衍射极限的数学基础
Algorithmic Foundations for the Diffraction Limit
摘要：
一个半多世纪以来，人们普遍认为（但从未严格证明）衍射物理学对光学系统的分辨率施加了某些基本限制。然而，我们对什么能解决什么不能解决的理解从来没有超越启发性的论点，更糟糕的是，这些论点似乎是矛盾的。在这项工作中，我们通过研究衍射极限作为一个统计逆问题来弥补这一差距，并根据连接到可证明的算法学习混合模型，我们严格证明了解决紧密间隔的点源所需的统计和算法复杂性的上限和下限。特别是，我们表明，有一个相变的样本复杂性从多项式到指数。令人惊讶的是，我们表明，这不会发生在阿贝极限，这一直被认为是真正的衍射极限。
英文摘要：
For more than a century and a half it has been widely-believed (but was never rigorously shown) that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system. However our understanding of what exactly can and cannot be resolved has never risen above heuristic arguments which, even worse, appear contradictory. In this work we remedy this gap by studying the diffraction limit as a statistical inverse problem and, based on connections to provable algorithms for learning mixture models, we rigorously prove upper and lower bounds on the statistical and algorithmic complexity needed to resolve closely spaced point sources. In particular we show that there is a phase transition where the sample complexity goes from polynomial to exponential. Surprisingly, we show that this does not occur at the Abbe limit, which has long been presumed to be the true diffraction limit.
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