 
Summary: IMA Journal of Applied Mathematics (2008) 73, 496538
doi:10.1093/imamat/hxn010
Advance Access publication on April 29, 2008
The discrete diffraction transform
I. SEDELNIKOV, A. AVERBUCH AND Y. SHKOLNISKY
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel
[Received on 8 December 2006; accepted on 4 March 2008]
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 Xray projections,
that holds for the 2D discrete Radon transform (DRT), is also proved. We prove that the DDT is rapidly
computable and invertible.
Keywords: diffraction tomography; discrete diffraction transform; Radon transform.
1. Introduction
Ultrasound imaging is an example of diffracted tomography (see Kak & Slaney, 2001). Xray tomog
raphy, mathematically described by the continuous Radon transform, is an example of nondiffracted
