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 we recently proposed. 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.

中文翻译：

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线性时间算法的辛辛变换的完全unit离散表示

偏微分变换（MT）是有时被称为线性典范变换的傅立叶变换的泛化，是现代光学中普遍使用的一种工具，例如，当计算近轴光学系统中的光束的变换时。MT也是我们最近提出的焦散的几何光学建模的重要组成部分。特别地，该应用依赖于近身身份MT（NIMT）。但是，到目前为止使用的NIMT近似值并不完全是一元的，并且会导致数值不稳定。在这里，我们开发了一个精确的单一MT，并将其近似以获得一个离散的NIMT，NIMT也为单一的并且可以在线性时间内进行计算。我们证明了离散NIMT在迭代时收敛于离散MT，从而使NIMT可以计算出不一定是同一身份的MT。然后，我们将通过一系列示例来演示新算法。