Commutative reduced biquaternions and their Fourier transform for signal and image processing applications

Digital signal and image processing using reduced biquaternions (RBs) are introduced in this paper. RBs are an extension of the complex numbers, following the doubling procedure. Two useful representations of RBs (e₁-e₂ form and matrix representation) are discussed in this paper. Besides, we propose a new representation of RBs (the polar form) to calculate the multiplication and conjugation of RBs easily. Furthermore, we define a unique and suitable RB norm and its conjugate. These definitions are similar and compatible with the complex numbers. The efficient algorithms of the discrete reduced biquaternion Fourier transform (DRBFT), convolution (DRBCV), correlation (DRBCR), and phase-only correlation are discussed in this paper. In addition, linear-time-invariant and symmetric multichannel complex systems can be easily analyzed by RBs. For color image processing, we define a simplified RB polar form to represent the color image. This representation is useful to process color images in the brightness-hue-saturation color space. Many different types of color template matching and color-sensitive edge detection (brightness, hue, saturation, and chromaticity matched edges) can be performed simultaneously by RBs.