Mathematical Problems of Thermoacoustic and Compton Camera Imaging

The results presented in this dissertation concern two different types of tomographic imaging. The first part of the dissertation is devoted to the time reversal method for approximate reconstruction of images in thermoacoustic tomography. A thorough numerical study of the method is presented. Error estimates of the time reversal approximation are provided. In the second part of the dissertation a type of emission tomography, called Compton camera imaging is considered. The mathematical problem arising in Compton camera imaging is the inversion of the cone transform. We present three methods for inversion of this transform in two dimensions. Numerical examples of reconstructions by these methods are also provided. Lastly, we turn to a problem of significance in homeland security, namely the detection of geometrically small, low emission sources in the presence of a large background radiation. We consider the use of Compton type detectors for this purpose and describe an efficient method for detection of such sources. Numerical examples demonstrating this method are also provided.