The evolution of image halftoning, from its analog roots to contemporary digital methodologies, encapsulates a fascinating journey marked by technological advancements and creative innovations. Yet the theoretical understanding of halftoning is much more recent. In this article, we explore various approaches towards shedding light on the design of halftoning approaches and why they work. We discuss both halftoning in a continuous domain and on a pixel grid. We start by reviewing the mathematical foundation of the so-called electrostatic halftoning method, which departed from the heuristic of considering the back dots of the halftoned image as charged particles attracted by the grey values of the image in combination with mutual repulsion. Such an attraction-repulsion model can be mathematically represented via an energy functional in a reproducing kernel Hilbert space allowing for a rigorous analysis of the resulting optimization problem as well as a convergence analysis in a suitable topology. A second class of methods that we discuss in detail is the class of error diffusion schemes, arguably among the most popular halftoning techniques due to their ability to work directly on a pixel grid and their ease of application. The main idea of these schemes is to choose the locations of the black pixels via a recurrence relation designed to agree with the image in terms of the local averages. We discuss some recent mathematical understanding of these methods that is based on a connection to Sigma-Delta quantizers, a popular class of algorithms for analog-to-digital conversion.

翻译：图像半色调化从模拟起源到当代数字方法的发展历程，展现了一段由技术进步与创造性创新交织而成的迷人旅程。然而，对半色调化的理论理解却迟得多。本文旨在探讨阐明半色调化方法设计原理及其有效性机制的各种研究路径。我们既讨论连续域上的半色调化，也探讨像素网格上的半色调化。首先回顾所谓静电半色调方法的数学基础——该方法突破了传统启发式思路，将半色调图像中的黑点视为受图像灰度值吸引并相互排斥的带电粒子。此类吸引-排斥模型可通过再生核希尔伯特空间中的能量泛函进行数学表述，从而能够对所得优化问题进行严格分析，并在适当拓扑结构下进行收敛性分析。我们详细讨论的第二类方法是误差扩散方案，这类方法因其能直接在像素网格上运行且易于实施，堪称最流行的半色调技术之一。此类方案的核心思想是通过递推关系选择黑像素位置，该关系设计为在局部平均值层面与原始图像保持一致。我们基于其与Σ-Δ量化器（一类流行的模数转换算法）的关联性，探讨这些方法的最新数学理解。