離散線性完整轉換及其應用

Abstract

在這篇論文中，我們首先由Hermite-高斯微分方程式討論離散Hermite函數(DHFs)，提出具有放大縮小能力的置中離散Hermite高斯函數(CDDHFs)以及位移且具有放大縮小能力的置中離散Hermite高斯函數(SDDHFs)。 同時，我們利用提出的置中離散Hermite高斯函數實現離散線性轉換，例如散線性完整轉換(DLCT)與離散Hilbert轉換；線性完整轉換(LCT)是傅立葉轉換(FT)、分數傅立葉轉換(FrFT)、菲涅耳轉換(Fresnel transform)與放大縮小運算(scaling operation)的一般式，因此線性完整轉換在訊號處理上十分具有吸引力；然而，以往的論文只討論連續取樣方式實現的DLCT，在這種連續取樣方式下，LCT固有的加成與可逆特性將無法成立；因此，我們利用提出利用CDDHFs實現離散線性完整轉換，該實現的離散線性完整轉換具有加成與可逆特性，而且不需要超取樣；更進一步，也利用提出的DLCT實現相關的解析訊號和Hilbert轉換定義，並應用於加密的單邊帶通訊。 再者，我們將DHFs延伸至一般化Hermite高斯函數，我們定義一般化Hermite高斯函數的微分方程式，並說明與standard和elegant高斯函數的關連；進一步，我們利用光學的模式轉換，將離散的一般化Hermite高斯束模式轉換至離散的一般化Laguerre和Ince高斯束模式；最後，我們推導從離散Hermite高斯束縳換至離散Laguerre高斯束的轉換係數之快速演算法，並提供特徵點偵測和影像重建等應用。

In this dissertation, we first provide a discussion of discrete Hermite functions (DHFs) starting from the Hermite-Gaussian differential equation. The proposed center dilated discrete Hermite functions (CDDHFs) have good ability in discrete scalable Hermite expansions. Whereas, the shifted dilated discrete Hermite functions (SDDHFs) are a shifting extension version of CDDHFs. Then, we use the developed DHFs to realize the discrete linear transform, such as the discrete linear canonical transform (DLCT) and discrete canonical Hilbert transform. The linear canonical transform (LCT) is an attractive transform because it generalizes Fourier transform (FT), fractional FT (FrFT), Fresnel transform, and scaling operation as its special cases. However, in earlier reference papers, they only discuss the sampled-continuous approach to realized DLCT. Under such sampled-continuous approach, the LCT inherent additivity and reversibility properties cannot be held. Therefore, we define a novel DLCT by means of eigen-decomposition in dilatable eigenspace based on the CDDHFs. The implemented DLCT possess additivity and reversibility properties while with no oversampling involved; meanwhile, the proposed DLCT has very good approximation to continuous LCT. Moreover, we use the proposed DLCT to realize canonical analytic signal (CAS) and canonical Hilbert transform (CHT). The proposed CAS and CHT have several practical applications, such as the scalable edge detection and secure single-sideband communication. Further, we generalize the DHFs to discrete “generalized” Hermite Gaussian functions. We provide a compact differential equation model for the generalized Hermite Gaussian functions and show the relations between standard and elegant Hermite Gaussian functions. Afterward, we extend the discrete “generalized” Hermite Gaussian mode to Laguerre and Ince Gaussian modes by using the mode conversion in optics. We also derive fast algorithm for the transformation coefficients to compute the discrete Laguerre Gaussian functions from discrete Hermite Gaussian functions. The applications of discrete Laguerre Gaussian functions in circular pattern keypoints selection and image reconstruction are also demonstrated.