Error diffusion method reduces boundary artifacts in digital halftoning
Towards Digital Halftoning on Closed Manifolds--An Error Diffusion Scheme for the $2D$ Torus based on Sigma-Delta Quantization along the Rank-one Lattice
Information Theory
Summary
Digital halftoning is a way to turn smooth images into black-and-white patterns while keeping the important details visible. Traditional methods can create unwanted edge effects when working on images that loop around like a donut shape. The authors studied this problem using a special looping pattern that arranges all pixels in one line, reducing mismatches at the edges. This new approach improves the quality of the halftoning and lowers visible boundary problems compared to usual grid methods.
What this means in practice
- •For image processing engineers: Create halftoned images on wrap-around surfaces with fewer visible edge artifacts by applying a rank-one lattice error diffusion scheme.
- •For digital printing teams: Improve print quality when using halftoning techniques on designs that require seamless repetition or looping patterns.
Authors
Felix Krahmer, Alessandro Lupoli
Abstract
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.