The discrete diffraction transform

In this paper, we define a discrete analogue of the continuous diffracted projection. We define the discrete diffraction transform (DDT) as a collection of the discrete diffracted projections (DDPs) taken at specific set of angles along specific set of lines. The 'DDP' is defined to be a discrete transform that is similar in its properties to the continuous diffracted projection. We prove that when the DDT is applied to 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. A similar statement, where diffracted projections are replaced by the X-ray projections, that holds for the 2D discrete Radon transform (DRT), is also proved. We prove that the DDT is rapidly computable and invertible.