Quaternion Fourier Transform-Based Alpha-Rooting Color Image Enhancement in 2 Algebras: Commutative and Non-Commutative

In this paper, we describe new methods of color image enhancement alpha-rooting by the 2D quaternion discrete Fourier transform (QDFT) in the commutative algebra, or the (2,2)-model. In this model, the concept of the convolution is unique, which is very important when transforming tasks with color images into the frequency domain. Also, there are only two types of the exponential function and therefore only two QDFTs. Both these transforms can be used to reduce the convolution to operation of multiplication. Illustrative examples on color image enhancement are given. Measures of the image enhancement and selection of the best parameters of alpha-rooting are described. A comparison with the traditional non-commutative quaternion algebra is discussed and shown that the (2,2)-model is more effective in image enhancement.