Nguyen, Linh V. (2010-08). Mathematical Problems of Thermoacoustic Tomography. Doctoral Dissertation. Thesis uri icon

abstract

  • Thermoacoustic tomography (TAT) is a newly emerging modality in biomedical imaging. It combines the good contrast of electromagnetic and good resolution of ultrasound imaging. The mathematical model of TAT is the observability problem for the wave equation: one observes the data on a hyper-surface and reconstructs the initial perturbation. In this dissertation, we consider several mathematical problems of TAT. The first problem is the inversion formulas. We provide a family of closed form inversion formulas to reconstruct the initial perturbation from the observed data. The second problem is the range description. We present the range description of the spherical mean Radon transform, which is an important transform in TAT. The next problem is the stability analysis for TAT. We prove that the reconstruction of the initial perturbation from observed data is not H?older stable if some observability condition is violated. The last problem is the speed determination. The question is whether the observed data uniquely determines the ultrasound speed and initial perturbation. We provide some initial results on this issue. They include the unique determination of the unknown constant speed, a weak local uniqueness, a characterization of the non-uniqueness, and a characterization of the kernel of the linearized operator.
  • Thermoacoustic tomography (TAT) is a newly emerging modality in biomedical
    imaging. It combines the good contrast of electromagnetic and good resolution of
    ultrasound imaging. The mathematical model of TAT is the observability problem
    for the wave equation: one observes the data on a hyper-surface and reconstructs the
    initial perturbation. In this dissertation, we consider several mathematical problems
    of TAT. The first problem is the inversion formulas. We provide a family of closed
    form inversion formulas to reconstruct the initial perturbation from the observed
    data. The second problem is the range description. We present the range description
    of the spherical mean Radon transform, which is an important transform in TAT. The
    next problem is the stability analysis for TAT. We prove that the reconstruction of
    the initial perturbation from observed data is not H?older stable if some observability
    condition is violated. The last problem is the speed determination. The question
    is whether the observed data uniquely determines the ultrasound speed and initial
    perturbation. We provide some initial results on this issue. They include the unique
    determination of the unknown constant speed, a weak local uniqueness, a characterization
    of the non-uniqueness, and a characterization of the kernel of the linearized
    operator.

publication date

  • August 2010