Some links between continuous and discrete Radon transform

The Filtered BackProjection is still questionable since many discrete versions have been derived from the continuous Radon formalism. From a continuous point of view, a previous work has made a link between continuous and discrete FBP versions denoted as Spline 0-FBP model leading to a regularization of the infinite Ramp filter by the Fourier transform of a trapezoidal shape. However, projections have to be oversampled (compared to the pixel size) to retrieve the theoretical properties of Sobolev and Spline spaces. Here we obtain a novel version of the Spline 0 FBP algorithm with a complete continuous/discrete correspondence using a specific discrete Radon transform, the Mojette transform. From a discrete point of view, the links toward the FBP algorithm are shaped with the morphological description and the extended use of discrete projection angles. The resulting equivalent FBP scheme uses a selected set of angles which covers all the possible discrete Katz's directions issued from the pixels of the (square) shape under reconstruction: this is implemented using the corresponding Farey's series. We present a new version of a discrete FBP method using a finite number of projections derived from discrete geometry considerations. This paper makes links between these two approaches.