In this paper, we provide a comprehensive theory of anti‐aliasing sampling patterns that explains and revises known results, and introduce a variational optimization framework to generate point patterns with any desired power spectra and anti‐aliasing properties. We start by deriving the exact spectral expression for expected error in reconstructing a function in terms of power spectra of sampling patterns, and analyzing how the shape of power spectra is related to anti‐aliasing properties. Based on this analysis, we then formulate the problem of generating anti‐aliasing sampling patterns as constrained variational optimization on power spectra. This allows us to not rely on any parametric form, and thus explore the whole space of realizable spectra. We show that the resulting optimized sampling patterns lead to reconstructions with less visible aliasing artifacts, while keeping low frequencies as clean as possible. Although we focus on image plane sampling, our theory and algorithms apply in any dimensions, and the variational optimization framework can be utilized in all problems where point pattern characteristics are given or optimized.

中文翻译：

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抗锯齿采样模式的综合理论和变分框架

在本文中，我们提供了解释和修改已知结果的抗锯齿采样模式的综合理论，并引入了一个变分优化框架来生成具有任何所需功率谱和抗锯齿特性的点模式。我们首先根据采样模式的功率谱推导出预期误差的准确谱表达式，并分析功率谱的形状如何与抗混叠特性相关。在此分析的基础上，我们将生成抗锯齿采样模式的问题表述为功率谱上的约束变分优化。这使我们可以不依赖任何参数形式，从而探索可实现光谱的整个空间。我们表明，由此产生的优化采样模式导致重建时可见混叠伪影较少，同时尽可能保持低频干净。尽管我们专注于图像平面采样，但我们的理论和算法适用于任何维度，并且变分优化框架可用于所有给定或优化点模式特征的问题。