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

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Abstract

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.

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Acknowledgements

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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  1. School of Physics and Astronomy, Monash University, Melbourne, Australia

    Matthew Ceko, Imants Svalbe & Timothy Petersen

  2. Scientific Technology, Melbourne, Australia

    Andrew Tirkel

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  1. Matthew Ceko
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  2. Imants Svalbe
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  3. Timothy Petersen
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  4. Andrew Tirkel
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Correspondence to Matthew Ceko.

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Ceko, M., Svalbe, I., Petersen, T. et al. Large Families of “Grey” Arrays with Perfect Auto-correlation and Optimal Cross-Correlation. J Math Imaging Vis 61, 237–248 (2019). https://doi.org/10.1007/s10851-018-0848-3

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  • DOI: https://doi.org/10.1007/s10851-018-0848-3

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