| Summary: | 博士 === 國立臺灣大學 === 電信工程學研究所 === 102 === 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.
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