Enhanced Digital Halftoning via Weighted Sigma-Delta Modulation

Felix Krahmer, Anna Veselovska

Research output: Contribution to journalArticlepeer-review


In this paper, we study error diffusion techniques for digital halftoning from the perspective of 1-bitquantization. We introduce a method to generate∑∆ schemes for two-dimensional signals as a weighted combination of their one-dimensional counterparts and show that various error diffusion schemes proposed in the literature can be represented in this framework via∑∆ 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∑∆ 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∑∆ schemes, we study the performance of the analogous weighted combinations of second-order∑∆ 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∑∆ 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.

Original languageEnglish
Pages (from-to)1727-1761
Number of pages35
JournalSIAM Journal on Imaging Sciences
Issue number3
StatePublished - 2023
Externally publishedYes


  • 1-bit quantization
  • Sigma-Delta
  • digital halftoning
  • error diffusion


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