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Title: | 應用四元數與退化四元數代數之彩色影像處理演算法 Color Image Processing Algorithms by Applying Quaternions and Reduced Biquaternions Algebra |
Authors: | Yu-Zhe Hsiao 蕭毓哲 |
Advisor: | 貝蘇章 |
Keyword: | 四元數,退化四元數,彩色影像處理, Quaternions,Reduced Biquaternions,Color Image Procesing, |
Publication Year : | 2016 |
Degree: | 博士 |
Abstract: | 本論文的目標是要展示四元數與退化四元數可應用於多維訊號處理,特別專注於彩色影像處理的部分。
近年來,歸功於四元數代數、退化四元數代數以及現代電腦科技的發展,許多以前被認為緩慢以及不切實際的彩色影像處理技巧變得十分地普及。因為彩色影像由三個色彩成份所組成 (即:紅色、綠色以及藍色),我們可將它的所有像素使用四元數或退化四元數編碼,並將整張彩色影像視為二維的四元數或退化四元數影像。許多彩色影像處理的工作,例如三維幾何轉換,色彩匹配,去雜訊以及陰影移除,與在傳統RGB色域相比,皆可在四元數與退化四元數領域更輕易地被處理。 在本論文中,首先我們會先回顧基本四元數代數、退化四元數代數的概念以及它們在頻率域的轉換。接著,我們會介紹作為離散四元數傅立葉轉換(DQFT)以及離散退化四元數傅立葉轉換(DRBQFT)的埃根函數,也就是二維埃爾米特-高斯函數(2D-HGF)。對於三種離散四元數傅立葉轉換以及兩種離散退化四元數傅立葉轉換,我們也推導了二維埃爾米特-高斯函數對這些轉換的埃根值。此外,二維埃爾米特-高斯函數與高斯-拉蓋爾圓諧函數之間的數學關係也會做介紹。有了前述的數學關係以及一些推導過程,高斯-拉蓋爾圓諧函數可被證明也是離散四元數傅立葉轉換以及離散退化四元數傅立葉轉換的埃根函數,其埃根值亦會做出彙整。 這些高斯-拉蓋爾圓諧函數可以作為彩色影像展開的基底函數。這些展開的係數可以被用來重建原始的彩色影像並作為旋轉不變特徵。高斯-拉蓋爾圓諧函數也可被應用於色彩匹配。我們也提出了需多創新的四元數與退化四元數的彩色影像處理技巧,例如用於紋理以及雜訊移除的四元數疊代濾波技術、用於空間仿射轉換的四元數分數延遲、用於彩色濾光片陣列解碼及色彩校正的四元數演算法以及基於四元數旋轉操作之亮度不變影像色彩校正演算法。 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. |
URI: | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/50417 |
DOI: | 10.6342/NTU201601409 |
Fulltext Rights: | 有償授權 |
Appears in Collections: | 電信工程學研究所 |
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ntu-105-1.pdf Restricted Access | 9.37 MB | Adobe PDF |
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