Journal of Mathematical Imaging and Vision

, Volume 61, Issue 2, pp 237–248 | Cite as

Large Families of “Grey” Arrays with Perfect Auto-correlation and Optimal Cross-Correlation

  • Matthew CekoEmail author
  • Imants Svalbe
  • Timothy Petersen
  • Andrew Tirkel


Large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation properties are highly valuable in many imaging and communications applications. We use the discrete Finite Radon Transform to construct \(p \times p\) arrays with “perfect” correlation properties, for any prime p. Array elements are restricted to the integers \(\{0,\pm 1, +2\}\). Each array exhibits perfect periodic auto-correlation, having peak correlation value \(p^2\), with all off-peak values being exactly zero. Each array contains just \(3(p-1)/2\) zero elements, the minimum number possible using this alphabet. Large families with size \(M = p^2-1\) of such arrays can be constructed. Each of the \(M(M-1)/2\) intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors. These family size M can be extended to multiples of \(p^2-1\) if we permit more than the three lowest cross-correlation levels.


Perfect arrays Low correlation arrays Discrete projection Finite Radon transform 



The School of Physics and Astronomy at Monash University, Australia, has supported and provided funds for this work. M.C. has the support of the Australian government’s Research Training Program (RTP) and the J.L. William scholarship from the School of Physics and Astronomy at Monash University.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Physics and AstronomyMonash UniversityMelbourneAustralia
  2. 2.Scientific TechnologyMelbourneAustralia

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