| Summary: | 碩士 === 臺灣大學 === 電信工程學研究所 === 98 === 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.
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