Quaternion and Octonion Color Image Processing with MATLAB

Abstract

The theory of complex numbers is well developed; complex numbers have been used in science and engineering for a long time and are still being used for solving many new problems. The arithmetic of these numbers generalizes the arithmetic of real numbers in the sense that, together with the operations of addition and multiplication by real numbers, the inverse number and the division are defined. Such a complete arithmetic exists for other numbers, which are called quaternions and octonions. Quaternions were first discovered by Hamilton in 1843 [6]. More recently, quaternions have been employed in bioinformatics, navigation systems [7], and image and video processing [8,9]. Octonions, which are defined as doubled quaternion numbers [34], have been used in signal and image processing, and we believe that they can also be used effectively for parallel processing many images.Recently, the theory of quaternion algebra has been used in the application of color science that processes the three color channels simultaneously [1]- [13]. Quaternion numbers found interesting applications in color image processing, such as image enhancement [19,31], watermarking [32], adaptive filtering [33], and prostate cancer Gleason grading [16]. The quaternion can be considered as a four-dimensional number with one real part and three imaginary parts. The imaginary dimensions are represented as i, j, and k, which are orthogonal to each other and to 1. In many cases, it is useful to transfer the calculations from the real space of signals and images to complex space, analyze and solve problems by using methods of the complex analysis (arithmetic), and then transfer the solution back to the real space. The transformation to the space of quaternions is also promising. Quaternion algebra for color imaging was first used by Pei and led to the description of new tools, such as the quaternion Fourier transforms and correlations for image processing by representing the red, green, and blue values at each pixel in the color image as single pure quaternion-valued pixels [10]. There are a number of studies on quaternions and quaternion operations and systems in 1

Multi-Focus Color Image Fusion Based on Quaternion Multi-Scale Singular Value Decomposition

“…The operations of the three imaginary parts are equivalent, which makes it very suitable for describing color images and expressing the internal connection of color channels. The three color channels of the image can be represented by three imaginary parts of quaternion (Chen et al, 2014;Xu et al, 2015;Grigoryan and Agaian, 2018). The general form of a quaternion is q = q a + q b i + q c j + q d k. It contains one real part q a and three imaginary parts q b i, q c j and q c k, if the real part q a of a quaternion q is zero, q is called a pure quaternion.…”

A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential systems

“…However, in some practical applications, signals are represented by more abstract structures, e.g. hypercomplex algebras [13,16,18,32]. A classic example is the use of them in the processing of color images (where there are at least three color components) [13,16], but also in the analysis of multispectral data (e.g.…”

“…hypercomplex algebras [13,16,18,32]. A classic example is the use of them in the processing of color images (where there are at least three color components) [13,16], but also in the analysis of multispectral data (e.g. in satellite images where not only visible light is recorded, but also other frequency ranges) [21,22].…”

“…Recently, hypercomplex algebras drew scientists' attention due to their numerous applications, among others in the study of neural networks [25,26,34], in the analysis of color and multispectral images [13,14,15,16,21,22,28], in biomedical signal processing [7,19], in fluid mechanics [8] or in general signal processing [18,32,33]. Quaternions may be used in few different ways -to describe a vector-valued signal (with three or four coordinates) of one variable, i.e.…”

Near orthogonal discrete quaternion Fourier transform components via an optimal frequency rescaling approach

“…Therefore, the discrete quaternion Fourier transform is the generalisation of the conventional discrete Fourier transform. As the discrete quaternion Fourier transform can be used for fusing various components of a multivariate signal to a quaternion‐valued signal [4 ], it is found that the discrete quaternion Fourier transform is applied in many science and engineering applications such as in the colour medical image processing application [5–8 ].…”