Ambartsoumian, Gaik (2006-08). Spherical radon transforms and mathematical problems of thermoacoustic tomography. Doctoral Dissertation. Thesis uri icon

abstract

  • 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.
  • 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.

publication date

  • August 2006