论文标题
正线性时间算法的Metaplic Transic的统一离散表示
Exactly unitary discrete representations of the metaplectic transform for linear-time algorithms
论文作者
论文摘要
Metaplectic Transform（MT）是傅立叶变换的概括有时称为线性规范变换，是一种在现代光学元件中使用的工具，例如，在计算携带光学系统中光束变换时。 MT也是作者最近提出的苛刻的几何透视建模的基本要素。特别是，此应用程序依赖于近乎认同的MT（NIMT）；但是，到目前为止使用的NIMT近似并不完全统一，并导致数值不稳定。在这里，我们开发了一个完全统一的离散MT，并近似它以获得一个也是统一并且可以在线性时间计算的离散NIMT。我们证明，离散的NIMT在迭代时会收敛到离散MT，从而允许NIMT计算不一定是近乎身份的MT。然后，我们通过一系列示例演示了新算法。
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.
