Study About The Approximate Reconstruction Methods Of Generalized Transform Of Radon Type

In this thesis, we first introduce and summarize some conclusions about the classical Radon transform, including characteristics of the Radon transform when certain special transform act on it, the properties of its range, and the sampling theory etc. All of these are the reference to the research of the generalized transform of Radon type. Then we utilize the theory of harmonic analysis such as the Fourier analysis theory and the functional analysis theory to study the approximate reconstruction of the generalized Radon transform based on the Radon transform. We mainly focus on two kinds of the generalized transform of Radon type: the exponential Radon transform and the generalized Radon transform of polynomial type. We study the approximate reconstruction about the two different generalized transform of Radon type and obtain the following results:(1) We discuss the numerical inversion of the exponential Radon transform: We expand the density function and its exponential Radon transform to the Fourier series with respect to their own variable respectively. Then we establish the numerical inversion method by means of seeking the relationship between the Fourier coefficient of the density function and the Fourier coefficient of its exponential Radon transform. We also give the numerical simulation of this method.(2) We innovationally study the singular value decomposition of the generalized Radon transform of polynomial type: In this thesis, the weight function is not the same of the weight in Xu Yuan's papers. Our weight function is identical to , whereμis the attenuation of the object to be reconstructed, x,ωis the inner product of x andω. We make as the linear operator from to , obtain its singular value decomposition using structured approach and get its approximate reconstruction further. This is one of the innovations made in this thesis.The thesis is organized as follows:In Chapter 1, we introduced some conclusions about the classical Radon transform and the present statement together with the development tendency of the generalized Radon transform.In Chapter 2, we gave some preliminaries.In Chapter 3, the numerical inversion of the exponential Radon transform is discussed.In Chapter 4, we study the singular value decomposition of generalized Radon transform of the polynomial type.