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IMA Journal of Applied Mathematics Advance Access published online on April 29, 2008
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxn010
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© The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The discrete diffraction transform

I. Sedelnikov, A. Averbuch{dagger} and Y. Shkolnisky

School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel

{dagger} Email: amir{at}math.tau.ac.il

Received on December 8, 2006; Accepted on March 4, 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 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.

Keywords: diffraction tomography; discrete diffraction transform; Radon transform.


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