Extended families of 2D arrays with near optimal auto and low cross-correlation

Abstract

Families of 2D arrays can be constructed where each array has perfect autocorrelation, and the cross-correlation between any pair of family members is optimally low. We exploit equivalent Hadamard matrices to construct many families of p p × p arrays, where p is any 4k-1 prime. From these families, we assemble extended families of arrays with members that exhibit perfect autocorrelation and next-to-optimally low cross-correlation. Pseudo-Hadamard matrices are used to construct extended families using p = 4k + 1 primes. An optimal family of 31 31 × 31 perfect arrays can provide copyright protection to uniquely stamp a robust, low-visibility watermark within every frame of each second of high-definition, 30 fps video. The extended families permit the embedding of many more perfect watermarks that have next-to-minimal cross-correlations.

Characterization of surface texture-measuring optical microscopes using a binary pseudo-random array standard

“…The first is that it immediately suggests lithographic processes for the fabrication of BPRA, which are ideally suited for the fabrication of binary patterns with small feature sizes. The other advantage is that the designer can draw on a large body of work on matrices with perfect autocorrelation, which have applications in several imaging and image processing applications [14][15][16][17] (here and in the following sections "autocorrelation" refers to the cyclical autocorrelation).…”