The Analysis Study And Application Of Some Integral Transforms

In recent years, X-ray computed tomography(CT) has became one of the most importantpart in the medical imaging field. CT plays a good supporting role in the diagnosis of thedisease, and improve the accuracy of the diagnosis of the disease, it is a safe, mature, which is achecking method commonly used in clinical. Because the concise features of the image, so it iswidely used in geography, astronomy, chemistry and other fields. Radon transform is the basicmathematical theory of the CT imaging. With the in-depth study of the scientific theories, ithas been transformed from the initial Radon transform to the research of the generalized Radontransform with the weight measure. Generalized Radon transform is the promotion and theformulation of a variety of situations of the Radon transform. The attenuation Radon transformand the exponential Radon transform is the most important and most promising promotionof the Radon transform. Thus, the for further to be studied of the problem of reconstructionand inverting formula of the Radon transform and the generalized Radon transform which isto determine the object function from the measurements of its transform has an importanttheoretical and application value. In this paper, we first analyze and summarize some results ofthe Radon transform. On this basis, we study the generalized Radon transforms and obtain sometheir results by use of harmonic analysis method, some mathematical method and numericalsolution etc. We use with weighted square integrable function square of operation theory toobtain some generalized Radon transform of singular value decomposition, furthermore to deriveinversion formulas and range characteristics of the generalized Radon transform.This thesis is composed of four chapters as follows:In Chapter1We systematically introduce research background, significance and basictheory of the computed tomography.In Chapter2We analyze and summarize some results of the Radon transform, introducesome content of the generalized Radon transform. In Chapter3We study analytical inversion the exponential Radon transform and reg-ularization problem of the approximate solution of the inverse to the data which is notsmooth.In Chapter4We study the inversion of the generalized Radon transform.