四元數與退化四元數在影像及訊號處理上之應用

Applications of Quaternions and Reduced Biquaternion for Image and Signal Processing

蕭毓哲(Yu-Zhe Hsiao)

指導教授 ： 貝蘇章

國立臺灣大學/電機資訊學院/電信工程學研究所/碩士(2010年)

https://doi.org/10.6342/NTU.2010.10122

摘要

在訊號以及影像處理的研究領域中，我們通常需要處理多維訊號的問題。但是，這些問題牽涉到艱深的數學因此難以解決。為了要清除我們遇到的這些障礙，我們訴諸於適合多維訊號處理的超複數代數。以彩色影像訊號處理的問題解決方法為例，ㄧ張彩色影像是一種多維訊號並且是由三個分離的頻道(R、G、B)所組成。由於網際網路的問世、電腦科技的發展以及數位相機的發明，我們有許多機會傳輸以及產生彩色影像。不過，通常沒有有效地方法能直接處理與分析彩色影像訊號。最普遍的方法是將彩色影像分解成三個通道的灰階影像(R、G、B)，然後用傳統的灰階影像處理演算法分別地處理它們。因為三個彩色通道之間有相關性(correlation)，上述基於灰階影像的方法由於是三個通道獨立地運作，所以不能表現出很好的效果。我們可以用四維的超複數來代表彩色影像並直接地處理彩色影像。四元數跟退化四元數就是兩種這樣的超複數，四元數構成不可交換性代數而退化四元數構成可交換性代數。四元數比退化四元數早問世，且具有明確的幾何意義。因為四元數的不可交換性，許多四元數的操作，諸如四元數的傅立葉轉換、捲積、相關以及奇異值分解都很複雜。而因為退化四元數的可交換性，退化四元數的傅立葉轉換、捲積、相關以及奇異值分解都比四元數的同類運算來的簡單。
在本篇論文當中，我們將會完整地介紹四元數以及退化四元數的基本概念以及它們在訊號以及影像處理上應用。我們將會複習四元數的傅立葉轉換、四元數的時頻分析、退化四元數的傅立葉轉換，並且將會提出一種基於四元數短時距傅立葉轉換的彩色影像邊緣偵測演算法。退化四元數的傅立葉轉換將會被使用於定義退化四元數的解析訊號，這將利於廣義訊號(improper signals)的分析。我們將會複習基於四元數的幾何操作，如旋轉、反射、剪變、膨脹，並提出這些操作用退化四元數修改過的版本。

關鍵字

四元數 ； 退化四元數 ； 訊號處理 ； 影像處理

並列摘要

In the research fields of signal and image processing, we often have to deal with problems of multi-dimensional signal processing. However, these problems involve difficult mathematics and they are therefore hard to be solved. In order to tackle with the obstacles that we encounter, we resort to hypercomplex algebra which is suitable for multi-dimensional signal processing. Take the problem solving of color image signal processing for illustration purpose, a color image is a kind of multi-dimensional signal and is composed of three separate channels (R,G,B). Thanks to the birth of internet, the advancement of modern computer technology, and the invention of digital camera, we have a lot of chances to transmit and produce color images. However, there are usually not efficient ways to process and analyze the color image directly. The most popular way to process a color image is to decompose it into three channel gray-level images (R,G,B) and cope with them separately by the traditional gray-level image processing algorithms. Because of the mutual correlations between these three color channels, the above gray-level based image processing method works independently and therefore does not perform well. We can use four-dimensional hypercomplex numbers to represent the color image and process the color image directly. Two such hypercomplex numbers are quaternions and reduced biquternions. The quaternions form a non-commutative algebra while the reduced biquaternions form a commutative algebra. Quaternions come into being earlier than reduced biquaternions and have clear geometric meaning. Due to the non-commutative property of the quaternions, many operations of the quaternions, such as quaternion Fourier transform, convolution, correlation, and singular value decomposition are very complicated. Owing to the commutative property of reduced biquaternions, the complexities of reduced biquaternion Fourier transform, convolution, correlation, and singular value decomposition are much simpler than those of quaternions.
In this thesis, we will introduce the concept of the quaternions and the reduced biquaternions and their applications in signal and image processing thoroughly. The quaternion Fourier transform, quaternion time-frequency analysis methods, reduced biquaternion Fourier transform will be reviewed and we will propose a quaternion short-term Fourier transform based color image edge detection algorithm. The reduced biquaternion Fourier transform will be utilized to define the reduced biquaternion analytic signal (RB-analytic signal), which is useful in the analysis of improper signals. The geometrical transformation operations based on quaternions, such as rotation, reflection, shear, and dilation will be reviewed and we will propose the modified versions of these operations for reduced biquaternions.

並列關鍵字

quaternion ； reduced biquaternion ； signal processing ； image processing

參考文獻

[C2-1] W. R. Hamilton, “Elements of Quaternion”, Longmans, Green, and Co., London,

[C2-2] I. Niven, “The roots of a quaternion”, Amer. Math. Monthly, vol.49, pp.386-388,

[C2-4] Stephen J. Sangwine and Nicolas Le Bihan, “Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form”, Advances in Applied Clifford Algebras, vol.20, no.1, pp.111-120, Mar. 2010.

[C3-2] Stephen J. Sangwine, “Fourier transforms of color images using quaternion, or hypercomplex, numbers”, Electronics Letters, vol. 32, no. 21, pp.1979-1980, Oct. 1996.

[C3-6] S. J. Sangwine, “ Color image edge detector based on quaternion convolution”, Electron, Lett., vol.34, no.10, pp.969-971, May 1998.