  In this paper, we study error diffusion techniques for digital halftoning
from the perspective of 1-bit Sigma-Delta quantization. We introduce a method
to generate Sigma-Delta schemes for two-dimensional signals as a weighted
combination of its one-dimensional counterparts and show that various error
diffusion schemes proposed in the literature can be represented in this
framework via Sigma-Delta schemes of first order. Under the model of
two-dimensional bandlimited signals, which is motivated by a mathematical model
of human visual perception, we derive quantitative error bounds for such
weighted Sigma-Delta schemes. We see these bounds as a step towards a
mathematical understanding of the good empirical performance of error
diffusion, even though they are formulated in the supremum norm, which is known
to not fully capture the visual similarity of images.
  Motivated by the correspondence between existing error diffusion algorithms
and first-order Sigma-Delta schemes, we study the performance of the analogous
weighted combinations of second-order Sigma-Delta schemes and show that they
exhibit a superior performance in terms of guaranteed error decay for
two-dimensional bandlimited signals. In extensive numerical simulations for
real world images, we demonstrate that with some modifications to enhance
stability this superior performance also translates to the problem of digital
halftoning.
  More concretely, we find that certain second-order weighted Sigma-Delta
schemes exhibit competitive performance for digital halftoning of real world
images in terms of the Feature Similarity Index (FSIM), a state-of-the-art
measure for image quality assessment.
