Towards Digital Halftoning on Closed Manifolds--An Error Diffusion Scheme for the $2D$ Torus based on Sigma-Delta Quantization along the Rank-one Lattice

Digital halftoning aims to represent continuous-tone images by binary patterns while preserving their visually relevant low-frequency content. Among the many available approaches, error-diffusion methods implement noise shaping through causal feedback filters and can be interpreted as two-dimensional versions of the signal quantization paradigm Sigma--Delta modulation. On closed domains, however, the terminal state of the underlying recurrence relation need not match the initial one, producing boundary artifacts. We study this problem for bandlimited functions on the two-dimensional torus. By arranging all pixels along a single closed rank-one lattice, we replace the multiple mismatches associated with separately processed rows and columns with a single terminal contribution, while retaining exact reconstruction. For a uniform lattice with \(N=M^2+1\) points, we obtain first- and second-order error bounds of order \(N^{-1/2}\) and \(N^{-1}\). A suitable constant update eliminates the terminal mismatch and reduces the spatial localization of the error without changing these asymptotic orders. For fixed-direction rank-one lattices, the corrected first- and second-order reconstructions instead achieve rates \(N^{-1}\) and \(N^{-2}\). Numerical experiments illustrate a reduction in boundary artifacts compared with classical schemes applied on the Cartesian grid.