Conical Radon transforms and their applications in tomography

University Of Texas At Arlington, Arlington TX

Investigators

Abstract

This project addresses a number of mathematical problems of image reconstruction in several emerging fields of imaging. Specific topics include single scattering optical tomography and gamma-ray emission tomography, which are used in medicine for diagnostics and treatment monitoring of various diseases. The project also considers Compton camera imaging, which is used for detection of radiation sources in homeland security and in radio astronomy. The outcomes of this project have potential impact for modern healthcare technologies, national security, exploration of space, and several industrial applications. A special emphasis in this project is made on training students from groups historically underrepresented in STEM fields, including women and minorities. The project is devoted to the study of broken ray (BRT) and conical (CRT) Radon transforms, and their applications to several modern imaging modalities. BRT is a transform that maps a function defined on a plane to its integrals along a certain family of broken rays. CRT maps a function defined in the 3D space to its integrals over a family of conical surfaces. These quantities often correspond to measurements of an imaging device, and the process of image reconstruction depends on the possibility of stable inversion of these transforms. The central problems addressed by the investigators include: the description of injectivity sets of BRT/CRT and formulas for their inversion, the study of the microlocal properties of these transforms, the description of their range, and the investigation of incomplete data problems. The methodology to attack such problems utilizes various combinations of techniques from Fourier analysis, partial differential equations, integral equations and microlocal analysis.