Octonion Discrete Fourier Transform : Fast Algorithms

 The color image from one of the color models, for instance the RGB model, can be transformed into the quaternion algebra and be represented as one quaternion image which allows to process simultaneously of all color components of the image. The color image can be also considered in different models with transformation to the octonion space with following processing in the 8-D frequency domain.  In this work, we describe the algorithm for the 2-D two-side octonion DFT (ODFT), by using two-side 2-D quaternion DFTs (QDFT).  The calculation of the transform is reduced to calculation of two 2-D QDFT which has fast algorithms. The octonion algebra with the Fourier transform can be used in color imaging as the 2-D ODFT, which found effective applications in color imaging, medical imaging, in image filtration, image enhancement.  The octonion 2-D DFT can be used not only in color imaging, but in gray-scale imaging as well, and for that there are many models of transferring one or a few gray-scale images into the octonion space.