The Discrete Diffraction Transform

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 similar statement, where diffracted projections are replaced by the X-ray projections, holds in the case of the DRT. We prove that the discrete diffraction transform is rapidly computable and invertible. Some of the underlying ideas came from the definition of the discrete 2D Radon transform (DRT). Unlike the discrete Radon transform, though, this transform cannot be used for reconstruction of the object from the set of rotated projections. Key word: diffraction tomography, discrete diffraction transform, Radon transform. AMS Subject Classfication: 44A12, 78A45