TY - JOUR

T1 - The discrete diffraction transform

AU - Sedelnikov, I.

AU - Averbuch, Amir

AU - Shkolnisky, Y.

N1 - Funding Information:
The third author was supported in part by the Eshkol Fellowship Grant from the Ministry of Science, Israel.

PY - 2008/6

Y1 - 2008/6

N2 - 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.

AB - 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.

KW - Diffraction tomography

KW - Discrete diffraction transform

KW - Radon transform

UR - http://www.scopus.com/inward/record.url?scp=44949245759&partnerID=8YFLogxK

U2 - 10.1093/imamat/hxn010

DO - 10.1093/imamat/hxn010

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:44949245759

SN - 0272-4960

VL - 73

SP - 496

EP - 538

JO - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

IS - 3

ER -