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

Abstract

Digital watermarking applications have a voracious demand for large sets of distinct 2D arrays of variable size that possess both strong auto-correlation and weak cross-correlation. We use the discrete Finite Radon Transform to construct “perfect” \(p \times p\) arrays, for p any prime. Here the array elements are comprised of 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, by design, contains just \(3(p-1)/2\) zero elements, the minimum number possible when using this “grey” alphabet. The grey alphabet and the low number of zero elements maximises the efficiency with which these perfect arrays can be embedded into discrete data. The most useful aspect of this work is that large families of such arrays can be constructed. Here the family size, M, is given by \(M = p^2-1\). Each of the \(M(M-1)/2\) intra-family periodic cross-correlations is guaranteed to have one of the three lowest possible merit factors for arrays with this alphabet. The merit factors here are given by \(v^2/(p^2-v^2)\), for \(v = 2\), 3 and 4. Whilst the strength of the auto-correlation rises with array size p as \(p^2\), the strength of the many (order \(p^4\)) cross-correlations between all M family members falls as \(1/p^2\).