The discrete Radon transform

Abstract

The Radon transform and its inversion are the mathematical keys that enable tomography. Radon transforms are defined for continuous objects with continuous projections at all angles in [0,π). In practice, however, we pre-filter discrete projections taken at a discrete set of angles and reconstruct a discrete object. Since we are approximating a continuous transform, it would seem that acquiring more projections at finer projection resolutions is the path to providing better reconstructions. Alternatively, a discrete Radon transform (DRT) and its inversion can be implemented. Then the angle set and the projection resolution are discrete having been predefined by the required resolution of the tomogram. DRT projections are not necessarily evenly spaced in [0, π), but are concentrated in directions which require more information due to the discrete square [or cubic] grid of the reconstruction space. A DRT, by design, removes the need for interpolation, speeding up the reconstruction process and gives the minimum number of projections required, reducing the acquisition time and minimizing the required radiation dose. This paper reviews the concept of a DRT and demonstrates how they can be used to reconstruct objects from X-ray projections more efficiently in terms of the number of projections and to enable speedier reconstruction. This idea has been studied as early as 1977 by Myron Katz. The work begun by Katz has continued and many methods using different DRT versions have been proposed for tomographic image reconstruction. Here, results using several of the prominent DRT formalisms are included to demonstrate the different techniques involved. The quality and artifact structure of the reconstructed images are compared and contrasted with that obtained using standard filtered back projection. Show more