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An explicit inversion formula for the exponential Radon transform using data from 180˚
Stockholm University, Faculty of Science, Department of Mathematics.
2004 (English)In: Arkiv för matematik, ISSN 0004-2080, Vol. 42, no 2, 253-262 p.Article in journal (Refereed) Published
Abstract [en]

We derive a direct inversion formula for the exponential Radon transform. Our formula requires only the values of the transform over an 180° range of angles. It is an explicit formula, except that it involves a holomorphic function for which an explicit expression has not been found. In practice, this function can be approximated by an easily computed polynomial of rather low degree.

Place, publisher, year, edition, pages
2004. Vol. 42, no 2, 253-262 p.
National Category
URN: urn:nbn:se:su:diva-23105DOI: 10.1007/BF02385485OAI: diva2:190168
Part of urn:nbn:se:su:diva-15Available from: 2003-10-02 Created: 2003-10-02 Last updated: 2009-10-21Bibliographically approved
In thesis
1. Topics in geometry, analysis and inverse problems
Open this publication in new window or tab >>Topics in geometry, analysis and inverse problems
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of three independent parts.

Part I: Polynomial amoebas

We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1.

Part II: Differential equations in the complex plane

We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform.

Part III: Radon transforms and tomography

This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Place, publisher, year, edition, pages
Matematiska institutionen, 2003. 123 p.
Laurent series, Harnack curves, differential equations, tomography
National Category
Algebra and Logic
urn:nbn:se:su:diva-15 (URN)91-7265-738-3 (ISBN)
Public defence
2003-10-10, sal 14, hus 5, Kräftriket, Stockholm, 10:00
Available from: 2003-10-02 Created: 2003-10-02Bibliographically approved

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Rullgård, Hans
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