Complex and Hypercomplex Numbers

TL;DR: The theory of complex and hypercomplex numbers is well developed and has been used in various fields of science and engineering. Quaternion and octonion numbers have been employed in image and video processing, navigation systems, and signal processing. Quaternion algebra has also been used in color science and image processing.

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