Applications of Quaternions and Reduced Biquaternions for Digital Signal and Color Image Processing
Date Issued
2004
Date
2004
Author(s)
Chang, Jahan
DOI
en-US
Abstract
Abstract
As the digital camera and the internet getting more and more common and popular, many color images are largely produced and transmitted everyday and everywhere. However, there does not exist tools to directly analyze the color image. The only way to perform color image processing is to decompose the color image into three channel gray images and analyze the three gray images separately and independently by the conventional gray image processing methods, such as Fourier transform, convolution, correlation and singular value decomposition, etc, until some researchers use the Hypercomplex numbers to represent the color image. By means of the Hypercomplex algebra, we can define the Hypercomplex Fourier transform, convolution and correlation, etc, for multi-dimensional signal processing and develop the efficient implementation algorithms for these operations. Therefore, we can use the Hypercomplex algebra and operations to directly process the color image without decomposing the color image into three channel gray images.
For color image processing, we can use the 4-dimensional Hypercomplex numbers to represent the color image. There are two possible Hypercomplex numbers, quaternions (non-commutative algebra) and reduced biquaternions (commutative algebra), can be used for color image processing. Quaternions are proposed earlier than reduced biquaternions and have clear geometric meaning. However, quaternions are not commutative. Due to this property, the complexity of the quaternion Fourier transform, convolution, correlation and singular value decomposion of a quaternion matrix are very complicated. On the other hand, although the reduced biquaternions are proposed later and are not familiar to the most engineers, the reduced biquaternions are commutative algebra. Therefore, the complexity of the reduced biquaternion Fourier transform, convolution, correlation and the singular value decomposition of a reduced biquaternion matrix are much simpler than the ones of quaternions. Besides, almost all the color image applications by using quaternions, such as Fourier transform of a color image, color image edge detection, color sensitive edge detection, color image correlation and color template matching etc, can be performed by using reduced biquaternions. Consequently, we think using reduced biquaternions to represent the color image is better using quaternions.
In this thesis, we will develop the efficient implementation algorithms of the quaternion and redcued biquaternion Fourier transform, convolution and correlation, and the efficient algorithms of the eigenvalue decomposition and singular value decomposition of a quaternion matrix or a reduced biquaternion matrix. Then we will compare the performance of the quaternions and reduced biquaternoins.
By means of these tools, we can directly analyze the color image without decomposing the color image into three channel image. By the Hyercomplex Fourier transform, we can analyze the property of a color image in frequency domain. By the Hypercomplex convolution and correlation, we can analyze the relations between two color images and perform the color template matching. By the singular value decomposition, we can perform the color image compression, enhancement, smoothing, and color image watermarking.
From the experimental results, we can find that using the Hypercomplex numbers, quaternions or reduced biquaternions, to represent the color image will produce many new color image applications. We believe that quaternions and reduced biquaternions are useflul for color image processing.
Subjects
數位影像處理
傅立葉轉換
超複數
數位信號處理
四元數
退化四元數
hypercomplex
Fourier transform, digital signal and image proces
singular value decomposition
quaternion
reduced biquaternion
Type
thesis
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