Inversion Of The Exponential Radon Transform And Study Of Its Harmonic Analysis Methods

Radon transform is the foundation of mathematics about Computerized Tomography andreconstruction problems. Both the Radon transform and the exponential Radon transform, aswell as the more general attenuated Radon transform, arise in applications to medical imaging.Specially, the exponential Radon transform is the mathematical description of practical problemssuch as single photon emission computerized tomography in medicine. It is then of interestto invert the transform which is to determine the object function from measurements of itstransform. In this thesis,we apply harmonic analysis methods to study some properties of theexponential Radon transform, and obtain inversion formulas and reconstruction algorithm ofthe exponential Radon transform.This thesis is composed of four chapters as follows:In chapter 1, we systematically introduce research backgrounds and developments of theexponential Radon transform.In chapter 2, we review and discuss some properties about the classical Radon transform,obtain some results of relations between the exponential Radon transform and the slant stacktransform.In chapter 3, we study some properties and relations about exponential Radon transformand Fourier transform of functions with complex variables,and establish an inversion formulaof the exponential Radon transform when the parameter is an imaginary.In chapter 4, we derive an inversion formula of the exponential Radon transform in generalcase, and study the reconstruction algorithms and implementation.