Range-behavior Analysis About The Generalized Transform Of Radon Type

The characterization of the ranges of the generalized transforms of Radon type is oneof the most important subjects in computerized tomography and integral geometry. Suchconditions have various purposes: completing incompleting data, detecting and correctingmeasurement errors and recovering unknown attenuation, ect. Thus, as soon as a newRadon type transform arises in an application, a quest for the range description begins.In this paper,we devote to a survey of some results on characterization of the range ofthe generalized transform of Radon type, such as classical Radon transform, circular Radontransform and exponential Radon transform. First We review and discuss some main resultsabout classical Radon transform,circular Radon transform, then give a new theorem tothe range conditions for exponential Radon transform and their equivalent relationshiptheorem, and prove them with a new method.The whole paper is divided into four chapters.In Chapter 1, we introduce some research backgrounds and developments of the gen-eralized transform of Radon type.In Chapter 2, we review and study some range results of the classical Radon transform,including Fourier slice theorem, moment condition. We extend the domain of the Radontransform by means of functional analytic methods and study its range.In Chapter 3, we discuss and give a complete range description for the circular Radontransform.In Chapter 4, we devote to the range of the exponential Radon transform. We establisha new range theorem and discuss the Paley-Wiener theorem for the exponential Radontransform.