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.