| Summary: | 博士 === 國立臺灣大學 === 電信工程學研究所 === 104 === The objective of this dissertation is to demonstrate how quaternion and reduced biquaternion algebra can be applied to multi-dimensional signal processing, in particular color image processing.
Thanks for the development of quaternion algebra, reduced biquaternion (RB) algebra, and modern computer technologies, many color image processing techniques that are considered slow and unrealistic become very popular in recent years. Since a color image has three components (red, green, and blue), we can encode its pixels to quaternions (or RB) and consider the whole color image as a two dimensional quaternion or RB image. Many tasks of color image processing, such as three dimensional geometrical transformations, color matching, and denoising can be done more easily in quaternion or RB domain rather than in RGB domain.
In this dissertation, the basic concepts of quaternion, RB algebra and their frequency domain transformations are reviewed. Then, we introduce the two dimensional Hermite-Gaussian functions (2D-HGFs) as the eigenfunction of discrete quaternion Fourier transform (DQFT) and discrete reduced biquaternion Fourier transform (DRBQFT). The eigenvalues of 2D-HGF for three types of DQFT and two types of DRBQFT are derived. After that, the mathematical relation between 2D-HGF and Gauss-Laguerre circular harmonic function (GLCHF) is given. From the aforementioned relation and some derivations, the GLCHF can be proved as the eigenfunction of DQFT/DRBQFT and its eigenvalues are summarized.
These GLCHFs can be used as the basis to perform color image expansion. The expansion coefficients can be used to reconstruct the original color image and as a rotation invariant feature. The GLCHFs can also be applied to color matching applications. We also proposed many novel quaternion and RB based color image processing techniques, such as quaternion iterative filtering for texture and noise removal, quaternion fractional delay for spatial Affine transformations, quaternion algorithm for color filtering array (CFA) demosaicing, quaternion based color correction method, and luminance-invariant color correction method based on quaternion rotation for color image.
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