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The discrete Radon transform (DRT) was defined by Abervuch as an analog of the continuous Radon transform for discrete data. Both the DRT and its inverse are computable in O(n(2) log n) operations for images of size n × n. In this paper, we demonstrate the applicability of the inverse DRT for the reconstruction of a 2-D object from its continuous projections. The DRT and its inverse are shown to model accurately the continuum as the number of samples increases. Numerical results for the reconstruction from parallel projections are presented. We also show that the inverse DRT can be used for reconstruction from fan-beam projections with equispaced detectors.

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In this paper we define a discrete analogue of the continuous diffracted projection. We define a discrete diffracted transform (DDT) as a collection of the discrete diffracted projections taken at specific set of angles along specific set of lines. We define 'discrete diffracted projection' to be a discrete transform that is similar in its properties to the continuous diffracted projection. We prove that when the DDT is applied on a set of samples of a continuous object, it approximates a set of continuous vertical diffracted projections of a horizontally sheared object and a set of continuous horizontal diffracted projections of a vertically sheared object. We prove that a similar statement, where diffracted projections are replaced by the X-ray projections, holds in the case of the discrete 2D Radon transform (DRT). We prove that the discrete diffraction transform is rapidly computable and invertible. Some of the underlying ideas came from the definition of DRT. Unlike the DRT, though, this transform cannot be used for reconstruction of the object from the set of rotated projections.