Sampling Matrices
Open this publication in new window or tab >>On the Theorem of Uniform Recovery of Random Sampling Matrices
2014 (English)In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 60, no 3, p. 1700-1710Article in journal (Refereed) Published
##### Abstract [en]

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m greater than or similar to s(ln s)(3) ln N. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m x N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, delta s, ensuring sparse recovery via l(1)-minimization. We show that delta(2s) < 4/root 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type delta(2s) < 2/3.

##### Keyword
Bounded orthogonal systems, compressive sensing, effective sparsity, l(1)-minimization, random sampling matrices, restricted isometry property
##### National Category
Signal Processing Other Mathematics Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-141831 (URN)10.1109/TIT.2014.2300092 (DOI)000331902400026 ()2-s2.0-84896839927 (Scopus ID)
##### Note

QC 20140228

Available from: 2014-02-25 Created: 2014-02-25 Last updated: 2017-12-05Bibliographically approved

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Cite
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