Global benign alteration for two-dimensional torus quantization

Design a global benign alteration, such as a single constant perturbation of the input image, that corrects the boundary errors of two-dimensional Sigma–Delta quantization on the torus when the required line- and column-dependent alterations differ.

Background

The paper explains that, for one-dimensional periodic quantization on the circle, adding a carefully selected constant to the signal can replace a localized terminal mismatch by a visually less disruptive global perturbation and can improve the reconstruction error. On the two-dimensional torus, however, applying row-wise or column-wise schemes may require different corrections for different trajectories.

The authors explicitly identify the unresolved difficulty of finding one global alteration that works across the entire two-dimensional image. Their rank-one lattice construction addresses the issue of multiple independently generated boundary contributions by using a single trajectory, but the problem of designing a global benign alteration for the more general Cartesian setting remains unresolved.