Hexagonal quantizers are not optimal for 2D data hiding

Data hiding using quantization has revealed as an effective way of taking into account side information at the encoder. When quantizing more than one host signal samples there are two choices: 1) using the Cartesian product of several one-dimensional quantizers, as made in Scalar Costa Scheme (SCS); or 2) performing vectorial quantization. The second option seems better, as rate-distortion theory affirms that higher dimensional quantizers yield improved performance due to better sphere-packing properties. Although the embedding problem does resemble that of rate-distortion, no attacks or host signal characteristics are usually considered when designing the quantizer in this way. We show that attacks worsen the performance of the a priori optimal lattice quantizer through a counterexample: the comparison under Gaussian distortion of hexagonal lattice quantization against bidimensional Distortion-Compensated Quantized Projection (DC-QP), a data hiding alternative based in quantizing a linear projection of the host signal. Apart from empirical comparisons, theoretical lower bounds on the probability of decoding error of hexagonal lattices under Gaussian host signal and attack are provided and compared to the already analyzed DC-QP method.