The Study About SPECT Image Reconstruction Under The Fan Beam Scanning Geometry

In the recent years, Single Photo Emission Computed Tomography (SPECT) has became indispensable part in the nuclear medicine field, plays the important role in the diagnosis of internal organs, has been widely used in the clinical medicine for the inspection. SPECT has been widely applied not only in the medicine field, but also in the mathematics even astronomy, chemistry and many other areas. As we know, Radon transform and its inversion formula is the mathematical basic of image reconstruction, which plays an important role in image reconstruction. With the in-depth study of the research, Radon transform has been unable to meet the needs of practical applications, it has been transformed to research of the exponential Radon transform and the attenuated Radon transform. In the past year, two mainly methods have been used to get the reconstruction:analytical algorithm and iterative algorithm. Analytical algorithm provides quick reconstruction and be more used in practircal application. Filtered back-project (FBP) algorithm, which is the classic algorithm in the analytical method, be used most widely in the practircal. At present, we commonly use parallel-beam, fan-beam and cone-beam scan to get the projection. The problem about the reconstruction under the parallel beam scan has been researched thoroughly by most scholars, more and more researchers have paid attention to the reconstruction under the fan-beam scan in SPECT recent years. In this paper we mainly discusses the reconstruction problem under the parallel-beam and fan-beam scan, we consider the inversion formula of the exponential Radon transform and the attenuated Radon transform. We propose two different methods to get the inversion formula of the exponential Radon transform. One is the classic FBP algorithm. Unlike other works, we calculate the partial derivative directly in the formula, so it does not contain the partial derivative operator in the final result, the final inversion formula also be briefer and more easily to be applied. The other one is to construct an intermediate function to obtain the inversion formula with the dual operator. We improved the inversion formula based on the Novikov’s formula for the attenuated Radon transform, it does not need to take the real part in the final result, so it may greatly simplify the data processing and calculating in the further work. At last we also present the numerical simulation.This thesis is composed of five chapters as follows:● In Chapter 1, we systematically introduce the background and significance of the re-search, introduce the result and the development trends in the future about the research at domestic and overseas.● In Chapter 2, we introduced the Hilbert transform and some related definitions of Radon transform and the basic lemmas.● In Chapter 3, we consider the inversion problem about the exponential Radon transform and the attenuated Radon transform under the parallel-beam geometry scanning.● In Chapter 4, we discusses the inversion problem about the exponential Radon trans-form and the attenuated Radon transform under the fan-beam geometry scanning, and implement the numerical simulation.● In Chapter 5, we sum up the problems which have been settled and to propose the problems which are not yet solved as the direction of future.