The Inverse Discrete Radon Transform with Applications to Tomographic Imaging Using Projection Data

Abstract

Publisher Summary This chapter deals with the situation when the Radon transform samples are on a regular grid. By viewing the Radon transform as a bivariate function, rather than as a parameterized univariate function as is usually done, the power of two-dimensional signal theory is applied to Radon transform theory. This approach permits the story of tomographic imaging, told by the varied and vast body of literature, to be unified and simplified. The chapter reviews and clarifies the requirements for a set of regularly spaced measurements to specify adequately the Radon transform. In order to provide insight into potential areas of applications of the radon transform in imaging, several areas where imaging is achieved from projection data are reviewed. The chapter discusses the historical background and review of reconstruction algorithms of tomography. Selected properties of the θu-coordinatized Radon transform are summarized and illustrated. The chapter elaborates on sampling the radon transform with parallel-beam projections and fan-beam projections. The existing questions regarding the amount of information in a finite number of projections are addressed and the processing required to extract this information is identified. It is shown to, within the standard approximations employed in sampling theory, that for Radon transforms adequately sampled on a rectangular grid an exact inverse discrete Radon transform exists. The effects of finite detector (source) size and motion are reviewed, and a systematic approach is again shown to be available through the unifying viewpoint provided by the application of two-dimensional signal processing techniques. The ultimate purpose of this chapter is to establish a firm theoretical foundation for tomographic imaging systems where the measurements are regularly spaced.