面向闭流形数字半调——基于秩一格点沿Sigma-Delta量化的二维环面误差扩散方案

中文摘要 AI 辅助

数字半调旨在通过二值模式表示连续色调图像，同时保留其视觉相关的低频内容。在众多可用方法中，误差扩散方法通过因果反馈滤波器实现噪声整形，并可被解释为信号量化范式Sigma-Delta调制的二维版本。然而，在闭域上，底层递推关系的终态不必与初态匹配，从而产生边界伪影。我们针对二维环面上的带限函数研究该问题。通过将所有像素沿单个闭合秩一格点排列，我们将分别处理行和列所产生的多个不匹配替换为单个终态贡献，同时保持精确重构。对于具有\(N=M^2+1\)个点的均匀格点，我们获得一阶和二阶误差界，分别为\(N^{-1/2}\)和\(N^{-1}\)阶。适当的常数更新消除了终态不匹配，并降低了误差的空间局部化，而不改变这些渐近阶。对于固定方向的秩一格点，修正后的一阶和二阶重构分别达到\(N^{-1}\)和\(N^{-2}\)的速率。数值实验表明，与在笛卡尔网格上应用的经典方案相比，边界伪影有所减少。

英文摘要

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.

发表机构

• Technische Universität Darmstadt（达姆施塔特工业大学）
• Technical University of Munich（慕尼黑工业大学）

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