Farey Sequences and Discrete Radon Transform Projection Angles

Abstract

Abstract This paper examines how the minimal set of digital projection angles for the Discrete Radon Transform (DRT) is selected from the known sequence of Farey fractions. A square array of prime size p defines a unique direction for each digital projection, m, through an integer ratio x m /y m . Here x m and y m define the nearest neighbour distance, d m , between projection samples under a modulus p sampling rule. We show the maximum gap length, d max , on square and hexagonal lattices is p /√3) and ≤ √ p respectively. The DRT angles are shown to replicate the entire Farey fraction sequence over the interval 1 ≤ d m ≤ √ p for a square lattice. For the interval √ p d m d max , the DRT set skips some Farey angles. The skipping of Farey angles is the result of a geometric restriction on the distance minimisation process. The DRT angle set varies significantly with p for those projections with d m near d max . This complicates the comparison of DRT projections over similar sized arrays. The detailed angle distributions of the DRT and the Farey sequences reflect, in different ways, the variable gaps between prime numbers. Thanks are due to Graham Farr, Computer Science, Monash University, for pointing us to the literature on finding minimal vectors in lattices. IS acknowledges support from the Centre for X-Ray Physics and Imaging for this project. AK is a Monash University postgraduate student in receipt of an Australian Postgraduate Award scholarship, provided through the Australian Government.