用于线性时间算法的元plectic变换的精确酉离散表示
Exactly unitary discrete representations of the metaplectic transform for linear-time algorithms
AI总结：
本文提出精确酉的离散元plectic变换及其近恒等近似，实现线性时间计算，并证明迭代收敛性，从而扩展了焦散光学建模等应用中MT的计算能力。
摘要：
元plectic变换（MT）是傅里叶变换的一种推广，有时被称为线性正则变换，是现代光学中普遍使用的工具，例如在计算傍轴光学系统中光束的变换时。MT也是作者最近提出的焦散几何光学建模的重要组成部分。特别是，该应用依赖于近恒等MT（NIMT）；然而，迄今为止使用的NIMT近似并非精确酉的，并会导致数值不稳定。在此，我们开发了一种精确酉的离散MT，并对其进行近似以获得一种同样是酉的、且可在线性时间内计算的离散NIMT。我们证明了离散NIMT在迭代时收敛于离散MT，从而使NIMT能够计算不一定接近恒等的MT。然后，我们通过一系列示例演示了新算法。
英文摘要：
The metaplectic transform (MT), a generalization of the Fourier transform sometimes called the linear canonical transform, is a tool used ubiquitously in modern optics, for example, when calculating the transformations of light beams in paraxial optical systems. The MT is also an essential ingredient of the geometrical-optics modeling of caustics that was recently proposed by the authors. In particular, this application relies on the near-identity MT (NIMT); however, the NIMT approximation used so far is not exactly unitary and leads to numerical instability. Here, we develop a discrete MT that is exactly unitary, and approximate it to obtain a discrete NIMT that is also unitary and can be computed in linear time. We prove that the discrete NIMT converges to the discrete MT when iterated, thereby allowing the NIMT to compute MTs that are not necessarily near-identity. We then demonstrate the new algorithms with a series of examples.
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