Algebraic construction of a new class of quasi-orthogonal arrays for steganography

Watermark recovery is often based on cross-correlating images with pseudo-noise sequences, as access to un-watermarked originals is not required. Successful recovery of these watermarks is determined by the (periodic or aperiodic) sequence auto- and cross-correlation properties. This paper presents several methods of extending the dimensionality of 1D sequences in order to utilize the advantages that this offers. A new type of 2D array construction is described, which meets the above requirements. They are constructed from 1D sequences that have good auto-correlation properties by appending rows of cyclic shifts of the original sequence. The sequence values, formed from the roots of unity, offer additional diversity and security over binary arrays. A family of such arrays is described which have low cross-correlation and can be folded and unfolded, rendering them robust to cryptographic attack. Row and column products of 1D Legendre sequences can also produce equally useful 2D arrays (with interesting properties resulting from the Fourier invariance of Legendre sequences). A metric to characterize all these 2D correlation based watermarks is proposed.