The spherical Radon transform (SRT) integrates a function over the set of all
spheres with a given set of centers. Such transforms play an important role in some
newly developing types of tomography as well as in several areas of mathematics
including approximation theory, integral geometry, inverse problems for PDEs, etc.
In Chapter I we give a brief description of thermoacoustic tomography (TAT or
TCT) and introduce the SRT.
In Chapter II we consider the injectivity problem for SRT. A major breakthrough
in the 2D case was made several years ago by M. Agranovsky and E. T. Quinto. Their
techniques involved microlocal analysis and known geometric properties of zeros of
harmonic polynomials in the plane. Since then there has been an active search for
alternative methods, which would be less restrictive in more general situations. We
provide some new results obtained by PDE techniques that essentially involve only
the finite speed of propagation and domain dependence for the wave equation.
In Chapter III we consider the transform that integrates a function supported
in the unit disk on the plane over circles centered at the boundary of this disk. As
is common for transforms of the Radon type, its range has an in finite co-dimension
in standard function spaces. Range descriptions for such transforms are known to be
very important for computed tomography, for instance when dealing with incomplete
data, error correction, and other issues. A complete range description for the circular Radon transform is obtained.
In Chapter IV we investigate implementation of the recently discovered exact
backprojection type inversion formulas for the case of spherical acquisition in 3D and
approximate inversion formulas in 2D. A numerical simulation of the data acquisition
with subsequent reconstructions is made for the Defrise phantom as well as for some
other phantoms. Both full and partial scan situations are considered.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1807 |
| Date | 02 June 2009 |
| Creators | Ambartsoumian, Gaik |
| Contributors | Kuchment, Peter |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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