Discrete Orthogonal Harmonic Transforms
Date Issued
2012
Date
2012
Author(s)
Liu, Chun-Lin
Abstract
In digital signal processing, the discrete orthogonal transforms are an important tool for signal analysis, signal representation, data compression, digital filtering, and signal detection. However, the existing discrete orthogonal transforms require the explicit definition of the discrete transform kernels. There are still lots of continuous orthogonal transforms, whose corresponding discrete orthogonal transforms are not found. Even they are found, the expressions are very complicated and not general.
In this thesis, a systematic method to computing the discrete orthogonal harmonic transforms is proposed. We first consider the one-dimensional signals and transforms. We can start from the differential equations of the continuous transform kernels, model them as the discrete eigen-problems, and then solve them numerically. As a result, we explore some transform kernels, such as the classical orthogonal functions, derived from the orthogonal polynomials, the prolate spheroidal wave functions, and the Schrodinger equation in quantum mechanics. The discrete equivalents of these functions are all verified by simulations. We also implement some discrete orthogonal harmonic transforms, such as the chromatic derivatives, the Airy transforms, the scale transforms, the generalized fractional Fourier transforms, and the linear canonical transforms. These discrete orthogonal harmonic transforms own different behaviors when they are the conventional Fourier-based transforms. For instance, the chromatic derivatives have the local approximation ability, the Airy transforms convert the signal to the domain which is inbetween the time domain and the frequency domain, the scale transforms are invariant to scale changes of the signals. We also explore the generalized fractional Fourier transforms and the eigenfunctions of the linear canonical transforms, in conjunction with their discrete implementations. These results can be used to analyze the information of the digital signals, directly in the discrete domain.
For the higher dimensional case, we consider the eigenfunctions of the two dimensional Fourier transforms and find a simple linear combination of the Hermite Gaussian functions to express the non-separable eigenfunctions. Changing the combination coefficients results in different eigenfunctions and the orthogonality is guaranteed. With this concept, in the two-dimensional case, the gyrator transforms, which are related to the rotated Hermite Gaussian function, and the Laguerre Gaussian transforms are implemented in the discrete domain and with perfect orthogonality. The gyrator transforms and the Laguerre Gaussian transforms are suitable for analyzing the discrete images. In addition, the rotational invariant features can be derived from the Laguerre Gaussian transforms. In the three-dimensional case, we solve the spherical harmonic oscillator wavefunctions and the spherical harmonic oscillator transforms in the discrete domain. They are suitable for analyzing three-dimensional volume data and for deriving the rotational invariant features. Finally, we also implement the two-dimensional non-separable linear canonical transforms. A fast and accurate algorithm is proposed based on decomposing the linear canonical transforms into many stages and utilizing the fast Fourier transform algorithm.
Subjects
Discrete Transforms
Harmonic Analysis
Fourier Transforms
Differential Equations
Orthogonality
Eigenvectors
Type
thesis
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